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Physica A 349 (2005) 60–132 Navier–Stokes revisited Howard Brenner Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA Received 27 August 2004 Communicated by J.V. Sergers Available online 24 November 2004 Abstract A revision of Newton’s law of viscosity appearing in the role of the deviatoric stress tensor in the Navier–Stokes equation is proposed for the case of compressible fluids, both gaseous and liquid. Explicitly, it is hypothesized that the velocity v appearing in the velocity gradient term rv in Newton’s rheological law be changed from the fluid’s mass-based velocity v m ; the latter being the velocity appearing in the continuity equation, to the fluid’s volume velocity v v ; the latter being a stand-in for the fluid’s volume current (volume flux density n v ). A similar v m ! v v alteration is proposed for the velocity v appearing in the no-slip tangential velocity boundary condition at solid surfaces. These proposed revisions are based upon both experiment and theory, including re-interpretation of the following three items: (i) experimental ‘‘near-continuum’’ thermophoretic and other low Reynolds number phoretic data for the movement of suspended particles in fluids under the influence of mass density gradients rr; caused either by temperature gradients in single-component fluids undergoing heat transfer or by species concentration gradients in inhomogeneous two-component mixtures undergoing mass transfer; (ii) the hierarchical re-ordering of the Burnett terms appearing in the Chapman–Enskog gas-kinetic theory perturbation expansion of the viscous stress tensor from one of being based upon small Knudsen numbers to one of being based upon small Mach numbers; (iii) Maxwell’s (1879) ubiquitous v m -based ‘‘thermal creep’’ or ‘‘thermal stress’’ slip boundary condition used in nonisothermal gas-kinetic theory models, recast in the form of a v v -based no-slip condition. The v v vs. v m dichotomy in the case of compressible fluids is shown to lead to a fundamental distinction between the fluid’s tracer velocity as recorded by monitoring the spatio-temporal trajectory of a small non-Brownian ARTICLE IN PRESS 0378-4371/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.10.034 Tel.: +1 617 253 6687; fax: +1 617 258 8224. E-mail address: [email protected] (H. Brenner).

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Physica A 349 (2005) 60–132

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Navier–Stokes revisited

Howard Brenner�

Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge,

MA 02139-4307, USA

Received 27 August 2004

Communicated by J.V. Sergers

Available online 24 November 2004


A revision of Newton’s law of viscosity appearing in the role of the deviatoric stress tensor

in the Navier–Stokes equation is proposed for the case of compressible fluids, both gaseous

and liquid. Explicitly, it is hypothesized that the velocity v appearing in the velocity gradient

term rv in Newton’s rheological law be changed from the fluid’s mass-based velocity vm; thelatter being the velocity appearing in the continuity equation, to the fluid’s volume velocity vv;the latter being a stand-in for the fluid’s volume current (volume flux density nv). A similar

vm ! vv alteration is proposed for the velocity v appearing in the no-slip tangential velocity

boundary condition at solid surfaces. These proposed revisions are based upon both

experiment and theory, including re-interpretation of the following three items: (i)

experimental ‘‘near-continuum’’ thermophoretic and other low Reynolds number phoretic

data for the movement of suspended particles in fluids under the influence of mass density

gradients rr; caused either by temperature gradients in single-component fluids undergoing

heat transfer or by species concentration gradients in inhomogeneous two-component

mixtures undergoing mass transfer; (ii) the hierarchical re-ordering of the Burnett terms

appearing in the Chapman–Enskog gas-kinetic theory perturbation expansion of the viscous

stress tensor from one of being based upon small Knudsen numbers to one of being based

upon small Mach numbers; (iii) Maxwell’s (1879) ubiquitous vm-based ‘‘thermal creep’’ or

‘‘thermal stress’’ slip boundary condition used in nonisothermal gas-kinetic theory models,

recast in the form of a vv-based no-slip condition. The vv vs. vm dichotomy in the case of

compressible fluids is shown to lead to a fundamental distinction between the fluid’s tracer

velocity as recorded by monitoring the spatio-temporal trajectory of a small non-Brownian

see front matter r 2004 Elsevier B.V. All rights reserved.


17 253 6687; fax: +1 617 258 8224.

dress: [email protected] (H. Brenner).

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H. Brenner / Physica A 349 (2005) 60–132 61

particle deliberately introduced into the fluid, and the fluid’s ‘‘optical’’ or ‘‘colorimetric’’

velocity as monitored, for example, by the introduction of a dye into the fluid or by some

photochromic- or fluorescence-based scheme in circumstances where the individual fluid

molecules are themselves responsive to being probed by light. Explicitly, it is argued that the

fluid’s tracer velocity, representing a strictly continuum nonmolecular notion, is vv; whereas itscolorimetric velocity, which measures the mean velocity of the molecules of which the fluid is

composed, is vm:r 2004 Elsevier B.V. All rights reserved.

PACS: 51.10.+y; 66.10.�x; 66.20.+d; 66.60.+a

Keywords: Navier–Stokes; No-slip; Rheology; Thermophoresis; Korteweg stress

1. Introduction

1.1. Background

This is the first in a projected two-part series of papers concerned with proposedmodifications to the Navier–Stokes–Fourier equations of continuum fluid mechanics(hereafter referred to by the acronym N–S–F equations) for compressible fluids, bothgases and liquids. The term ‘‘compressibility’’ as used here refers not to the usualeffects of pressure on fluid density, but rather to the effects on density of temperatureand/or composition (the latter in multicomponent mixtures) in systems wherepressure effects on density are assumed to be small relative to these other effects, e.g.liquids or essentially isobaric gases. The present paper, the first in the series, focuseson the seemingly less controversial aspects of the changes we propose, namely thoseconnected with: (i) purely viscous effects in fluids associated with the form of theconstitutive equation for the deviatoric stress appearing in Newton’s rheologicalviscosity law; and (ii) the no-slip tangential velocity boundary condition imposed atsolid surfaces. Experimental and other data will be presented in support of thehypothesized changes in these two items. All of these data pertain to phenomenainvolving creeping flow situations, where inertial effects in the momentum equationare negligible, as too are viscous dissipation effects appearing in the energy equation.Moreover, the data presented here are, for all practical purposes, limited to idealgases, although the underlying theory to be developed appears to bear no suchlimitation.The second paper in the projected series focuses upon further modifications in the

N–S–F equation set posed by the work of the late statistical mechanician Yuri L.Klimontovich in the inertial and viscous dissipative terms (as briefly discussed inSection 7). Currently, no experimental data exists to support these further changes.Moreover, these additional changes raise fundamental questions whose profundityand controversial nature greatly exceed the comparable level of contentiousnesslikely to be aroused by the changes to the N–S–F equations advocated herein. Assuch, it seemed prudent to clearly separate the issues associated with Klimontovich’swork from those discussed in the present paper, by simply postponing a discussion of

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their common features. Those readers whose appetites are likely to be whetted by thedeeper, even more fundamental, issues to be addressed in the second paper areencouraged to study in advance the key works of Klimontovich, briefly cited inSection 7.

1.2. Newton’s law of viscosity. Mass velocity vm vs. volume velocity vv

The Navier–Stokes equations, valid for Newtonian fluids (both gases and liquids),has been a fixture of continuum fluid mechanics ever since 1845 following theseemingly definitive work of Stokes [1] and others [2], who proposed the followingrheological constitutive expression for the fluid’s deviatoric or viscous stress T:

T ¼ 2mrvþ kIr . v ; (1.1)

in which the velocity v appearing therein is the mass velocity vm; namely the velocityappearing in the continuity equation

qr=qt þr . ðrvmÞ ¼ 0 : (1.2)

The overbar appearing in (1.1) denotes the symmetric and traceless portion of thedyadic which it surmounts, so that with D any dyadic, D ¼ ð1

2ÞðDþDTÞ � ð1


in which I is the dyadic idemfactor.Despite the essentially universal acceptance of the fact that [3]

v¼?vm ; (1.3)

we argue below that the velocity appearing in Newton’s viscosity law should, in fact,be the volume velocity, vv [4,5]:

v ¼ vv : (1.4)

Our arguments are based upon a trio of interrelated arguments: (i) ascribing acontinuum interpretation to experimental thermophoretic particle velocity data inthe small Knudsen number gaseous regime, thereby challenging the contemporaryview of such thermophoretic motion as a noncontinuum phenomenon (Section 3);(ii) re-scaling the hierarchical order of the Burnett thermal stress terms appearing inthe Chapman–Enskog Knudsen number expansion of gas kinetic theory [6,7],thereby changing their status from noncontinuum to continuum-level stresses on apar with the viscous terms appearing in the classical Navier–Stokes equations(Section 4); (iii) re-interpreting Maxwell’s thermal creep slip condition imposed uponvm into a no-slip condition imposed upon vv (Section 5). Individually and collectivelythese lend credibility to Eq. (1.4).The volume and mass velocities are not independent of one another, but are

related through the expression [4]

vv ¼ vm þ jv (1.5)

in which jv represents the diffusive flux of volume. [Eq. (1.5) represents thedecomposition of the volume flux nv into respective convective and diffusivecontributions, nmv and jv; where v ¼ 1=r is the specific volume.] In the case of

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single-component fluids undergoing heat transfer the constitutive equation for jv isgiven by the expression [4]

jv ¼ ar ln r ; (1.6)

wherein a ¼ k=rcp is the fluid’s thermometric diffusivity, with k the thermalconductivity and cp the isobaric specific heat. Eq. (1.6) is also applicable to the caseof two-component mixtures undergoing isothermal mass transfer [4], but with areplaced therein by the Fick’s law binary diffusion coefficient D: The respective heat-or mass-transfer cases to which (1.6) applies correspond to circumstances in which,for a fixed pressure, either r ¼ rðTÞ or r ¼ rðwÞ; with T the temperature and wherew denotes either one of the two species weight fractions in the isothermal binarymixture.Obviously jv ¼ 0 when the fluid is incompressible, corresponding to the case where

r is uniform throughout the fluid (for all time), and hence from Eq. (1.2) for whichr . vm ¼ 0: In such circumstances the velocities vm and vv coalesce, whence themodified Newton’s viscosity law (1.1) � (1.4) reverts to its conventional form, (1.1)� (1.3).

1.3. What velocity appears in the no-slip boundary condition?

Accompanying the vm vs. vv velocity issue with respect to the constitutive choicefor v appearing in the deviatoric Newton’s law stress expression for T; Eq. (1.1), is acomparable issue that arises in connection with the dynamical no-slip tangentialvelocity boundary condition

Is . ðv�UÞ ¼ 0 on qVs (1.7)

imposed at a solid surface qV s; with U the velocity of the solid at a point lyingon its surface. In the above, with n a unit normal vector on the surface, Is ¼ I� nn isthe unit surface dyadic or surface projection operator. While vm is universallyregarded as being the appropriate velocity v to insert into (1.7) under ordinarycircumstances, we nevertheless present experimental data as well as theoreticalkinetic theory results dating back to Maxwell [8] in 1879 (see Section 5) that implicitysupports the fact that the velocity appearing in (1.7) should be vv rather thanvm: On the other hand, the purely kinematical no-penetration, normal-velocityboundary condition,

n . ðvm �UÞ ¼ 0 on qVs ; (1.8)

retains its usual mass-based form.

1.4. The fluid’s tracer velocity

Let the spatio-temporal curve x ¼ xðx0; tÞ denote the trajectory through space of asmall (albeit non-Brownian) passive tracer particle entrained in the fluid that passesthough the space-fixed point x0 at time t ¼ 0 and is later observed to be present atsome other point x at time t: The current, purely kinematical, view of continuum

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fluid motion is that the fluid’s tracer or Lagrangian velocity vl ; defined by the relation

vl :¼qxqt

� �x0


and derived from the tracer’s trajectory, is identical to the fluid’s mass velocity vm

[9–11]. In opposition to the accepted view that vl?¼vm; we will argue later (see Section

6), based upon the same experimental data and gas-kinetic theory result invoked insupport of the relation v ¼ vv; that, in fact,

vl ¼ vv : (1.10)

For the time being we do not proffer any general theoretical reasons as to whyseemingly incontrovertible theoretical notions underlying the classical hypothesisthat v ¼ vm fail in the case of compressible fluids. Rather, we simply regard our mainresult, v ¼ vv; as being purely empirical. Our results do, however, point up thesingular nature of so-called ‘‘incompressible’’ fluids, a subject that has long attractedthe attention of theoreticians, and led, inter alia, to the development of the field of‘‘extended irreversible thermodynamics’’ [12]. Indeed, our results will be seen toimpact on the general subject of irreversible thermodynamics [13–16] as a whole,with regard to both the presence of nonnegative quadratic terms in the local rate ofentropy production and the Onsager symmetry relations [17] for linear constitutiveforce/flux relations.

2. ‘‘Pre-constitutive’’ transport equations

2.1. Generic physical laws

For simplicity in what follows we limit ourselves to single-component Newtonianfluids undergoing heat transfer. The basic equations governing transport in suchfluids [18] consist of: (i) the continuity Eq. (1.2); (ii) the Cauchy linear momentumequation (in the absence of external body forces),


Dt¼ r . P; P ¼ �Ip þ T; (2.1)

and (iii) the energy equation,


Dt¼ �r . ju þ r . ðP . vmÞ; e ¼ u þ v2m=2 : (2.2)

In the above,




þ vm .r (2.3)

denotes the material derivative, p is the thermodynamic pressure, P the pressuretensor, and e and u are, respectively, the specific (i.e., per unit mass) total andinternal energies. Moreover, ju is the diffuse internal energy current (i.e., fluxdensity), whose constitutive form remains to be specified. No distinction need be

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made between the diffusive total and internal energy fluxes, je and ju; since neitherkinetic nor potential energy (the latter present when time-dependent conservativeforces act—see Appendix A) can be transported diffusively, only convectively. Whilewe keep open the choice of velocity v appearing in the constitutive expression (1.1)for T contributing to ð2:12Þ; accounting for our use of the word ‘‘pre-constitutive’’ todescribe the above pair of Eqs. (2.1) and (2.2), no comparable freedom of choice ispossible for the other velocities appearing in these equations, all of which have beenchosen to be vm: These other velocities, three in number, refer to their respective rolesas: (i) the specific momentum density, say m; the latter appearing in the guise of vm inEq. ð2:11Þ [19,20]; (ii) the ‘‘kinetic energy’’, say v2k=2; the latter appearing in the guiseof v2m=2 in Eq. ð2:22Þ; and (iii) the ‘‘rate of working’’, say r . ðP . vwÞ the latterappearing in the guise of r . ðP . vmÞ in Eq. ð2:21Þ: As discussed in Appendix A, thislack of freedom in possibly choosing one or more of these three velocities, namelyðm; vk; vwÞ; to be vv; say, rather than vm; is imposed by the requirement that thecontinuity, momentum, and energy equations all remain invariant under choice ofreference frame.While it is commonly assumed [18] that the diffusive internal energy current ju is

given in single-component fluids by Fourier’s law, namely

ju ¼ q ; (2.4)


q ¼ �krT ; (2.5)

we have argued elsewhere [4,21] that in the case of compressible fluids the correctexpression should be given, rather, by the constitutive relation

ju ¼ q� pjv : (2.6)

To the extent that Eq. (2.6) is correct, the diffuse volume current jv transports notonly momentum diffusively, as implied by Eqs. (1.1) � (1.4)–(1.5), but, concurrently,it also carries internal energy diffusively, as embodied in the jv term appearing in Eq.(2.6), above any beyond the Fourier conduction term (2.5). In order to keep our‘‘pre-constitutive’’ options open for as long as possible, thereby eschewing aparticular constitutive choice for ju; i.e., Eq. (2.4) vs. (2.6), it proves convenient towrite the flux density ju appearing in the energy equation ð2:21Þ in the pre-constitutively neutral form,

ju ¼ q� pðv� vmÞ : (2.7)

In view of Eq. (1.5), this expression thus constitutively embodies both theconventional and modified formulas for ju; namely (2.4) or (2.6), according as eitherv ¼ vm or v ¼ vv:It also proves convenient to reformulate the energy equation (2.2) in terms of

enthalpy, rather than internal energy. By definition, u ¼ h � pv in which h is thespecific enthalpy and v ¼ 1=r is the specific volume. Application of the materialderivative operator (2.3) to this thermodynamic identity, together with use of boththe continuity equation (1.2) and the standard single-component thermodynamic

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relation [22]

dh ¼ cp dT þ v � Tqv


� �p

" #dp ; (2.8)

eventually transforms Eq. (2.2) into the form



qtþ vm .rT

� �¼ r . ðkrTÞ þ r . ½pðv� vmÞ �

q ln rq ln T

� �p



þ 2mrv:rvm þ kðr . vÞðr . vmÞ : ð2:9Þ

In the classic case, where v ¼ vm; this reduces to the standard equation [18, p. 589]governing the spatio-temporal evolution of the temperature field in single-component fluids.Additionally, when written out explicitly, the constitutively neutral linear

momentum equation (2.1), applicable to both the conventional and unconventionalchoices of v; becomes


qtþ vm .rvm

� �¼ �rp þ r . ð2mrvÞ þ rðkr . vÞ : (2.10)

In the subsequent context of analyzing the experimental ‘‘phoretic’’ data used tosupport our hypothesis that v ¼ vv; it will be seen [23–25] that the ‘‘dissipative’’terms, other than r . ðkrTÞ; appearing in the energy equation (2.9) and the inertialterms appearing in the momentum equation (2.10) prove to be negligible in thepresent class of phoretic experiments, whence it will turn out that these equationsare, respectively, replaced by the much more tractable expressions,



qtþ vm .rT

� �� kr2T (2.11)


�rp þ mr2vþ krr . v � 0 ; (2.12)

the latter constituting a ‘‘creeping flow’’ approximation (cf. [82]). Constant physicalproperties have been assumed in the above pair of equations, except of course for thedensity r:

2.1.1. Boundary and initial conditions

Irrespective of which of the two constitutive relations, v ¼ vm or v ¼ vv; isultimately identified as being the physically correct choice, the requisite velocityboundary conditions (1.7) and (1.8) to be imposed upon the independent variablesðvm; p;r;TÞ appearing in Eqs. (2.9) and (2.10), or in the simplified phoretic forms(2.11) and (2.12), are applicable for both choices of v:

2.1.2. Further simplifications

For the class of phoretic and transpiration problems that we subsequently addressit will be assumed as a satisfactory approximation that the ‘‘law of adiabatically

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additive volumes’’ [4] adequately describes the equation of state for the fluid,corresponding to the fact that ðqv=qTÞp is a constant at a fixed pressure. (The latterrepresents an obviously correct assumption in the case of single-component idealgases; moreover, in the case of liquids it is surely a valid approximation oversufficiently small temperature ranges). This assumption, in conjunction with the factthat the viscous dissipative terms in the energy transport equation (2.9) arenegligible, leads to the conclusion [4] that the volume velocity is solenoidal,

r . vv ¼ 0 ; (2.13)

despite the fact that r . vma0 in the case of compressible fluids. Eq. (2.13) is a purelykinematical relation, and hence holds irrespective of whether the dynamicalconstitutive choice made for v is correctly given by vm or vv [26,27].

2.1.3. The case v ¼ vv

As a consequence of (2.13) the bulk viscosity term disappears from Eq. (2.12)when the constitutive equation v ¼ vv applies (although this term will not generallydisappear in the classical case where v ¼ vm unless k ¼ 0). In such circumstances, Eq.(2.12) simply becomes

�rp þ mr2vv � 0 : (2.14)

Note in this case that Eqs. (2.13) and (2.14) are now identical in appearance to theclassical creeping flow equations (cf. Ref. [82]) for incompressible fluids, wherein vm

appears in these equations in place of vv: When considered in conjunction with theno-slip boundary condition (1.7) imposed upon vv; we see that the solutions ðvv; pÞ ofsuch compressible flow problems may be simply obtained from those already existingin the literature for conventional incompressible creeping flow solutions [28].

3. Thermophoretic motion. Experimental confirmation of the no-slip condition (1.7)

for v ¼ vv

The ultimate test of any physical theory lies in the agreement of its quantitativepredictions with experiment. In this context we advance the hypothesis that thevelocity v appearing in the pre-constitutive energy and momentum Eqs. (2.9) and(2.10) as well as in the no-slip boundary condition (1.7) is correctly given by v ¼ vv;rather than by its traditional form v ¼ vm: This hypothesis is based upon what will beshown, inter alia, to be its accord with existing thermophoretic [23] and other [24,25]phoretic data (the latter data being discussed in Section 7). Subsequently, in Section6, we tentatively advance a reason for the success of this constitutive choice, derivedfrom the empirical observation, demonstrated after the fact, that vv; rather than vm;constitutes the ‘‘tracer’’ (or Lagrangian) velocity vl of the fluid continuum. However,we neither require, nor do we proffer here, a rational explanation of the fact thatv ¼ vv; rather, we simply regard the latter constitutive expression as representing apurely empirical relation, one that when used in conjunction with the pre-constitutive momentum and energy transport equations (2.1) and (2.2) and

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boundary conditions of Sections 1 and 2, together with the continuity equation, fitsthe all of the experimental facts to which it is addressed (including, of course, thewealth of data existing for the incompressible case, for which circumstance vv and vm

are synonymous).Thermophoresis is a phenomenon whereby a generally heat-conducting, force-

and torque-free solid particle, typically spherical, suspended in a fluid (usually a gas)within which an externally imposed temperature gradient exists, is observed to movefrom the hotter to the colder regions of the fluid; that is, the particle moves againstthe temperature gradient [29–32]. Recognition of this phenomenon was apparentlyfirst recorded in the literature by Tyndall [33,34] in 1870. The first successfulquantitative explanation of thermophoresis, limited to gases, was offered by Epstein[35] in 1929, who, building upon Maxwell’s [8] explanation of the phenomenon ofthermal transpiration some 50 years earlier, also involving fluid motion in gasesanimated by a temperature gradient [cf. 24], attributed thermophoretic particlemotion in the so-called ‘‘near-continuum’’ range of Knudsen numbers ðKn51Þ tosmall noncontinuum effects, resulting in Maxwell slip (‘‘thermal creep.’’) at theparticle surface (see Section 5). Maxwell’s argument ascribes this vm slip to the actionof thermal stresses existing in the gas proximate to the surface (see Section 4).Despite many embellishments since Epstein’s [35] original analysis, the explanationof the thermophoretic movement of aerosol particles in gases for Kn51 remains,today, universally accepted as a strictly noncontinuum phenomenon, since no suchmotion, either of the particle or fluid, is predicted by N–S–F equations whenconsidered in conjunction with the traditional no-slip boundary condition. However,since the thermophoretic velocity of a macroscopic non-Brownian particle in thenear-continuum region is observed to be independent of its size, this noncontinuumview of the phenomenon appears inconsistent to us. Explicitly, for a given gaspressure, and hence a given mean-free path l; the non-Brownian particle’sthermophoretic velocity U is observed to be independent of the Knudsen number,Kn ¼ l=a; based upon the sphere radius a: (In the latter, l ¼ 1=nps2 according toelementary kinetic theory [18], with n the number density of molecules and s themolecular collision diameter.) And since the Kn value is the sole determinant ofwhether the observed phenomenon is, or is not, due to noncontinuum effects, suchparticle size-and, hence, Knudsen number-independence appears inconsistent withthe latter possibility.Epstein’s [35] noncontinuum thermal creep interpretation of thermophoretic

motion in the small Knudsen number regime was based upon his use of thetraditional relation v ¼ vm in the constitutive equation (1.1) for the deviatoric stressT; together with his adoption of Maxwell slip [8] [cf. Eq. (5.2)] of the mass velocity vm

at the particle surface in lieu of the traditional no-slip condition (1.7) � (1.3). Weoffer here an alternative, strictly continuum interpretation of thermophoresis basedupon use of the constitutive equation v ¼ vv; in both (1.1) and (1.7), the latterconnoting no slip of the volume velocity vv at the sphere surface. As in theaccompanying thermophoresis paper [23; see also Appendix B], we address theproblem of a laterally unbounded single-component fluid, either gas or liquid,confined between two parallel walls separated by a distance L; with the hot wall

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situated at the coordinate x ¼ 0 maintained for all time at a temperature T0; and thecold wall situated at x ¼ L maintained at TL: A solid sphere of radius a ða=L51Þand thermal conductivity ks; present in the fluid, is situated sufficiently far fromeither wall such that particle-wall hydrodynamic interaction effects can be neglected.Gravity effects are also assumed negligible. The thermal boundary conditions, asusual, entail continuity of both the temperature and the normal heat flux componentat the sphere surface. Using the mass, momentum, and internal energy equations (thelatter within both the fluid and solid sphere) of Section 2, we seek to calculate thevelocity U of the force- and couple-free sphere for circumstances in which the fluidmotion may be regarded as quasistatic. In what follows in the next two subsections,this calculation is effected for both constitutive choices, vm and vv; of the velocity v:Ultimately the respective predictions of U for the two cases are compared withexperiment.

3.1. The case v ¼ vm

Upon use of the traditional vm formulation of the N–S–F equations and no-slipboundary condition imposed upon vm in (1.7) it readily follows [23], inter alia, fromthe trio of mass, momentum, and energy equations, in conjuction with the appropriatethermal boundary conditions, that vm ¼ 0 and p ¼ const: ð8xÞ; corresponding to theabsence of fluid motion and concomitant sphere motion, U ¼ 0: After all, the problemis one of steady-state, convection-free, heat conduction through the fluid as well asthrough the sphere’s interior, governed in both phases by the classical heat conductionequation, r2T ¼ 0: As such, it is unsurprising that no force acts on the sphere thatwould otherwise serve to animate it. It was this failure of the traditional continuumN–S–F equations and boundary conditions to predict the observed thermophoreticmotion of the sphere that led Epstein [35], later followed by others (see the review inRef. [23]), to seek a noncontinuum slip-based explanation for this particle motion.Apart from the nonuniform temperature field TðxÞ characterizing this pure heat-conduction problem, the only other interesting physical feature worthy of note—onethat would, perhaps, not normally come to mind—is the fact that the undisturbedtemperature gradient, jrT j ¼ ðT0 � TLÞ=L; existing in the quiescent isobaric fluidnecessarily creates a corresponding density gradient, rr; owing to the fluid’s isobaricequation of state, r ¼ rðTÞ: Nevertheless, according to conventional N–S–F theory,the existence of this density gradient is predicted to be without physical effect asregards steady-state momentum and energy transport. As will be seen, it is in thisrespect, namely the absence of physical consequences stemming from the local thermalexpansion of the fluid, that the traditional constitutive choice, v ¼ vm; in Eqs. (1.1) and(1.7) ultimately proves to be unsatisfactory in explaining the origin of the fluid-mechanical forces serving to animate the sphere.

3.2. The case v ¼ vv

With this alternative choice of consitutive equation, the governing equationsand boundary conditions outlined in Sections 1 and 2 now yield [23] nontrivial

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physical results, wherein Ua0; stemming from the presence of a temperature-induced density gradient. In effect, the departure from the traditional N–S–F resultdescribed in the preceding paragraph, wherein U ¼ 0; is a consequence of theappearance of the density gradient in the constitutive equation (1.6) for thediffusive volume flux, which, through Eqs. (1.4) and (1.5), serves to couple vv to rr;thereby ultimately linking the particle’s thermophoretic motion to the fluid’sthermal expansion. The mathematical details underlying the calculation of U

are set forth in a companion paper [23; but see also Appendix B], which incor-porates several simplifying assumptions in order to reduce the algebraic effortrequired to solve these now strongly coupled transport equations. In addi-tion to assuming constant equilibrium and transport properties for cp; k; ks

and m; Eq. (2.13) was further assumed to be applicable. In circumstanceswhere the particle is nonconducting, corresponding to the case where ks ¼ 0; it isshown in Appendix B that the calculation of U can be effected trivially by the use ofFaxen’s law, without the need to literally solve the requisite boundary-valueproblem!In any event, the more detailed and general calculation [23] based upon the

constitutive relation v ¼ vv; addressing the case where ksa0; eventually yields thefollowing expression for the thermophoretic velocity of the sphere:

U ¼ �ab

1þ ðks=2kÞrT (3.1)

in which b ¼ ðq ln v=qTÞp � �ðq ln r=qTÞp is the fluid’s coefficient of thermalexpansion. Furthermore, rT is the temperature gradient in the neighborhoodof the sphere that would exist in the sphere’s absence from the fluid. It is given by theexpression rT ¼ �xðT0 � TLÞ=L; in which x is a unit vector in the x-direction,perpendicular to the walls. We note that the product, ab � kðqv=qTÞp=cp; appearingin Eq. (3.1) is a temperature-independent constant, since each of the three thermaltransport and equilibrium properties appearing on the right-hand side of the producthave, individually, been supposed constant. According to its derivation, Eq. (3.1) isequally applicable to all fluids, whether gas or liquid.

3.3. Comparison of Eq. (3.1) with experimental data

3.3.1. Gases

In the case of nonpolar, generally polyatomic, ideal gases, the gas’s thermometricdiffusivity appearing in Eq. (3.1) can be expressed via the Eucken equation [18] asa ¼ ð9� 5g�1Þu=4; with u ¼ m=r the fluid’s kinematic viscosity and g ¼ cp=cv thefluid’s specific heat ratio. In addition, b ¼ 1=T for ideal gases. Accordingly, in suchcircumstances Eq. (3.1) becomes

U ¼ �C0s


1þ ðks=2kÞur ln T ; (3.2)

in which C0s ¼ Cð9� 5g�1Þ=4 is an Oð1Þ dimensionless coefficient. The values of g for

ideal monatomic and diatomic gases are, respectively, g ¼ 53and g ¼ 7

5; whence it

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follows that:

C0s ¼

1:5 ðmonatomic gasesÞ ;

1:36 ðdiatomic gasesÞ :


The Eucken relation entering into (3.3) is an approximate relation. More precisely,with Pr ¼ cpm=k the fluid’s Prandtl number, it follows that a ¼ u=Pr; representing anexact relation between a and u: Expressed more accurately, one thus has for idealgases that C0

s ¼ 1=Pr:Experimentally, in the case of gases, the thermophoretic velocity U of a spherical

particle relative to the walls (on whose surfaces the prescribed temperatures T0 andTL are time-independent constants) is given over the entire range of Knudsennumbers, Kn ¼ l=a; from the ‘‘near continuum’’ regime, Kn51; to the ‘‘free-molecule’’ regime, Knb1; by the semi-empirical data correlation,

U ¼Csðk=ks þ CtKnÞ

ð1þ 3CmKnÞðk=ks þ 1=2þ CtKnÞur ln T : (3.4)

This generally Knudsen number-dependent formula, due originally to Brock [36],builds upon the prior work of Epstein [35] (see the correlations in Refs. [36–40]), andserves to organize a vast amount of experimental thermophoretic data for gases. Thedimensionless coefficients Cm; and Ct; are manifestations of noncontinuumhydrodynamic isothermal ‘‘velocity slip’’ and ‘‘thermal jump’’ effects, respectively,whereas Cs is Maxwell’s, nonisothermal slip-velocity thermal creep coefficient. Thelatter is assumed in Maxwell’s [8] model of the phenomenon (see Section 5) toconstitute a noncontinuum effect arising from the so-called thermal stresses. [In ourview, on which we will subsequently elaborate, this would appear to be aninconsistency, since Cs is invariably taken to be a constant, independent of Knudsennumber. This independence serves to distinguish its effects from those of the othertwo terms appearing in (3.4) involving the pair of noncontinuum coefficients, Cm andCt; each of which, being multiplied by the Knudsen number, thus disappears fromconsideration in the Kn ¼ 0 continuum limit.] A compilation is available [39] of‘‘best fit’’ experimental values for these three semi-empirical coefficients, correspond-ing to the values Cs ¼ 1:17;Cm ¼ 1:14; and Ct ¼ 2:18:In the free-molecule limit, Kn ! 1; Eq. (3.4) shows that U ! 0: This was to be

expected intuitively, since the spacing between molecules is so large on averagecompared with the size of the sphere that the molecules do not ‘‘see’’ the latter andhence pass it by unimpeded, failing thereby to animate the sphere.In the continuum limit, Kn ! 0; of interest to us in connection with the test of our

continuum hypothesis v ¼ vv; Eq. (3.4) adopts the form

U ¼ �Cs


ð1þ ks=2kÞur ln T : (3.5)

Following its original introduction in 1879 [8], the nature and magnitude ofMaxwell’s slip coefficient Cs; the only one of the three noncontinuum coefficientsnow contributing to U; has attracted the attention of a number of theoreticians and

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experimentalists. As discussed in the accompanying paper [23], theoretical values ofCs have ranged from Maxwell’s [8] original estimate of 3

4for Maxwell molecules to

Derjaguin et al.’s [38] predicted value of 32; with experimental values falling in the

range 0.89–1.17, the latter figure constituting the most recent ‘‘best-fit’’ value [39,40].These latter, experimentally based, values are assumed applicable to bothmonatomic and polyatomic molecules, despite the fact, historically, the theoriesdating back to Maxwell [8], and underlying the generic equation (3.4) were basedentirely upon, and therefore presumably limited to, monatomic Maxwell molecules.Our theoretical formula (3.2) � (3.3) for gaseous continua obviously accords well

with the experimental data correlation (3.5) for the ‘‘near-continuum’’ regime. This,despite the fact that our formula is based upon purely continuum-mechanical no-sliparguments, which suppose that v ¼ vv: In contrast, Epstein’s [35] theory, whichunderlies Eq. (3.5), is based upon noncontinuum thermal creep arguments, whilesupposing that v ¼ vm; albeit with a Maxwell tangential slip velocity boundarycondition imposed upon vm (in contrast with the traditional case, where no vm slip isnormally assumed to occur). Subsequently, in Section 5, the agreement between thesetwo very different theoretical approaches, Epstein’s and ours, is reconciled, where itis pointed out that the attribution by Epstein and others of noncontinuum Knudsen-based behavior to Maxwell’s slip condition appears to be unwarranted, amisconception not actually due Maxwell himself, but rather to those who followedhim [41–44].

3.3.2. Liquids

Our companion paper [23] also compares our theoretical vv-based equation (3.1)with the experimental thermophoretic data of McNab and Meisen [45] for liquids,the only such ‘single-particle’ liquid-phase data known to us. Without repeatingwhat is said in Ref. [23], suffice it to say here that Eq. (3.1) accords satisfactorily withthese limited data. Clearly, liquids are incapable of displaying noncontinuumbehavior with respect to rationalizing the thermophoretic movement of macroscopic(non-Brownian) particles. As such, there exists no possibility that the thermo-phoretic behavior observed by McNab and Meisen in liquids can be explained byother than a continuum mechanism [46]. Since, as we have pointed out, classicalcontinuum arguments, based on the assumption that v ¼ vm; fail to predict anythermophoretic motion whatsoever, the agreement of our nontraditional v ¼ vv

continuum model with the liquid-phase data of McNab and Meisen [45] wouldappear to enhance the credibility of our nontraditional volume velocity hypothesis.An alternative theory of thermophoresis in liquids has recently been put forward

by Semenov and Schimpf [47]; see also Refs. [135,136]. In common with ourapproach, their expression for the thermopheretic velocity of a ‘‘solute molecule’’contains the same thermal expansion proportionality factor b½1þ ðks=2kÞ�1rT asappears in our Eq. (3.1) [see Eq. (8.9)]. Moreover, in agreement with our predictionsfor liquids, their thermophoretic velocity too is independent of particle size.However, in contrast with our analysis, where the ‘‘solute’’ particle plays a passiverole relative to the solvent (at least in the nonconducting case), while simply beingentrained in a fluid already in (volumetric) motion owing to the temperature gradient

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(see Section 6), in the Semenov–Schimpf model the physicochemical properties of thesolute particle enter explicitly into their expression for U: Their results are principallyemployed to calculate Soret and thermal diffusivites in polymer solutions, a topicthat we too have addressed elsewhere [48] based upon our Eq. (3.1), and wherein wehave compared our respective predictions. Further commentary on their model vs.ours is offered in Section 8.

4. A modified Newton’s law of viscosity. Theoretical confirmation of the deviatoric

stress relation for v ¼ vv based upon the re-scaled Burnett equations

To the extent that the appropriate constitutive velocity v appearing in Eq. (1.1) forthe viscous stress is given by vv rather than by its more traditional value vm;Newton’sviscosity law (1.1) departs from its usually assumed form, resulting in the followingequation for the deviatoric stress [49]:

T ¼ Tm þ 2mrjv ; (4.1)

in which Eq. (1.5) has been used. Here,

Tm ¼ 2mrvm (4.2)

denotes the traditional form of Newton’s rheological law. The revised relation (4.1)plays a major role in the detailed thermophoretic calculations of Ref. [23] (see alsoAppendix B for the nonconducting case) summarized in the preceding section. But inthose calculations the choice of the velocity v appeared not only in Newton’sviscosity relation (1.1), but also in the no-slip boundary condition (1.7). As such, it isnot inappropriate to ask: ‘‘Is there any independent evidence of the correctness of therevised rheological law (4.1) that does not rely on the solution of a particularboundary-value problem, and hence on the validity of the nonstandard no-slipboundary condition (1.7), such as entered into the thermophoretically based indirectevidence provided in Section 3 for the viability of the constitutive relation v ¼ vv?’’The answer is ‘‘Yes’’! Direct evidence does indeed exist, based upon the well-knownresults of Burnett [7] in the kinetic theory of gases, as set forth in standard texts [6]on the subject.Extracting physically relevant results from the kinetic theory of gases involves

solving the Boltzmann equation for ideal, single-component, monatomic gases byasymptotically expanding the solutions thereof for the deviatoric stress field T andheat flux vector q in powers of the Knudsen number for Kn51: These expansionstake the form [6]

T ¼ T0 þ KnT1 þ Kn2 T2 þ OðKn3Þ ; (4.3)

with a similar expansion for q: The leading-order, OðKn0Þ � Oð1Þ; so-called‘‘continuum’’ term in this expansion corresponds to the Euler equation of idealfluid theory, namely T0 ¼ 0 and q0 ¼ 0; according to which only convectivemomentum and energy transport mechanisms exist. Equivalently, diffuse ormolecular transport mechanisms are absent. The next, OðKnÞ; term in the expansion,

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the so-called ‘‘near-continuum’’ contribution to the stress tensor, yields Newton’slaw of viscosity, namely KnT1 ¼ Tm as well as Fourier’s law of heat conduction,Kn q1 ¼ �krT : Proceeding further in the hierarchy, the so-called ‘‘noncontinuum’’OðKn2Þ terms in the expansion of the deviatoric stress [50] correspond to a sequenceof generally nonlinear Burnett terms [6,7], representing the corrections to theclassical vm-based Newtonian deviatoric stress tensor (4.2) [51–54]. Upon incorpor-ating these Burnett terms into the expression for the deviatoric stress, it provesconvenient to write

T ¼ Tm þ Tþ þ OðKn3Þ ; (4.4)

with Tþ � Kn 2 T2; which is of OðKn2Þ; designated here as being the ‘‘extra’’deviatoric stress, above and beyond the Newton’s law Tm-level.Kogan et al. [55] and Bobylev [56] (see also the recent work of Yariv and Brenner

[57], briefly summarized near the end of the present section) each discuss thehierarchical ordering of the six OðKn2Þ Burnett terms contributing to Tþ (see [6, p.286]) with respect to their relative orders-of-magnitude in relation to the near-continuum OðKnÞ Newtonian term, Eq. (4.2). The Kogan–Bobylev discussion of thepertinent scaling issues leads to the following conclusion: For Kn51; and forcircumstances in which both a characteristic Reynolds number Re � LU=n and acharacteristic nondimensional temperature gradient LjrT=T j; each based uponsome characteristic length L of the system, are both of Oð1Þ; two terms among the sixappearing in the complete Burnett sequence of stresses [6,7] are, in fact, actually ofOðKnÞ; rather than of OðKn2Þ; and hence should properly be classified hierarchicallyas belonging to the traditional N–S–F equation set. In drawing the distinctionbetween their modified view [55–57] of the scaling of the Burnett terms and the moretraditional view [6,7] thereof, the velocity U appearing in their definition of Re usedin the scaling is taken to be the fluid-mechanical velocity vm rather than the velocityof sound, c:These two re-scaled Burnett deviatoric stress terms are given explicitly by the


Tþ ¼ �m2

rTðK1rrT þ K2T

�1rTrTÞ ; (4.5)

wherein K1 and K2 are Oð1Þ dimensionless constants whose respective values dependupon the particular choice of intermolecular potential used in evaluating theBoltzmann collision integral. As the two terms in Eq. (4.5) contributing to thedeviatoric stress both depend exclusively upon temperature gradients in the gas, theyare referred to in the literature as ‘‘thermal stresses,’’ a concept dating back toMaxwell [8]. As discussed below, this pair of thermal stresses should not be regardedas OðKn2Þ noncontinuum terms, but rather as near-continuum OðKnÞ terms in theterminology of gas-kinetic theory, since they are of the same order as thoseappearing in the N–S–F equation set. Of course, in fluid-mechanical parlance, boththe Euler and N–S–F equations are regarded as strictly continuum-level linearmomentum equations! As such, in sorting out the issues, one must carefullydistinguish between the respective gas-kinetic and fluid-mechanical notions of what

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constitutes a continuum. This issue is further discussed in the next section in relationto Maxwell’s thermal stress-induced, so-called slip, boundary condition.For monatomic Maxwell molecules [6,7,58] the values of the two constants are,

respectively, K1 ¼ 3 and K2 ¼ 3d ln m=d ln T : Accordingly, Eq. (4.5) adopts theform [59]

Tþ ¼ �3mrT


Now, m ¼ ru; in which u is the kinematic viscosity. Furthermore, for single-component ideal gases the relation between density and temperature at constantpressure is such that r ¼ C=T ; where C is a constant. Thereby, it readily follows thatfor isobaric situations,

Tþ ¼ 3mrður ln rÞ:

However, for an ideal monatomic gas, one has from the Eucken relation [18] thatu ¼ ð2=3Þa: Accordingly, with use of the constitutive equation (1.6) for jv applicableto the present single-component heat-transfer case, the preceding equation for theextra stress tensor becomes, exactly,

Tþ ¼ 2mrjv : (4.6)

Upon inserting the latter into (4.4) and comparing the resulting expression withEq. (4.1), it is seen that the relation thereby obtained for T via gas-kinetic theoryarguments is identical to that obtained from our purely continuum-mechanicalrelation (4.1), the latter simply having been hypothesized on the basis of theagreement of the v ¼ vv-based predictions derived therefrom with thermophoretic(and other) experimental data. By any reasonable criterion this exact agreementbetween two such disparate derivations—one based upon purely theoreticalmolecular-level arguments, and the other upon purely experimental continuum-level arguments—both yielding exactly the same rheological constitutive expressionfor the stress tensor, is so striking as to provide seemingly unequivocal corroborationof the correctness of the proposed continuum-mechanical constitutive equation (1.1)� (1.4) as well as of the no-slip boundary condition (1.7) entering into thecalculation, certainly for gases. Concomitantly, on the gas-kinetic side of the ledger,this agreement would appear to provide compelling evidence in support of theKogan/Bobylev [55,56] argument that the hierarchical perturbation order of the twoBurnett thermal stress terms (4.5) is such that they should indeed be regarded asrepresenting near-continuum OðKnÞ; but not noncontinuum OðKn2Þ; terms. That is,in fluid-mechanical terminology the pair of Burnett thermal stresses (4.5) should belabeled as being strictly continuum-level terms, on a par with those in the N–S–Fequation set.As subsequently discussed, this altered view of the classical gas-kinetic theory

Burnett hierarchy has important consequences in a variety of contexts, including are-interpretation of Maxwell’s slip condition (Section 5), currently regarded as amanifestation of noncontinuum OðKn2Þ effects. Equally striking is the fact that—tothe extent that our modified constitutive equation v ¼ vv; together with Eqs. (1.5)

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and (1.6), is accepted as being correct on the basis of experimental confirmation—such experiments may be regarded as also formally confirming key aspects ofBurnett’s gas-kinetic theory calculations. On the other hand, a number of tantalizingtheoretical issues remain unresolved with respect to the foundations of gas-kinetictheory. Why, for example, does it appear on the basis of the K1 and K2 valuesappearing in Eq. (4.5) that it is only for Maxwell molecules that the constitutiveequation (1.6) holds for monatomic gases? And, more generally, how does the notionof a volume velocity enter into kinetic theory?In any event, and irrespective of the latter issue, a formal rationalization of the

retention of only the two ‘‘thermal stress’’ terms in the Burnett expansion, whiledisregarding the other four terms appearing therein, is provided in Ref. [57]. Thatrationalization is based upon the fact that classical Chapman–Enskog, smallKnudsen number expansion of the Boltzmann equation is not uniformly valid. Thisderives from the fact that with v a characteristic velocity, the Knudsen number isrelated to the Mach and Reynolds numbers, M ¼ v=c and Re ¼ Lv=u; respectively,by the expression Kn ¼ M=Re: The conventional scaling arguments used inobtaining the Chapman–Enskog perturbation expansion for the Kn51 near-continuum OðKnÞ case are commonly understood as applying in the dual limits,M ¼ Oð1Þ and Reb1; that is, where v ¼ OðcÞ [6]. However, one could, alternatively,achieve the limiting Kn51 near-continuum criterion by instead considering the dualpair of inequalities, M51 and Re ¼ Oð1Þ; corresponding to the case where v ¼

OðvmÞ5c: And it is precisely for those circumstances in which this dual combinationof limits (rather than the single limit Kn51) is met that one would expect a linearizedrheological law like Newton’s law of viscosity to be valid. When each member of thecomplete set of six Burnett terms is now individually rescaled according to thesealtered criteria, one finds that the two thermal stress terms, designated by Tþ in Eq.(4.5), indeed become of the same order as the classical viscous stress terms (4.2) withrespect to their hierarchical OðMÞ ordering in the perturbation expansion. It is thisformal small Mach number re-scaling [57] that singles out for special attention onlythe two thermal stress terms among the six Burnett terms, the remaining ones allbeing of higher order in Mach number. Explicitly, in place of Eq. (4.3) one wouldnow write

T ¼ Tð0Þ þ MTð1Þ þ M2Tð2Þ þ OðM3Þ; Re ¼ Oð1Þ ; (4.7)

valid for M51; with MTð1Þ given by Eq. (4.1).Written out explicitly in the light of (1.5), Eq. (4.1) becomes

T ¼ 2mrvv : (4.8)

This relation has, thus far in this section, been independently confirmed to applyonly to the case of single-component monatomic ideal gases. However, to the extentthat Eq. (1.1) � (1.4) applies quite generally, all Newtonian fluids [liquids andpolyatomic gases, both single- and two-component, the latter with a replaced in (1.6)by the binary diffusivity D], would be expected to obey Eq. (4.8), at least incircumstances where bulk viscosity effects are absent (the latter due either to the factthat k is zero or that r . vv ¼ 0) [60].

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In summary of the results of this section, while the existence of thermal stresses asembodied in Eq. (4.5) or any of its subsequent variants has, in the past, beenregarded as a noncontinuum, Knudsen-type phenomenon, this attribution does notappear to be correct, as shown by the above analysis. By way of comparison, in thecase of solids, the comparable existence of thermoelastic stresses [61] in linearlyelastic media obeying Hooke’s law, and stemming from thermal expansion, has longbeen recognized as a strictly continuum-level phenomenon. That it has taken so longto recognize the existence of comparable continuum-level thermal stress effects influids, stemming from the same thermal expansion effects as exist in solids, is surelyattributable to the focus upon mass rather than volume in connection withdeformation in fluid continua. In solids it is the geometry of continuous deformationrather than the displacement of mass, viewed as a continuum, that commands one’sattention when addressing the fundamental issue of the movement of the respectivecontinua through space [62]. Dilatation, a feature common of both solids and fluids,as embodied in the role of b in governing the deviatoric stress, constitutes one aspectof such deformation, the major one in the case of thermal expansion effects. In thissense, the movement of volume [4], whether regarded as static as in the case of solidsor kinetic as in the case of fluids, constitutes the common theme connecting the twodistinct types of continua. These come together in the notion of Korteweg stresses[63], a strictly continuum-level concept discussed in Section 8, where it is the densitygradient or equivalent specific volume gradient, rather than the temperaturegradient, which serves to unify the notion of thermal stresses in solid and fluidcontinua.

5. Maxwell’s vm-slip boundary condition viewed alternatively as a no-slip condition

imposed upon vv

Maxwell’s [8] well-known thermal creep slip-velocity boundary condition imposedupon vm at a solid surface plays a fundamental role in the application of gas-kinetictheory to actual physical problems. Its importance stems from the fact that theBoltzmann equation, by itself, offers no insight into the boundary condition to beimposed upon the tangential velocity at such surfaces [64, pp. 52–56,98–102]. As aresult, the use of Maxwell’s slip condition is ubiquitous in applications. It is relevant,for example, in analyses of gaseous phoretic phenomena, including thermophoresis,thermal transpiration, and diffusiophoresis (the latter when this slip condition isextended from temperature to species concentration in isothermal binary systems[25,65]), as well to analyses of other baroeffect phenomena [66]. This sectionaddresses, inter alia, the background behind Maxwell’s thermal creep condition,including the approximations inherent in its original derivation. Our goal in doing sois, in part, to demonstrate that the current view of Maxwell slip as a residualmanifestation of Burnett-level OðKn2Þ noncontinuum behavior is a misinterpretationof its true status in the asymptotic Knudsen number perturbation-expansionhierarchy. Rather, we argue that his thermal creep condition is actually an OðKnÞ

near-continuum term in the obfuscating language of statistical mechanics. On the

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other hand, when expressed in the terminology of fluid mechanics it constitutes astrictly continuum-level term, on a par with the N–S–F equations (either the originalor modified versions thereof). Having thus relegated thermal creep to the role of anordinary continuum fluid-mechanical boundary condition, we argue that whatappears to be Maxwell ‘‘slip’’ of the mass velocity vm corresponds exactly to ‘‘no-slip’’ of the volume velocity vv; that is, Maxwell’s boundary condition is really a no-slip fluid-mechanical continuum-level boundary condition in disguise. In turn, thisrecognition offers further quantitative corroboration of the viability of ournontraditional constitutive relation v ¼ vv; independently of that offered in Sections3 and 4.The review of Maxwell slip which follows is carried out in the context of our

proposed no-slip v ¼ vv continuum boundary condition,

Is . ðvv �UÞ ¼ 0 on qVs ; (5.1)

derived from Eqs. (1.7) � (1.4) and used in our solution [23] of the thermophoreticproblem in the accompanying paper, as well as in the elementary Faxen’s law-basedversion thereof derived in Appendix B. Explicitly, in what follows, it is shown in thecase of single-component heat-transfer problems in gases that the Maxwell’sboundary condition [8],

Is . ðvm �UÞ ¼ Csurs ln T on qV s ; (5.2)

with rs ¼ Is .r the surface-gradient operator and Cs Maxwell’s slip coefficient, is, infact, constitutively identical to our Eq. (5.1). Whereas Eq. (5.2) is currently regardedin the literature as a noncontinuum OðKn2Þ Burnett-level-derived relation, our Eq.(5.1) is a strictly continuum relation in the language of fluid mechanics. Indeed, thefact that these relations prove to be constitutively identical in their common domainof validity, namely ideal monatomic gases, adds credibility to the viability of ourfundamental relations, Eqs. (1.4)–(1.6), as well as to that of our hypothesis regardingthe proper no-slip boundary condition to be imposed at solid surfaces, Eq. (5.1).

5.1. Chronology of the Maxwell slip condition

In an attempt to explain Crookes’s (1876) observation [67] of the rotation of a‘‘windmill’’ in a partially evacuated radiometer, Maxwell [8], in 1879, introduced thehypothesis of a temperature-gradient-induced slip imposed upon vm at solid–gasboundaries [68]. From Maxwell’s own description [8] of the evolution of his thinkingabout the problem, the derivation of his slip boundary condition (5.2) evolved in twodistinct stages and timelines. Our subsequent chronological review thereof serves toclarify the issue of whether or not Maxwell’s boundary condition is indeed to beviewed as an OðKn2Þ noncontinuum effect owing to its Burnett-like origin, or as anOðKnÞ near-continuum condition (as we believe Maxwell himself clearly under-stood).In the first stage of his celebrated paper, directed exclusively at qualitatively

rationalizing the observed behavior of Crookes’s radiometer as a strictly slip-basedphenomenon, Maxwell introduced the notion of thermal stresses existing in the

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interior of the gas, molecularly far from any surfaces. It is useful here to recall thatthe notion of stresses in a fluid (gas) originally involved only viscous stresses, arisingfrom mass–velocity gradients, as embodied in Newton’s law of viscosity, Eq. (1.1) �(1.3). Indeed, one of the early triumphs of gas-kinetic theory was Maxwell’stheoretical derivation in 1860 [69] of Newton’s law of viscosity (accompanied by thelater, experimentally confirmed and surprising, conclusion that the viscosity of a gasis independent of pressure—albeit a statement limited at the time to Maxwellmolecules). In his calculation, Maxwell used his own pre-Boltzmann molecularcollision model, involving so-called Maxwell molecules [58], the latter repelling oneanother pairwise with a force inversely proportional to the fifth power of theirseparation distance, while devoid of an attractive component.These thermal stresses, above and beyond the purely viscous stresses Tm given by

Eq. (4.2), and arising from the mass–velocity, were demonstrated by Maxwell tostem from small temperature gradients existing within the gas. By adopting anelementary perturbation scheme based upon the relative smallness of these gradients,Maxwell [8] succeeded in deriving the following formula for the deviatoric thermalstress T0þ in the gas, namely the stress above and beyond the Newtonian viscousstress tensor (4.2):

T0þ ¼ �3m2

rTrrT þ


2Ir2T þ OðT�1jrT j2Þ

� �: (5.3)

Comparison of the above with the Burnett thermal stress tensor (4.5) for the case ofMaxwell molecules shows that Maxwell’s [8] calculation represented only anapproximation to Burnett’s later [6,7], more accurate, calculations, which alsoincluded nonlinear temperature gradient effects, not accounted for by Maxwellowing to the latter’s assumption of ‘‘smallness’’ of the temperature gradient. It waswith the derivation of Eq. (5.3), valid in the interior of the gas, that the first phase ofMaxwell’s eventual slip velocity calculation ended. That is, no attempt was made byMaxwell at the time to use the knowledge embodied in Eq. (5.3) to estimate theappropriate form of the tangential velocity boundary condition at a solid surface,although he recognized that motion of the gas along a solid surface on which atemperature gradient existed would somehow be affected by these thermal stresses.The second stage of Maxwell’s derivation of his slip condition, Eq. (5.2), was

accomplished some months later. In an appendix added in May 1879 [70] to theoriginal version of his paper [8], and ending with the derivation of Eq. (5.2), Maxwellwrites: ‘‘In the paper as sent in to the Royal Society [read on April 11, 1878], I made no

attempt to express the conditions which must be satisfied by a gas in contact with a solid

body, for I thought it very unlikely that any equations I could write down would be a

satisfactory representation of the actual conditions, especially as it is almost certain

that the stratum of gas nearest to a solid body is in a very different condition from the

rest of the gas’’. He then goes on to further state that: ‘‘One of the referees, however,pointed out that it was desirable to make the attempt.... I have therefore added the

following calculations, which are carried out to the same degree of approximation as

those for the interior of the gas.’’ In his appendix, Maxwell then goes on to point outthat it was only after hearing Reynolds read his paper on thermal transpiration

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[71,72] before the Royal Society that he ‘‘...began to reconsider the surface conditions

of a gas....’’ Maxwell then goes on to describe his scheme as being ‘‘...purely

statistical...,’’ in addition to commenting that he is treating ‘‘...a gas which is not

highly rarified [sic], and in which the space-variations within distances comparable to l[essentially the mean-free path l; see below] are not very great.’’The slip-velocity formula that Maxwell ultimately developed is given by the

following expression, in which x lies normal to the solid surface qV s; and v is thetangential velocity component in the y-direction, lying in the tangent plane of thesurface:

v � Gqv





� ��3




qy¼ 0 on qV s : (5.4)

In the above

G ¼2



f� 1

� �l ; (5.5)

in which G is a coefficient of slipping, with l the mean-free path of the gas molecules,and 1� f the fraction of molecules reflected by the surface, f then being the fractionabsorbed. Maxwell’s equation (5.4) is based directly upon his approximate thermalstress equation (5.3) [73]. It is only after setting G ¼ 0 in this approximate Maxwellmolecule model that the classical Maxwell thermal creep formula (5.2) obtains,wherein Cs ¼ 3=4: Given the scaling argument discussed in Section 4 demonstratingthat the Burnett thermal stress terms constitute a continuum-level phenomenon (inthe language of fluid mechanics), it is apparent that Maxwell’s slip-velocity formula,originating from his version (5.3) of these thermal stresses, must also be a continuumresult, unrelated to OðKn2Þ noncontinuum phenomena.To bring to fruition the process of demonstrating the equality of the Maxwell slip

formula (5.2) with our no-slip volume-velocity continuum formula (5.1), substituteEqs. (1.5) and (1.6) into (5.1), and invoke the Eucken formula [18] preceding Eq.(3.2) so as to obtain the following formula, formally equivalent in physical content toEq. (5.1):

Is . ðvm �UÞ ¼ C0snrs ln T on qV s ; (5.6)

in which C0s is the coefficient defined following Eq. (3.2). Eq. (3.3) shows that C0

s ¼32

for monatomic gases. Clearly, our continuum equation (5.6) is constitutivelyidentical to Maxwell’s slip formula (5.2). Maxwell’s value of 34 for Cs differs from ourpredicted value of 3

2in Eq. (3.3) for monatomic gases. However, Maxwell’s result

applies only to Maxwell molecules, whereas our result bears no such limitation. Andit is known [6] that the Maxwell’s intermolecular force model does not generally yieldpredictions that accord well with experiments. Moreover, Maxwell’s Cs result alsodepends to some extent on the details of the collision model adopted for thereflection of the molecules from the wall (cf. Ref. [139, pp. 367–400]). As such, thedisparity in the respective values of the two slip coefficients is not regarded asserious, especially given Talbot et al.’s [39] ‘‘best fit’’ value of Cs ¼ 1:17 whichstraddles the two, together with the fact Maxwell’s thermal transpiration calculation

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[8] based upon his Cs ¼34

value underpredicts the experimentally-observedthermomolecular pressures by a factor of about 2 [24].In comparing our proposed boundary condition with that posed by Maxwell it

needs to be stressed that Maxwell’s result has been shown to hold only formonatomic gases. In contrast, our more general version,

Is . ðvm �UÞ ¼ ars ln r on qV s ; (5.7)

of Maxwell’s vm-based slip formula, representing the precursor of Eq. (5.6) withoutinvoking the Eucken relation, holds also for polyatomic gases [cf. Eq. (3.3)], andpresumably for liquids too, although the latter remains to be studied experimentally.Indeed, as pointed out in Section 8 in connection with our discussion of Kortewegstresses, in terms of our more generic formulation,

Is . ðvm �UÞ ¼ Is . jv on qV s ; (5.8)

of the slip boundary condition (5.1) [with jv given by Eq. (1.6) for single-componentgases and by its a ) D two-component counterpart for isothermal binary mixtures],it is apparent that the preceding relation should be equally applicable to the case ofbinary diffusion (a fact which was subsequently confirmed by our analysis [25] ofexperimental isothermal gaseous diffusiophoretic data in such systems). In anyevent, it now appears clear that to refer to Maxwell’s boundary condition as a ‘‘slip’’condition is to obfuscate the fact that the physical boundary condition to be appliedalong a solid surface, nonisothermal or not, is simply the standard continuum one ofno slip, albeit in disguise, since it is imposed upon vv rather than vm!

5.2. How can use of the wrong velocity field (in conjunction with the correct boundary

conditions) lead to the correct global physical result?

Various nonisothermal experiments such as thermal transpiration and thermo-phoresis have been rationalized [8,35] during the past 125 years by invokingMaxwell’s boundary condition (5.2), which, as we have indicated, is the physicallycorrect no-slip boundary condition to be applied at solid surfaces, albeit formulatedin terms of vm rather than vv: However, from the perspective of vv being the correctvelocity to use for v in the generic pre-constitutive equations of Sections 1 and 2,researchers solving these phoretic problems in the past have, according to ourinterpretation, used the wrong momentum and energy transport equations, namelythose based upon the choice v ¼ vm [74]. For consistency, these transport equationsshould have added Maxwell’s thermal stress approximation, T0þ; Eq. (5.3), or, moreaccurately, the Burnett thermal stress version thereof, Eq. (4.5), to the classicalviscous stress contribution Tm so as to obtain the correct deviatoric stress T; Eq.(4.4), to be used in the creeping flow relation, �rp þ r . T ¼ 0: Moreover, theseequations should not have been based upon the assumption of incompressibility,r . vm ¼ 0; as was originally supposed by Maxwell [8] in his analysis of thermaltranspiration and later adopted by Epstein [35] in the latter’s analysis ofthermophoretic movement. Nevertheless, our more recent recalculations of thephysical forces (or, equivalently, the thermophoretic particle velocity U [23] and the

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thermal transpiration thermomolecular pressure difference Dp [24] using ourmodified theory, have yielded the same results obtained by these prior researchers(at least to within a factor of 2 based upon Maxwell’s Cs ¼

34slip coefficient), results

which accord with experiment. But if these prior researchers indeed solved the wrongset of transport equations (albeit subject to the correct boundary conditions), howcan it be that their final results were correct, in the sense of agreeing with experimentsas well as with our results [23,24]? The answer will be seen to lie in the fact that theirresults were fortuitous, valid only in circumstances where the temperature gradientsinvolved are small (when appropriately scaled), so that neither the nonlinear terms inthe Burnett thermal stress formula (4.5) nor the incompressibility assumption affectthe outcome of the calculation. However, in more general circumstances, wherenonlinear effects are retained in calculations of this genre, the scheme based upon theincorrect constitutive formulation (1.3) will presumably lead to erroneous conclu-sions relative to those based upon the constitutive relation (1.4).Appendix C shows explicitly how and why these inherently wrong transport

equations can nevertheless lead fortuitously to the correct result, at least to leadingorder in the externally imposed temperature gradient. A main point to note in thisclass of thermal problems is that the convective fluxes appearing in the momentumand energy equations always prove to be negligible relative to the diffusive fluxeswhen the temperature gradients are sufficiently small. The calculations presented inAppendix C, applicable to both the thermophoretic [23] and thermal transpiration[24] cases, are based upon the fact that the respective velocities vm and vv sought inthe two schemes are related by Eq. (1.5), in which the diffuse volume flux is given by(1.6), namely jv ¼ ar ln r or, equivalently, by jv ¼ �ðk=cpÞrv: Inasmuch as we havesupposed, based upon the thermal law of adiabatically additive volumes, thatðqv=qTÞp ¼ const; it follows that:

jv ¼ �k




� �p

rT ; (5.9)

in which, as discussed, the multiplier of rT appearing therein is taken to be atemperature- and pressure-independent constant.

5.3. Further comments on the Maxwell slip condition

5.3.1. Knudsen boundary layers

It is invariably assumed in the literature, inappropriately in our view [75], thatMaxwell arrived at his slip condition (5.2) by introducing noncontinuum argumentsbased, in effect, upon the notion that a hypothetical Knudsen boundary layer or sub-layer exists in proximity to a solid surface qV s: This notion likely stems fromMaxwell’s [8] remark: ‘‘...that the stratum of gas nearest to a solid body is in a very

different state than the rest of the gas.’’ However, introducing a noncontinuumOðKn2Þ argument, while setting G ¼ 0 in Eq. (5.4) in an attempt to rationalizeMaxwell’s slip condition lacks consistency, since the slip coefficient Cs is independentof the Knudsen number [in contrast to the limiting behavior displayed by the other

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two noncontinuum slip effects appearing in Eq. (3.4)]. In particular, the termsmultiplied by G; characterized by a length term a; say, involve a derivative withrespect to x; the distance normal to the surface. Since G is proportional to l; theoperator Gq=qx is of Oðl=aÞ � OðKnÞ: This contribution, of course, vanishes in thelimit as Kn ! 0; whereas the Maxwell slip term in (5.4) does not. As such, the ‘‘slip’’manifested in Eq. (5.2) does not vanish in the Kn ¼ 0 continuum fluid-mechanicallimit, in disagreement with all contemporary views of the no-slip boundary conditionfor fluid continua, isothermal or not. Accordingly, it would appear that Maxwell slipcannot be attributed to a hypothetical noncontinuum, Knudsen boundary layer.Moreover, it appears clear from Maxwell’s discussion of the slip issue in

connection with Eqs. (5.4) and (5.5), that the ‘‘stratum of gas’’ in the neighborhoodof the surface, to which he refers as being physically different from that in the bulk, isassociated with the true slip coefficient G rather than with the thermal stress term inhis Eq. (5.4). Thus, setting G ¼ 0 in the latter equation in order to obtain (5.2)eliminates the possibility of Knudsen-based slip in the physical meaning of the word‘‘slip.’’ [In the contemporary literature, G is associated with what is now known as‘‘velocity slip,’’ a true noncontinuum temperature-gradient-independent phenomen-on, corresponding to the appearance of the coefficient Cm in Eq. (3.4)].

5.3.2. Maxwell slip violates the First law of thermodynamics

Last but not least it is pertinent to note that Maxwell’s slip condition (5.2), whenused in conjunction with the conventional incompressible N–S–F equation set, asMaxwell has done in his treatment [8] of the thermal transpiration problem, violatesthe First law of thermodynamics. This violation stems from the fact that were slip tooccur at a rigid immobile mass-impermeable surface, work would be performed bythe surroundings across this surface upon the fluid confined within. But this isimpossible because under such circumstances no mechanism exists by which thesurroundings can do work on the fluid! As such, this fundamental thermodynamicfact furnishes yet another reason for rejecting the notion of the Maxwell boundarycondition (5.2) as representing physical state of slip; that is, while the boundarycondition itself is constitutively correct, it does not connote slip of the actual physicalvelocity, vv: This point is further elaborated upon in Appendix D, where the First lawviolation is demonstrated in a more general context transcending our phoreticapplications.

6. The fluid’s tracer (Lagrangian) velocity vl : Rationalization of the constitutive

relation v ¼ vv

This section addresses what might appear superficially to be a kinematical ratherthan dynamical issue pertaining to the fundamental notion of the velocity of the fluidat a point in space occupied by the continuum. Whereas it is trivial to understand thenotion of the velocity of a material object, particularly the velocity of a rigid body,and especially if the body is envisioned simply as an effectively pointsize non-Brownian object, the same cannot be said of the ease of understanding the concept of

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fluid velocity at a point in space. After all, a continuum does not physically exist as acontinuous material entity. Rather, it is an abstraction since the matter of which thecontinuum is composed consists of discrete material entities, molecules with emptyspaces separating them. And these spaces, devoid of matter, can be quite vast in thecase of low-pressure gases. Thus, if a continuum does not exist physically, what is theentity whose velocity we believe that we are measuring? While one’s first responsemight be to answer that one is measuring the mean velocity of the molecules in theneighborhood of the point of interest, this cannot really be so obvious. After all,given that Maxwell’s notion of mobile molecules was not introduced into physicsuntil 1860, what in 1755, 100 years earlier, did Euler [76] believe was the physicalinterpretation of the velocity that appeared in his continuity equation (1.2) which hehad then derived? Similarly, in 1845, what did Stokes [1] understand to be theexperimental realization of the velocity appearing in his formulation of Newton’sviscosity law, Eq. (1.1)?While it might appear at first that the issues being addressed are purely

philosophical, and hence without physical import, we believe that the reader willultimately think otherwise. For example, we raise the question of whether, even in asingle-species fluid, the optical-colorimetric measurement of the spatio-temporalmotion through space of a small photochromically colored or dyed region of amoving fluid [11] will generally record the same velocity as would result fromcomparably monitoring the motion of a ‘‘tracer’’, a small (albeit non-Brownian)solid particle deliberately introduced into the fluid [9]. And if these twomeasurements disagree, which, if either, should be understood as ‘‘the’’ velocity ofthe fluid at the spatial point in question? Though both experimental methods are inwide use today, it is not obvious a priori that these two distinct schemes willnecessarily yield the same fluid velocity under all circumstances. In this context,imagine, for example, a (non-Brownian) particle placed into an ideal fluidundergoing a steady motion that is uniform far from the particle. According toD’Alembert’s paradox [77] such a particle experiences no force, whence it willpresumably remain in place as the fluid flows around it. Clearly, such a particlewould make a poor tracer of the undisturbed uniform fluid motion. (I am grateful toDr. Ehud Yariv for this example.) On the other hand, were one to photochromically‘‘color the molecules’’, so to speak, of such an ideal fluid in some small fluid region,motion of the fluid as defined by the movement of the resulting color through spacewould presumably be observed (almost certainly, registering mass motion, vm).While one might argue that no such inviscid fluid exists, thus rendering the questionmoot, this begs the fundamental question.On the other hand, a rigid particle to which fluid adheres [and hence on whose

surface the no-slip condition (1.7) prevails] has the potential to serve as a tracer ofthe fluid motion, provided that other qualifications are met, including ‘‘passivity’’ ofthe particle, an attribute which will subsequently be defined more precisely. In anyevent, a truly critical set of experiments aimed at testing the hypothesis of equalitybetween the fluid’s mass velocity vm (presumably the velocity being measured duringa colorimetric molecular-tagging velocimetry experiment [11]) and the fluid’s tracervelocity [9], say vl ; appears to have not yet been performed, at least in circumstances

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where a sensible difference between them might be expected based upon our modifiedN–S–F equations (1.4) and (2.6). Nevertheless, on the basis of what is to follow,these velocities would appear to be equal only in the case of incompressible fluidmotions, r . vm ¼ 0; wherein the density is uniform.The instantaneous tracer or Lagrangian velocity vlðx; tÞ of a small [78] particle

(whose center of volume, say, is) situated at time t at some position x in space wouldappear to be defined by Eq. (1.9), with the curve x ¼ xðx0; tÞ constituting thetrajectory of the particle through space and time. One of the key words in thisdefinition of a tracer is the word ‘‘particle’’. In continuum mechanics [79] this isinvariably interpreted as being a differentially sized ‘‘material fluid particle’’, apurely hypothetical entity. However, since the continuum is composed of discretemolecules, the notion of such a material particle is devoid of strict operationalsignificance owing to the fact that individual molecules are free to enter and leavesuch a body through its surface, despite the constancy of the material particle’s totalmass during its journey through space. As such, the definition (1.9) possesses anunambiguous operational significance only in the case of a real physical particle,namely a small (albeit non-Brownian) rigid object deliberately introduced into thefluid and subsequently monitored in space and time in order to establish theparticle’s trajectory x ¼ xðx0; tÞ and hence therefrom the ‘‘velocity’’ of theundisturbed fluid at a point of space. More will be said of this later. For the timebeing, the important point that we wish to make here is to propose the tentativekinematical hypothesis that the fluid’s tracer velocity, as measured by the precedingparticle trajectory scheme, is identical to its volume velocity. Explicitly, as suggestedby Eq. (1.10), our hypothesis is that the fluid’s tracer velocity vl is identical to thefluid’s volume velocity vv:The case for the equality vl ¼ vv can only be made experimentally, using real

particles, not hypothetical material fluid particles. That is, it is not possible to provethis relation purely theoretically, either by continuum-mechanical or statistical-mechanical arguments applied only to the fluid. Indeed, the very definition of vl ;requiring the deliberate introduction of a (real) foreign object into a hypotheticalfluid continuum, implies the impossibility of a purely kinematical pointwisecontinuum-mechanical definition of the undisturbed fluid’s tracer velocity. As such,Eq. (1.9) is not to be regarded as the definition of vl ; but rather as representing themechanism by which the trajectory, x ¼ xðx0; tÞ; of a hypothetical ‘‘fluid particle’’ x0moving through space can be determined once the tracer velocity field vlðx; tÞ is

already known independently. In this paper, we pose but a single experiment insupport of the hypothesis (1.10), namely one using a thermophoretically animatedparticle as the tracer. A subsequent paper [25] addresses a similar experiment basedupon the diffusiophoretic velocity of a particle, the analysis and interpretation ofwhich further supports the hypothesis. While the pertinent isolated-particle datacurrently available in support of Eq. (1.10) are thus quite limited in terms ofquantity, it will be seen, however, that the purely kinematical equality (1.10)possesses considerable intuitive appeal in a dynamical sense, in terms of providing arationale for the otherwise seemingly empirical constitutive relation, v ¼ vv to beemployed in Eqs. (1.1) and (1.7). That is, if Eq. (1.10) is, in fact, true, then so also is

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the resulting expression v ¼ vl ; derived from Eq. (1.4), the latter representing adynamical relation quantifying the diffuse transport of momentum through space inconnection with Eq. (1.1).This brings us to the real motivation underlying the writing of this section, namely

that of offering a rational explanation for the seemingly empirical fact that the fluidcontinuum’s ‘‘velocity’’ v appears to be given generally by the fluid’s volume velocityvv; rather than by its mass velocity vm: As suggested above, we intend to do this byintroducing the additional notion of the tracer velocity of the fluid vl (to which weshall also refer as the fluid’s Lagrangian velocity, accounting thereby for theappended subscript), and subsequently arguing that the fluid’s tracer velocity isidentical to its volume velocity, as set forth in Eq. (1.10). A tracer is a material objectthat moves through space, but not through the fluid. Indeed, the tracer moves with

the fluid, being (by definition) conveyed piggy-back by the latter. The tracer thusmoves through ‘‘space’’ in a manner no different from that of an elementary masspoint moving though space. As such, repeating the closing comments of thepreceding paragraph, it is intuitively obvious as to why the tracer velocity vl shouldqualify as the fluid’s ‘‘velocity’’. And since we intend to experimentally demonstratethe viability of (1.10), albeit only to the data-limited extent currently possible, thereason behind the seemingly empirical constitutive choice (1.4) for the fluid’s velocitywill then be seen to be transparent.

6.1. Measurement of the fluid’s tracer velocity

During the course of seeking to experimentally demonstrate the contention (1.10),it is necessary at the outset to first address in detail the question of how, at least inprinciple, the fluid’s tracer velocity field vl ½xðx0; tÞ; t; ‘‘defined’’ by Eq. (1.9), is to bemeasured. By way of reviewing the generic background underlying the experimentalmeasurement of any field variable (of which the fluid’s velocity is but an example),and as prelude to such an undertaking, we note that the operational or experimentaldefinition of any continuum field variable at a point x in space involves inserting anappropriate pointsize entity into the field at that point, and subsequently performingsome measurement on that entity, such that the magnitude (and direction, ifrelevant) of the entity’s response quantifies the strength (and, possibly, direction-ality) of the undisturbed field at that point. Of course, the entity needs to be of such anature as to not sensibly perturb the very field that one is objectively attempting tomeasure; hence, the requirement, inter alia, that the responsive entity be effectively‘‘pointsize’’ in its dimensions. But it must not be so small as to be able to interactresponsively with the individual sub-continuum (i.e., molecular) units of which thehypothetical continuum is composed; otherwise, its response will not be to thehypothetical ‘‘continuum’’ as a meaningful macroscopic physical entity in its ownright, but, rather, to the discrete microscopic sub-continuum molecular objects ofwhich the continuum is composed.From a practical point of view, no such pointsize entities exist. Accordingly,

application of this scheme in practice, at least for steady fluid motions [80], involvesthe following sequential steps: (i) successively inserting, in the neighborhood of the

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same specified point x of interest where one wishes to measure the field, a series ofotherwise identical particles of ever-decreasing size; (ii) measuring and recordingtheir respective responses; (iii) subsequently extrapolating these measurements to thelimit of zero particle size; and (iv) repeating this sequence of steps with a series ofother geometrically, physically, or physicochemically different particles, so as to beassured that the resulting, zero-size measurement obtained is independent of thephysical properties of the particle, and hence dependent only upon those of theentraining fluid itself. In circumstances where particles of different shape or differentphysicochemical properties might yield different extrapolated measurements, onemust identify among those different particles that class or group of extrapolatedmeasurements characterized by having registered the same zero-size values of vl : Forexample, in the case of the thermophoretic experiments described in Section 3, thelatter, perhaps seemingly abstruse point, becomes pertinent with respect to thethermal conductivity ks of the tracer particle when measuring the fluid’s tracervelocity vl ; cf. the subsequent remarks made in connection with Eqs. (6.1) and (6.2).Thus, in the particular case where the fluid’s tracer velocity field vlðx; tÞ constitutes

the continuum field of interest, one carries out the above-cited scheme (at least inprinciple) by sequentially introducing a series of ever smaller rigid particles, say,spherical in shape, and of diminishing radii a (all, however, sufficiently large such asto manifest no sensible Brownian motion) into the flowing fluid, with their respectivecenters of volume initially situated in the neighborhood of the point x of interest.Subsequent measurement of the vector displacement Dx of each such particlethrough space during an appropriately small time interval Dt (using, say, streakphotography or some equivalent particle-tracer velocimetry measurement [9]), thenfurnishes the particle’s radius-dependent vector velocity, U ¼ Ufag; where U :¼limDt!0 Dx=Dt � dx=dt at the point x: Extrapolation of these velocity data to zerosphere radius then furnishes the pointsize tracer velocity, Uf0g � ðqx=qtÞx0 ;representing the fluid’s tracer velocity vl with which the undisturbed fluid is movingat the point x in question. Repetition of this procedure at other points of the fluidserves to map out the entire fluid tracer velocity field. As earlier mentioned, it isimportant that the fluid adhere to the particle, so that the no-slip boundarycondition, imposed upon vm or vv; whichever is appropriate, is satisfied.

6.2. Tracer velocity in thermophoretically-animated flows

The experimentally confirmed equation (3.1) reveals, rather surprisingly, that anon-Brownian thermally animated sphere moves (relative to the steady temperaturefield) with a velocity that is independent of its radius [81,82], at least in the effectivelyzero Knudsen number continuum regime! An identical phenomenon is observed incompilations of data derived from comparable (isothermal) diffusiophoreticexperiments [25,37,83], where the movement of suspended aerosol particles [relativeto the local mass velocity, cf. Eq. (7.1)] in binary gas mixtures arises from theexistence of species-concentration-induced density gradients rr; with r ¼ rðwÞ: Thefundamental and profound significance of this size-independence, occurring in allphoretic experiments performed under effectively zero-Knudsen number circum-

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stances, appears to have escaped the attention of previous investigators. Thissize-independence, down to the limit of an effectively zero-size particle (a ‘‘tracer’’),will be seen to provide the experimental basis for our hypothesis regardingthe equality vl ¼ vv of the tracer and volume velocities of the particle-freefluid itself.In the limiting case, ks=k51; of an effectively insulated or nonconducting sphere,

Eq. (3.1) (see the derivation applicable to this nonconducting case in Appendix B)governing the thermophoretic velocity of a particle reduces to the form

U ¼ �abrT : (6.1)

The experimental conditions under which this equation applies are precisely thosewhich qualify the experimentally confirmed thermophoretic particle velocity (6.1) tobe identified as the undisturbed fluid’s tracer velocity, vl � Uf0g; namely the fluid’sintrinsic ‘‘velocity’’ through space in the absence of the particle. Specifically, thefollowing requirements are met during the thermophoretic experiments underlying(6.1): (i) owing to its applicability in the Kn ¼ 0; continuum limit, Eq. (6.1) fulfills therequirement of representing the velocity of a non-Brownian particle—that is, aparticle sufficiently large such as to be unresponsive to direct molecular-levelphenomena, and hence to be capable of responding only to continuum-levelphenomena; (ii) the, in-principle, extrapolation of a sequence of variable sphere-sizemeasurements to the hypothetical zero-size limit proves to be unnecessary in presentcircumstances, since the experimental measurements confirming Eq. (6.1) reveal theparticle velocity to be independent of size, and hence not requiring sizeextrapolation; (iii) the ‘‘passivity’’ requirement demanded of the tracer particle inorder that its measured velocity qualify as the fluid’s tracer velocity vl is fulfilled bythe choice of a nonconducting particle (i.e., ks=k51), whereby, consistent with thenature of this passivity requirement, Eq. (6.1) shows the particle velocity to dependonly upon the properties of the fluid through which the particle moves,independently of any physical attributes of the particle itself [84]; (iv) an additionaland important feature further supporting Eq. (6.1) as the (particle property-independent) tracer velocity of the fluid is the fact that in the zero particleconductivity case Eq. (6.1) holds irrespective of particle shape (i.e., for nonsphericalparticles), and independently of the orientation of such particles relative to thedirection of the temperature gradient, as is demonstrated to be the case in theaccompanying thermophoretic paper [23], as well as in Appendix B. And it is only inthis effectively zero conductivity case, ks=k51; that particle shape- and orientation-independence obtains!Given these circumstances together with the fact that vl � Uf0g we may write

vl ¼ �abrT in place of Eq. (6.1), since the experimentally confirmed particlevelocity (6.1) has now been established as being the tracer velocity of the gas relativeto the container walls. Inasmuch as the (undisturbed) temperature field is also steadyrelative to the container walls, this temperature field is then, by definition, not being‘‘convected’’ relative to the constant-temperature walls. As such, since the velocity vl ;has been measured relative to these walls, this requires that vm ¼ 0 with respect tothis same reference frame. Accordingly, one can infer from the experimentally

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observed absence of wall effects on particle motion [effects that, if present, wouldotherwise be manifested in Eqs. (3.4), (3.5) or (6.1) by the explicit appearance thereinof the ratio a=L of sphere radius a to wall separation distance L] that the walls playno explicit hydrodynamic role in thermophoretic migration phenomena. Accord-ingly, the relative motion of the undisturbed fluid velocity field with respect to thetemperature field can be expressed objectively in the following reference-frame-independent form [23]:

vl � vm ¼ �abrT : (6.2)

According to this strictly fluid-related, particle-free relation, the fluid’s tracer andmass velocities, vl and vm; are indeed different, at least in the presence of atemperature gradient!The strictly kinematical measurement of the motion of an entrained particle

manifested in Eq. (6.1) has nothing whatsover to do with the dynamicalcircumstances (i.e., temperature gradients in the present thermophoretic case, ormolar species concentration- and molecular weight-gradients in the gaseousdiffusiophoretic case [25]) that animate the fluid’s undisturbed motion vl existingin the tracer’s absence. This is true whether the dynamical forces causing the pre-existing fluid motion vl into which the particle is subsequently introduced be causedby temperature gradients, pressure gradients, buoyant gravitational forces (i.e.,natural convection [85]), or otherwise. After all, the entrained particle plays no rolein creating the pre-existing undisturbed fluid motion vl that exists prior to theparticle’s introduction into the fluid. That the dynamical circumstances giving rise tothe entrained particle’s motion are kinematically irrelevant is evidenced by the factthat a fluid-mechanical experimentalist, concerned exclusively with kinematicalvelocity measurements, need not have at her/his disposal a thermomometer,manometer, gravitometer, or any other similar device in order to objectivelymeasure the movement of tracer particles, and, hence, to subsequently report hisfindings of the respective magnitude and direction of the undisturbed fluid velocityfield, as measured by the zero-size tracer’s response to that field. In short, he/sheneed have no idea whatsoever as to the physical cause(s) of the fluid motion that he isattempting to measure (see, however, the comments in Ref. [84] with regard to theneed to perform replicate experiments using tracers of varied materials and shapes).Indeed, the less he knows about the origin of the fluid motion upon which he isreporting, the less biased will he be when interpreting his purely kinematicalobservations, and, concomitantly, the more objectively will his results be viewed byindependent referees reviewing his work.The preceding interpretation of the thermophoretic data leads to the seemingly

paradoxical conclusion that the (particle-free) fluid is ‘‘moving’’, vla0; despite thefact that the fluid is apparently ‘‘at rest’’, vm ¼ 0: That is, the fluid was already inmotion with velocity vl prior to the introduction into it of any foreign (tracer)particle! How can this be? Resolution of this paradox lies in the fact that the vectorfield vm is a ‘‘velocity’’ in name only, but not in a true physical sense. In this context,it is important to note that the sole Eulerian continuum flow field parameter, otherthan r; capable (at least in principle) of being directly measured by performing an

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experiment at the space-fixed point x; is not the velocity vector vmðx; tÞ; but is, rather,the mass flux density vector, nmðx; tÞ: As earlier discussed in section 1 (see Ref. [5]),the latter is defined such that the extensive quantity dS . nm constitutes theinstantaneous rate, at time t; at which mass flows across a space-fixed directedelement of surface area dS centered at x: In contrast with nm; the indirectly derivedintensive vector field, vm :¼ nm=r; is merely a defined quantity, essentially nothingmore than a mathematical symbol, to which one may, or may not, seek to assign aphysical interpretation. Velocity, on the other hand, viewed as a physical concept inthe classical Newtonian sense of referring to the movement of a real material objectthrough space, pertains to the motion of a pointsize object, initially situated at apoint in space, say, x0; at time t ¼ 0; and subsequently traversing a definitetrajectory, x ¼ xðx0; tÞ; as discussed in connection with Eq. (1.9).In contrast with vm; the fluid’s tracer velocity vl clearly fulfills the physical

notion of a ‘‘velocity’’ in the sense of an object of indisputable time-independentintegrity executing a deterministic trajectory through space, since the object withwhich this velocity is experimentally identified is, in fact, a material entity,namely the pointsize tracer, a foreign body deliberately introduced into the fluidin order to monitor the latter’s motion. As such, it is clearly not arbitrary toidentify the tracer velocity vl as being ‘‘the’’ velocity of the fluid continuum.The velocity vl would therefore appear to qualify as being the appropriate purelykinematical ‘‘fluid velocity’’ v referred to in textbooks in the context of fluidmovement through space (although it is not the velocity appearing in the con-tinuity equation, nor is it the fluid’s specific momentum, both of which correspondto vm).

6.3. Confirmation of the hypothesis embodied by Eq. (1.10)

Eqs. (1.5) and (1.6) combine in the present single-component heat-transfer case toyield vv � vm ¼ �ar ln v: In circumstances where the thermal law of adiabaticallyadditive volumes applies, whereupon viscous dissipative effects are necessarilynegligible, and in which the effects of pressure on specific volume are small comparedwith those of temperature, so that v ¼ vðTÞ; the preceding expression in thisparagraph adopts the form

vv � vm ¼ �abrT : (6.3)

Comparison of the latter with the experimentally confirmed tracer relation (6.2)shows that the hypothesized equality vl ¼ vv is indeed valid. As Eq. (6.2) has beenexperimentally established as applicable to both gases and liquids (albeit with lessconfidence in its validity for liquids owing to the current dearth of appropriateliquid-phase thermophoretic data), we conclude, tentatively, that the equality (1.10)is general in scope. Admittedly, this has thus far been confirmed only for thethermophoretic case. However, a similar analysis of experimental isothermaldiffusiophoretic data, coupled with a theory of that phenomenon based upon theconstitutive relation v ¼ vv applicable to the binary mass-transfer case [25], furthersupports (1.10) [see also Eq. (7.1)].

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6.4. Further discussion of the notion of a tracer velocity

The notion of the tracer velocity, vlðx; tÞ � vlðx0; tÞ; of a fluid (the latter functional-dependence notation involving x0 serving to emphasize the particle-like trajectoryaspect of the fluid’s tracer velocity) is a strictly continuum concept. As such, nodirect gas-kinetic molecular interpretation [86] of vl appears possible. This is equallytrue of the fluid’s volume velocity, vvðx; tÞ: Despite the fact that particle-tracervelocimetry is a widely practiced and seemingly elementary experimental techniquefor measuring fluid motion, given our prior discussion the possible limitations oftracer velocimetry appear not to have been scrutinized with the fundamental carethat they deserve, indeed require, in order for such measurements to be regarded asaccurately monitoring the undisturbed fluid ‘‘motion’’.As a case in point, consider Eq. (3.1) or, more properly, the formally equivalent


U ¼ ðvmÞ0 �ab

1þ ðks=2kÞðrTÞ0 ; (6.4)

representing the velocity U of a non-Brownian force and torque-free spherical particle(not necessarily a ‘‘tracer’’, owing to the particle’s nonzero conductivity) immersed in anonisothermal fluid whose undisturbed mass velocity and temperature fields are,respectively, vm and T ; the particle being supposed situated far from any boundaries.The subscript ‘‘0’’ connotes evaluation at the point in the fluid presently occupied by(say, the center of volume of) the particle. In circumstances where the particle’sconductivity is small compared with that of the fluid ðks=k51Þ; the particle can serve asa tracer of the undisturbed fluid motion, even in the presence of temperature gradients,with the fluid’s tracer velocity at the point ‘‘0’’ given by the expressionU ¼ vm � abrT ;as in Eq. (6.2). On the other hand, in the opposite case, where ks=kb1; one finds thatU ¼ vm; whence the thermophoretically animated particle no longer serves as a faithfulmonitor of the undisturbed fluid’s tracer motion, through space. Note that the particlevelocity U is size-independent in both cases, although in the highly conducting particlecase the particle velocity U will no longer be particle shape- and orientation-independent [23], an important point; that is, in the highly conducting particle case theformula U ¼ vm would be valid only for a spherical particle. (Presumably, theexperimental observation of this departure from comparable results recorded withother classes of particles would enable a thoughtful and rigorous experimentalist todiscover that this particular class of particles was not ‘‘passive.’’) Additionally, it isimportant to note that both limiting results hold only when the no-slip condition, Eq.(5.1), imposed upon vv; obtains. The ideal, inviscid fluid example discussed earlieremphasizes the importance of the no-slip condition in tracer experiments designed tomonitor the intrinsic motion of the undisturbed, particle-free fluid.

6.5. Concluding remarks

It is important to recognize that the experimentally observed difference invelocities between vl and vm in Eq. (6.2) is not simply an artifact of the particular

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class of phoretic thermal experiments that brought this velocity difference to light.Stated explicitly, the distinction is intrinsic to the very nature of inhomogeneouscontinuum fluid mechanics itself, and arises from the existence of a density gradient,rr; rather than from a temperature gradient in the single-component case (or aspecies concentration difference in the isothermal binary diffusion case). Explicitly,from Eqs. (1.5) � (1.6) and (1.10) the generic tracer/mass velocity difference is

vl � vm ¼ Dvr ln r ; (6.5)

with Dv equal to either a or D; thereby revealing the density gradient to be thefundamental source of the velocity disparity for both thermophoretic anddiffusiophoretic particle motions. As such, the driving force animating thermo-phoretic particle motion, for example, should not be attributed to the temperaturegradient; for, as is clear from Eq. (3.1), were the fluid’s thermal expansivity b to bezero, neither the fluid nor the entrained particle would move, temperature gradientsnotwithstanding. Thus, it is the existence of the density gradient rr; rather than thetemperature gradient rT ; to which the fluid’s tracer motion should be attributed. Inthis same context, the thermal stress Tþ (Section 4) should, more fundamentally, betermed a Korteweg stress (as subsequently discussed in Section 8); for were b ¼ 0 atsome intermediate point in a fluid within which the temperature varied, as is possiblein the case of some liquids (e.g. water at 4 �C), no thermal stress would presumablyexist at that point despite the presence there of a temperature gradient.At least in the zero Knudsen number continuum fluid regime, the latter

observation regarding expansivity belies the commonly held belief [43] that a freelysuspended particle moves through a gas because those molecules striking the particlefrom its hotter side are more energetic, and hence possess greater momentum, thanthose on its colder side, resulting thereby in the transfer of momentum from the gasto the particle, thus causing movement of the particle through the suspending fluidtowards the lower temperature region. (This argument may, however, possess somemeasure of validity in the large Knudsen number, noncontinuum regime.) Rather,the ultimate source of the particle motion resides in the fact that the fluid itself ismoving ðvl � vma0Þ relative to the space-fixed temperature field and, consequently,relative to the space-fixed walls. Since no force acts upon the unrestrainedthermophoretic particle, from the vantage point of an observer fixed in the walls(for which vm ¼ 0), the particle is simply entrained—that is, carried along piggy-back—by the moving fluid ðvla0Þ [except, possibly, for the ‘‘slip’’ effect seen in Eq.(3.1) due to particle’s nonzero thermal conductivity ks; causing the sphere to lag thefluid]. Indeed, were one to accept this discredited ‘‘hot-side/cold-side’’ argument,when the sphere radius became vanishingly small, no particle ‘‘sides’’ would remainto permit such a distinction, whence a very small particle would presumably notmove! Yet, based upon its size-independence, the theoretically confirmed expression(3.1) shows, at least in the continuum limit, that the particle would indeed continueto move through space relative to the temperature gradient, were its radius tobecome vanishingly small.Finally, as practical matter, we note that in circumstances where the fluid’s

undisturbed mass velocity satisfies the inequality vð0Þm

�� ��b jð0Þv ¼ abrT ð0Þ�� ��; Eq. (6.4)

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shows that U � vð0Þm ; in which case the experimentally measured particle velocity U

would closely approximate the fluid’s tracer velocity, vl :

7. Further evidence in support of the v ¼ vv hypothesis

Direct measurement of the fluid’s volume velocity (namely the flux density ofvolume nv [5]) at a point of a fluid appears infeasible on purely theoretical grounds,much less as a practical possibility. Accordingly, any evidence offered in support ofthe fundamental hypothesis (1.4), as well as of the accompanying supplementary no-slip boundary condition hypothesis (1.7) associated therewith, must necessarily takean indirect form.In this context, we note that the evidence offered thus far in support of the above two

interrelated hypotheses derives from experimental thermophoretic data for gases and, toa much more limited extent, comparable data for liquids. Such corroboration involvesthe observed agreement, discussed in Section 3, between the theoretically derivedequation (3.1) [23] and the experimentally based semi-empirical thermophoretic velocitycorrelation (3.5) for gases, as well as with the raw liquid-phase velocity data of McNaband Meisen [45]. Such agreement provides continuum-mechanical support for theviability of the relation v ¼ vv in the context of the no-slip condition (1.7). On themolecular side of the ledger, our derivation in Section 4 of the nontraditional Newtonianviscosity constitutive equation (4.8) from the Burnett solution [6,7] of the Boltzmannequation (for monatomic Maxwell molecules) provides independent, molecularly based,noncontinuum-mechanical evidence in support of the volume velocity hypothesis v ¼ vv

in the context of Eq. (1.1). This latter verification is accomplished independently of thesupplementary no-slip boundary condition (1.7), whose validity (or lack thereof) doesnot enter into the question of the fluid’s rheological equation of state for the deviatoricstress T: Reciprocally, our demonstration in Section 5 of the fact that the experimentallyand theoretically based Maxwell slip velocity formula (5.2) for ideal monatomic gasesaccords constitutively with our proposed no-slip volume velocity boundary condition(5.1), offers additional independent evidence in support of our proposed nontraditionalno-slip boundary condition (1.7) � (1.4).What follows below is a summary of further, independent evidence offered in

affirmation of the hypothesis (1.4) (or, less stringently in several cases, at least of thefact that vavm) and/or the proposed no-slip boundary condition (1.7)/(5.1) [87].Each such evidentiary item is further documented in detail elsewhere. The evidenceoffered below is of a tripartite nature, consisting of experimental, molecular, andcontinuum-mechanical documentation. Each of these three distinct categories isseparately discussed below.

7.1. Additional experimental evidence

7.1.1. Diffusiophoresis

Diffusiophoretic particle motion [25], occurring with a velocity U� ðvmÞ0 relativeto the fluid’s undisturbed mass motion vm existing at the point presently occupied by

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the center ‘‘0’’ of the particle, is the exact analog of the thermophoretic particlemotion discussed in Section 3 [see also Eq. (6.4)]; now, however, the density gradientanimating the particle, as embodied in the generic Korteweg stress [cf. Eq. (8.1)], iscaused by a binary species density gradient in an isothermal fluid rather than by atemperature gradient in a single-species fluid. In circumstances where the effects ofpressure gradient effects on the fluid’s density are small compared with those causedby species concentration gradients, one obtains the following theoretical formula forthe diffusiophoretic velocity of a finite-size solid particle whose physicochemical‘‘passivity’’ does not permit diffusion of either species through its interior, noradsorption onto its surface [29]:

U� vm ¼ Dr ln r : (7.1)

The subscript ‘‘0’’ has been omitted for simplicity in the above; rð� r0Þ is the densityof the undisturbed fluid in the neighborhood of the particle, and D is the mixture’sFick’s law diffusivity. This formula, which is applicable to both gases and liquids,was derived [25] on the basis of the pre-constitutive equations set forth in Sections 1and 2, with the velocity v assumed given by the nontraditional constitutive equation(1.4), together with the comparable nontraditional boundary condition (1.7).Additionally, the energy equation (2.9) has been replaced by the elementary binaryspecies transport equation [18],


Dtþ r . ji ¼ 0 ði ¼ 1; 2Þ ; (7.2)

with wi the mass fraction of species i; and in which the diffusive species current isassumed given constitutively by Fick’s law, ji ¼ �rDrwi; valid in the case ofisothermal isobaric systems [88]. Additionally, the law of additive volumes forthermodynamically ideal mixtures, ðqv=qw1Þp;T ¼ const, was assumed applicable,together with the functional diffusivity dependence for gases demanded by gas-kinetic theory [18], namely D ¼ D�=r with D� a density-independent constant [89].Eq. (7.1), based upon our nontraditional fluid-mechanical hypothesis (1.4), is

shown elsewhere [25] to agree well with experimental diffusiophoretic data for binarygas mixtures in the continuum regime. It also accords with other theoreticalapproaches to the rationalization of diffusiophoretic particle motion based upon thetraditional Navier–Stokes–Fick formulation derived from the classical velocitychoice (1.3) (and the inconsistent assumption that the mixture is incompressible,r . vm ¼ 0). Moreover, in these latter approaches, Maxwell’s slip boundary condition(5.2) was replaced by the so-called Kramers–Kistemaker [65] ‘‘concentration-slip’’boundary condition for gases, the latter representing the (empirical) analog [90] ofMaxwell’s Eq. (5.2) for gases, with the temperature gradient replaced by a species-gradient counterpart. Similar to the issues outlined in Section 5 and documented inAppendix C, it is fortuitous that these N–S–F analyses (‘‘F’’ here referring to Fickrather than Fourier) furnish theoretical results that accord with experiment.Eq. (7.1) shows the diffusiophoretic particle velocity to be independent of particle

size. Moreover, as a result of the particle’s supposed passivity, U is also independentof particle shape and, hence, of the particle’s orientation relative to the space-fixed

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species concentration gradient [25]. Accordingly, in the sense of particle-trackingvelocimetry experiments, such a particle plays the role of a passive tracer of thefluid’s undisturbed tracer velocity vl in the sense that U � vl : The latter conclusionaccords with the generic theoretical formula (6.5) upon there replacing the symbol Dv

by the binary diffusion coefficient D:

7.1.2. Thermal transpiration

The phenomenon of thermal transpiration, discovered experimentally by OsborneReynolds [71] in 1879, was, historically, the first ‘‘phoretic’’ phenomenon to beexplained using Maxwell’s slip condition (5.2). Indeed, as discussed in Section 5, itwas to explain Reynolds’ observations that Maxwell created the celebratedAppendix to his 1879 paper [8,70], wherein Eq. (5.2) was first derived andsubsequently applied to Reynolds’ data.The so-called ‘‘thermolecular pressure’’ associated with thermal transpiration

phenomena arises during steady-state heat transfer in horizontal, single-component,gas-filled, capillary tubes (radius a; length L) of large aspect ratio ð� ¼ a=L51Þpossessing insulated side-walls, and whose two closed heat-conducting ends aremaintained at different temperatures. According to the traditional N–S–F equationsbased upon Eq. (1.3) and satisfying the traditional no-slip boundary condition (1.7)� (1.3) on the side-walls and ends, the pressure should be uniform throughout thecapillary tube owing to the predicted absence of mass motion, vm ¼ 0; andconcomitant absence of pressure gradients based upon the traditional creeping flowequation (2.12) with v ¼ vm and r . vm ¼ 0: Explicitly, the only transportphenomenon predicted to occur in such circumstances on the basis of the N–S–Fequations is that of one-dimensional steady-state heat conduction between the hotand cold reservoirs (respectively, maintained at temperatures Th and Tc) situated atthe opposite ends of the capillary, with the resulting local temperature gradient rT

at each point of the fluid accompanied by a corresponding local density gradient rrin the supposedly isobaric system. However, upon performing this experiment with agas, Reynolds [71,91] observed a pressure difference between the two ends of thecapillary, with the pressure being highest at the hotter end ðph4pcÞ: The observedthermomolecular pressure difference ph � pc40 for gases has since been confirmedby others (cf. the review in Ref. [24] encompassing a wide variety of gases). In fact,thermal transpiration enjoys a number of current practical uses [92].As the thermomolecular pressure difference was not explicable by conventional

N–S–F continuum fluid-mechanical arguments, Maxwell attempted to the explainthe phenomenon on the basis of his slip hypothesis (5.2), and, in so doing,concomitantly give quantitative credibility to his earlier qualitative speculationsregarding the workings of Crookes’s radiometer [67]. It was in this context thatMaxwell, in the Appendix to his 1879 paper [8], first introduced his notion of thermalslip along the insulated side walls of the capillary tube, still, however, regarding theconventional incompressible N–S–F equations as correctly quantifying the fluidmotion ensuing within the capillary tube interior, as in Section 5. Maxwell’s analysishas since proved successful [24] in explaining the observed pressure difference for(monatomic) gases [93], at least to within a factor of two. However, as in all other

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applications which invoke Maxwell slip (e.g. thermophoresis and diffusiophoresis),thermal transpiration is regarded as a noncontinuum OðKn2Þ phenomenon despiteuse of the conventional N–S–F continuum equations in its rationalization. Again,this viewpoint appears to be inconsistent. In contrast, Bielenberg and Brenner [24],using the nontraditional equation set and no-slip boundary condition of Sections 1and 2 based upon Eq. (1.4), arrive in the case of gases at the same thermomolecularpressure difference, ph � pc ¼ 8Csðm2=rTa2ÞðTh � TcÞ; as is given by Maxwell’soriginal 1879 formula [8] for monatomic gases (albeit with our Cs ¼

32appearing in

place of Maxwell’s Cs ¼34) in capillary tubes of large aspect ratio. In the preceding, a

is the capillary radius and rT � const. throughout the tube owing to the assumptionof a small temperature difference, Th � Tc; and a concomitantly sensibly uniformpressure thus prevailing throughout the ideal-gas-filled capillary tube. A moregeneral formula than that given here, based upon our theory, and presumably alsovalid for liquids, is given in Ref. [24].That the two thermomolecular pressure formulas, ours and Maxwell’s, for the

thermal transpiration case should agree despite Maxwell’s apparently inconsistentuse of the traditional incompressible N–S–F equations can be explained in the samemanner as set forth generically for single-component gases in connection with Table2 in Appendix C. While agreement thus obtains in this case between Maxwell’sv ¼ vm noncontinuum slip scheme and our nontraditional v ¼ vv continuum scheme,the use of the Maxwell slip condition in conjunction with the incompressible N–S–Fequations results in a violation of the first law of thermodynamics in the presentthermal transpiration case, as earlier noted in Section 5 (see Appendix D). Thisviolation owes to the nonzero mechanical work predicted by Maxwell’s theory tohave been done on the confined fluid by the ‘‘surroundings’’ [through the action ofthe Tm ¼ 2mrvm viscous stresses existing at the wall in conjunction with theMaxwell’s thermal creep condition (5.2) prevailing there], with the resultingmechanical energy thereby supposedly introduced into the fluid ultimately beingdissipated via viscous action occurring in the fluid’s interior. In turn, as aconsequence of this dissipation, and for the steady-state circumstances characterizingthermal transpiration, more internal energy would leave the capillary at its cold endthan entered at its hot end. Such behavior constitutes a clear-cut violation of basicthermodynamic principles, given that no heat or work from the surroundings entersthe fluid through the rigid immobile insulated side walls. In contrast, no suchcontradiction arises in our scheme where no slip of the true fluid velocity vl(or vv),based upon Eqs. (1.4)/(1.10), occurs at the capillary walls!Summarizing the conclusions of this subsection, the agreement of our unconven-

tional continuum theory with thermal transpiration experiments, as detailed in Ref.[24], is offered as further evidence of the viability of Eq. (1.4). Conversely, theresulting violation of thermodynamic laws implicit in Maxwell’s scheme is offered asfurther evidence of the fact that the latter’s thermal-stress slip theory cannot beregarded as a correct physical explanation of the phenomenon of thermaltranspiration, despite fortuitously furnishing the mathematically correct result forthe thermomolecular pressure difference. Our thermal transpiration theory,emphasized above for gases, is presumably equally applicable to liquids [24], for

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which physical state no other theory currently exists. Insofar as we are aware, noliquid-phase experimental data bearing on thermomolecular pressure is yet availableto test this prediction. This poses an obvious challenge for experimentalists.

7.2. Additional ‘‘molecular’’ evidence

7.2.1. Nelson’s stochastic mechanics [94]

Nelson’s [95] stochastic-mechanical Brownian motion model of fluid-physics isbased upon the Ito calculus [96]. Heretofore, Nelson’s work has been employedconceptually only in the context of posing an alternative deterministic interpretationto the conventional probabilistic Copenhagen interpretation of the foundations ofquantum mechanics. However, when addressed instead to ordinary fluid motion inviscous continua, Nelson’s model appears to implicitly affirm the purely kinematicalaspects of our theory. Explicitly, Nelson’s relation between the fluid’s respectivetracer and mass velocities, vl and vm; as derived from his model of fluid motion is, inour notation [97],

vl ¼ vm þ Dr ln r : (7.3)

Nelson’s purely kinematical Eq. (7.3) is equivalent to a composite of our Eqs.(1.5)–(1.6) and (1.10), independently of the dynamical aspects of our theory, thelatter as embodied in the use of either of Eqs. (1.3) or (1.4)] in Eq. (1.1) or (1.7).Nelson designates Dr ln r as an ‘‘osmotic velocity’’, with D presumably a self-diffusion coefficient, the analog of Einstein’s Brownian diffusion coefficient [98–100]appearing in the Ito calculus formula for single-component stochastic processes.Nelson’s version of Eq. (7.3) traces its quantum-mechanical roots back to

Schrodinger’s [101] search for a deterministic, nonprobabilistic foundation forquantum mechanics, one based upon more classical Newtonian-like mechanicalconcepts. Nelson’s original Brownian motion-based theory [95], including its relationto quantum mechanics [102,103], has since been elaborated by Garbaczewski et al.[104,105], as well as by others [106]. From their common perspective the osmoticterm in Eq. (7.3) constitutes an extended version of the Einstein [107]–Smoluchowski[108] Brownian motion theory for a foreign (colloidal) particle dispersed in a solvent,representing an essentially purely kinematical theory [109,110]. In Nelson’s analysisthe material entity undergoing the diffusive process quantified by the coefficient D

effectively serves as a ‘‘Brownian tracer’’ of the undisturbed fluid motion (7.3). Theexistence of Nelson’s osmotic velocity is attributed to what Garbaczewski [104] termsa ‘‘Brownian recoil’’ effect of the ‘solvent’ fluid upon the ‘solute’ tracer particle.As regards the physical interpretation of Nelson’s ‘‘forward drift velocity’’ vl ;

Garbaczewski [104] states that (with x a generic point of the fluid): ‘‘If the diffusion

pertains to massive [point] particles, we have a natural physical interpretation of the

forward drift as the mean velocity of particles leaving x at time t [along samplepaths]. . . :’’ In support of this view he goes on to further state that: ‘‘The mean

position evaluated along sample paths of outgoing particles, a time Dt after they left x

at t, is xþ vlðx; tÞDt:’’ Clearly, this interpretation of Nelson’s forward drift velocity,represented by the left-hand side of Eq. (7.3), accords with the requirements

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demanded of a ‘‘particle’’ in order that the latter qualify as a ‘‘tracer’’ of theundisturbed fluid motion in the sense described earlier, namely necessitating that thetracer be: (i) non-Brownian (i.e., ‘‘massive’’ relative to the masses of individual fluidmolecules); (ii) ‘‘pointsize’’ in the particle-size-extrapolated limit; and (iii) inert orpassive.In summary, the apparent agreement of Nelson’s extended Brownian motion-

based kinematical formula (7.3) with our nontraditional kinematical view of thefluid’s tracer motion appears to add to the credibility of our volume velocityhypothesis (1.4).

7.2.2. Klimontovich’s statistical mechanics

The late Russian statistical-mechanician, Yuri L. Klimontovich (1924–2002) [111]proposed a fundamental modification of the Boltzmann equation which involvedadding to the traditional collision integral pertaining to transport processesoccurring in velocity space an additional dissipative self-diffusion-type terminvolving transport processes occurring in physical-space [112]. According to itsauthor, inclusion of the latter term results in a unified description of kinetic,hydrodynamic, and diffusional processes, presumably valid for all Knudsennumbers. The equations governing gas dynamics are then derived from thepreceding generalized kinetic equation without, however, invoking the standardsmall Knudsen number perturbation theory arguments [6,7]. In the case where bodyforces are absent and for the single-component fluid case, the results obtained in thismanner by Klimontovich are identical to our modified Eqs. (4.1) for the deviatoricstress and (2.6) for the internal energy flux, in which the quantity jv appearing inthese two equations is given by Eq. (1.6). Klimontovich’s theory is based upon theassumption that the fluid’s thermometric diffusivity a; kinematic viscosity u; and self-diffusivity D are all, equal, conditions not fully realized in practice. InKlimontovich’s work, the quantity that we have termed jv is identified by him asconstituting a self-diffusion flux rather than a volume flux.As our continuum-mechanical results were hypothesized and compared with

existing data prior to becoming aware of Klimontovich’s statistical-mechanical work(which, incidentally, bears no obvious relation to the Maxwell-Burnett nonconti-nuum notion of thermal stresses in gases) we regarded and continue to regard theagreement between the two theories as offering confirmation of both theories. This isespecially true in view of the fact that Eq. (1.5) in the form vm ¼ vv � ar ln r (ournotation) also appears naturally in Klimontovich’s work, with vm implicitlyidentified therein as being the fluid’s mass velocity by virtue of its appearance inthe latter’s continuity equation (1.2) (again, our notation). Klimontovich uses thesymbol u for what we have here identified as the volume velocity vv; although nocomparable volume-related physical identification is assigned by him to u: While thetwo modified theories agree exactly with regard to the common equations citedabove, they differ nevertheless in one important respect. Explicitly, in Klimonto-vich’s momentum and energy equations, the three ‘‘velocities’’ ðm; vk; vwÞ upon whichwe focused in Appendix A and identified therein as being vm; are, in fact, identified inhis work as being vv: This is amusing because in the original version of our present

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theory we made exactly the same identification as did Klimontovich for thesethree velocities. However, we subsequently had occasion to question this view,owing to the fact that it appears to violate Newtonian mechanics, as per thediscussion of Appendix A. That is, Klimontovich’s proposal in regard to the threevelocities appears to be in conflict with the basic principles of classical mechanics.This becomes especially clear upon including in the momentum equation bodyforces (real and fictitious), as we have done in Appendix A. We confess to beingconflicted about our altered choice of vm over vv; an issue which we plan toreview in the near future. In any event, the issue is moot at the present timein the sense that, owing to the fact that these velocities prove to be negligible inthe creeping flow problems underlying the phoretic phenomena addressed, thechoice does not impact upon the experimental confirmation of the several hypo-thesis that we have effected regarding the proposed modifications of the N–S–Fequations.

7.3. Additional continuum-mechanical ‘‘evidence’’

7.3.1. Fitts on diffuse internal energy flow [14]

Our modified theory of transport in fluid continua centers directly on the twinissues of the constitutive equation (1.4) and the related no-slip boundary condition(1.7)/(5.1), both appearing superficially to pertain only to momentum transport.However, a third transport-related issue enters indirectly into consideration, namelythat embodied in the pre-constitutive equation (2.6) involving the internal energycurrent ju; whose explicit constitutive form is implicitly required in the basic internalenergy transport equation (2.9). In particular, the nontraditional precursorthermodynamic work term �pjv appearing therein, arising from the diffusivetransport of volume, is not present in any of the standard texts on transportprocesses or irreversible thermodynamics, the sole exception (of which we are aware)occurring in the book of Fitts [14]. In place of what we have denoted as pjv in Eq.(2.6) there appears in Fitts’s expression for the internal energy flux density [see Fitts’Eqs. (3-21) and (3-25)] the term p

PNi¼1 viji [113], with vi the particle specific volume of

species i: In isobaric isothermal multicomponent mixtures [4,13] one has that vv ¼PNi¼1 vini; with ni ¼ nmwi þ ji the Eulerian flux density of species i: Inasmuch as

r ¼PN

i¼1 wivi; it thus follows from Eq. (1.5) in conjunction with the definition nm ¼

rvm of the mass velocity that

jv ¼XN


viji : (7.4)

Fitts’ extra energy flux term is clearly identical to our nontraditional pjv term,although in his analysis this term arises only in the context of multicomponent fluidmixtures, being absent in the case of single-component fluids. This contrasts with ourown work [4,21], where a diffusional volume flux contribution �pjv adds to thethermodynamic rate of working at a material surface, even in the case of pure fluids(for which jv ¼ ar ln r). Nevertheless, despite this issue of its domain of

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applicability, we regard the analysis of Fitts (and, to a lesser extent, that of theclosely related work of de Groot and Mazur [113]) as offering an independentmeasure of support for our nontraditional N–S–F theme.

7.3.2. Prior speculation in the literature regarding the equality vl ¼ vv (for

multicomponent mixtures)

The evidence to be presented in this subsection cannot strictly be countedas ‘‘legally admissible’’ in the same quantitative sense as in the examples offeredabove. Nevertheless, the musings of other researchers whose work, describedbelow, relates peripherally to our nontraditional view, would appear to constitutesomething more than mere hearsay. In this context, we refer here to remarks byOnsager [114] and Haase [115–117], independently made in connection withtheir respective discussions of isothermal diffusion phenomena, and apparentlyimplying the validity of Eq. (1.10), namely vl ¼ vv: (Of course, in the context oftheir pronouncements on the latter equation, both authors already understoodthat, generally, vvavm in multicomponent fluids owing to species concentra-tion-induced mass–density gradients. As such, it can be confidently asserted thateach appreciated the fundamental fact that vlavm; one of the main items inour theory.)The hints found thus far in the literature pertaining to our proposed equality,

vl ¼ vv; each involve the role of the volume velocity in multicomponent diffusionproblems. Thus, Onsager [114], upon recognizing the strictly relative, rather thanabsolute, nature of the diffusional process occurring among the constituent species ina mixture, refers to what he calls a ‘‘hydrodynamic velocity’’ in such problems. Hethen proceeds to identify, albeit only implicitly, the latter velocity with what is nowcalled the volume velocity [4,13–15], a name with which he was apparentlyunfamiliar at the time, as would appear from the context of his remarks. Haase[15,115], on the other hand, refers explicitly to the volume-average velocity vv inmulticomponent mixtures as being the ‘‘convective velocity’’, presumably vl ; clearlydistinguishing the latter from the mass-average velocity vm of the mixture, andobviously understanding that the former, rather than the latter, represents thephysical velocity with which the fluid moves through space. (That is, by the word‘‘convection’’ one understands the conveyance through space of an entity entrainedin a flowing fluid. Hence, the phrase ‘‘convective velocity’’ on the part of Haasewould appear to refer implicitly to the movement of a tracer entrained in the fluid,rather than to the movement of a hypothetical ‘‘material particle’’, the latter movingat the mass-average velocity vm:) The contexts of each of these two authors waskinematic rather than dynamic, although Onsager’s use of the word ‘‘hydro-dynamic’’ suggests an intuitive understanding of the dynamical issues owing to themass flow engendered by the molecular diffusion processes. Interestingly, Maxwell[118] himself refers to experimentally: ‘‘. . .measuring [sic] the velocity by its volume

which passes through a unit area rather than by the distance traveled by a molecule in

unit of time’’, apparently regarding these as being equivalent, although he may haveintended these remarks to apply only in the uninteresting case of incompressiblefluids, wherein vl ¼ vm:

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7.3.3. Streater et al. on the N– S– F equations

Streater’s ongoing theoretical studies [119–123] of single-component nonisother-mal fluids (termed by him ‘‘compressible Navier– Stokes with temperature’’, to whichtopic he subsequently refers using the acronym C-N-S-T) raise questions similar tothose raised here, albeit from a very different physico-mathematical perspective,namely ‘‘statistical dynamics’’ [123,124], a field of inquiry closely related to the fieldof ‘‘information dynamics’’. Grasselli and Streater [120], like us, argue that problemsexist with respect to the accepted form of the Navier–Stokes equations. Thisimportant point is emphasized in their opening paragraph by the statement that: ‘‘An

unusual feature of the system is that the Euler continuity equation acquires a diffusion

term. This bulk diffusion does not appear in the standard theory [124], but has arisen in

some other work [125]. It is likely that our equations are more stable and physically

more accurate than the usual Navier– Stokes equations.’’ While the actual form oftheir diffusive term differs slightly from ours, this is likely accounted for by the factthat the Grasselli–Streater ‘‘hopping rule’’ [120], quantifying the process of‘‘thermalization’’ upon which their detailed calculations are based, is somewhat adhoc. This leads also to the apparent inability of these authors to explicitly relate theirthermalization parameter l directly to a physical property of the fluid, such as thefluid’s thermometric diffusivity. Grasselli and Streater’s [120] diffusive term spillsover into their momentum and energy equations. However, their ad hocthermalization model does not permit a direct comparison of their momentumand energy transport equations with ours.

8. Discussion

This section addresses, inter alia, an apparent relation existing between theMaxwell–Burnett temperature-gradient stress and Korteweg’s density-gradientstress. Furthermore, we contrast our present body-force-free phoretic phenomenawith another class of ‘‘phoretic’’ motions animated by body forces appearing in thelinear momentum equation. The latter class, illustrated by electrophoresis, whilesuperficially appearing to involve ‘‘slip’’ of vm at the particle surface, actuallyinvolves ‘‘velocity jump’’ across a thin boundary layer (the region in which the bodyforces are sensible) accompanied by no slip of vm at the physical surface of theparticle. The important body-force distinction existing between these two classes ofparticle locomotion is used to compare and contrast our model of thermophoresis inliquids with an alternative theory of the phenomenon due to Semenov and Schimpf[47]. Additionally, several potentially practical applications of the present theory arediscussed, extending beyond the realm of strictly phoretic phenomena.

8.1. Korteweg stresses

In 1901 Korteweg [63] introduced the notion of a density gradient-inducedmechanical stress tensor, above and beyond the standard Newtonian viscous stresstensor, Eq. (4.2). After a long period of relative dormancy, the subject of Korteweg

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stresses was resurrected in connection with interfacial phenomena. In the case ofimmiscible fluids this occurred in connection with the Cahn–Hilliard stress tensor[126,27], where the interface between the two bulk fluid phases is regarded notas a singular surface but rather as a diffuse interfacial region of effectively finitethickness, whose local equilibrium and transport properties, particularly thedensity r; vary continuously across the interfacial transition region. Similar diffuseinterface uses for the Korteweg stress were later adopted by Joseph and others[26,127,128] in relation to the phenomenological possibility of a transient interfacialtension arising between a pair of miscible fluids (of different densities) broughtsuddenly into intimate contact, thereby creating a fairly steep initial densitygradient ‘‘interface’’ between them. Since the two miscible bulk fluids subsequentlyundergo mutual diffusion, any such interfacial tension must necessarily be transientin duration.The phenomenological form proposed by Korteweg [63] for his (traceless and

symmetric) stress tensor is

TK ¼ I½gr2r� aðrrÞ2 � bðrrÞðrrÞ þ drrr ; (8.1)

where the four independent Korteweg coefficients ða;b; g; dÞ (his symbols) appearingabove are, in the absence of an appropriate molecular theory, to be determinedexperimentally, including their algebraic signs. It is interesting to note that theconstitutive form of the above stress is identical to the extra stress given by Eq. (4.6)jointly with Eq. (1.6) (generalized to be applicable to both liquids and gases, as wellas to single- and two-component mixtures, the latter upon replacing a by Dv).Indeed, upon identifying this Tþ extra stress with the Korteweg’s TK stress, andupon supposing Dv to be a constant [129], the phenomenological coefficientsappearing in Eq. (8.1) are found to be given by the following expressions:

a ¼ �2



; b ¼ 2Dvmr2

; g ¼ �2



; d ¼ 2Dvmr

: (8.2)

Given Maxwell’s prior thermal stress analysis as set forth in Eq. (5.3) and aslater amended by Burnett [cf. Eq. (4.5)], together with the fact that these thermalstresses are, in reality, due to density gradients as discussed in Section 6, it wouldappear in the case of single-species fluids that the Maxwell–Burnett thermal stressand the Korteweg density-gradient stress have much in common. Although thethermal stress analysis of Sections 4 and 5 was limited to gases, Korteweg’s stressappears to be bear no such restriction. As such, it is presumed by its users[27,126–128] to be applicable to liquids. In any event, both types of stresses representthe extra stress, Tþ ¼ 2mrjv; above and beyond the classical Newtonian viscousstress tensor, Tm ¼ 2mrvm: It is interesting to note that despite their commondensity-gradient ancestry, the Korteweg and Maxwell–Burnett stresses are regardedin the literature as being continuum and noncontinuum stresses, respectively,without any rational reason being advanced for this distinction. Indeed, accordingto the thermal stress arguments advanced in Section 4, such a distinction has nobasis in fact.

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8.2. Body-force-driven vs. body-force-free phoretic particle motions

The modified Navier–Stokes equation (2.1) � (1.1) � (1.4) proposed here in placeof its traditional formulation is


Dt¼ �rp þ mr2vv ; (8.3)

at least in circumstances where, for simplicity, the viscosity is taken to be a constantand where the quasi-compressibility condition,

r . vv ¼ 0 ; (8.4)

is further supposed applicable [130]. Additionally, the proposed no-slip boundarycondition to be used in the solution of transport problems is taken to be (5.1). Inarguing in favor of Eq. (8.3), we have restricted attention exclusively tocircumstances in which f ¼ 0; and in which there is no externally imposedundisturbed pressure gradient, rpð0Þ ¼ 0 (which would otherwise have resulted ina forced-convective motion vð0Þm of the fluid). As such, the body-force-free phoreticfluid motion which ensues is necessarily animated exclusively by mass–densitygradients, rr: In this context it is crucial to note that our main motivation inproposing (8.3) was not to explain thermophoretic [23] or diffusiophoretic [25]motion; rather, it was just the opposite, namely to use knowledge of these phoreticphenomena to demonstrate that our hypothesized modification (8.3) of theNavier–Stokes equation as well as of the no-slip boundary condition, Eq. (5.1),were consistent with these experimental data.Acceptance of the modified theory as a working hypothesis enables one to make

predictions regarding phoretic and related phenomena (such as thermal diffusion[48]) that is applicable to liquids, despite the fact that for all practical purposes onlygaseous experimental data were used in demonstrating the consistency of the theory.We emphasize this fact because there does exist a body of liquid-phase thermaldiffusion data, including a comprehensive theory thereof due to Semenov andSchimpf [47], briefly discussed at the end of Section 3, and further discussed below,that attempts to rationalize at least some of these experimental data [131]. Yet, aswill be seen, their liquid-phase thermophoresis/thermal diffusion theory does notbear directly upon the validity of our modified fluid-mechanical equations (8.3) �(5.1) because their work refers to thermophoretic particle motion animated by theaction of body forces (albeit differing sensibly from zero only in the immediateneighborhood of the particles), whereas our thermophoretic theory is based uponcircumstances in which such body forces are everywhere precluded, including any‘‘boundary-layer’’ region proximate to the particle.As a prelude to illustrating the fundamental differences existing between these

respective body-force-free and body-force-driven modes of thermophoretic motion,below we contrast our analysis of body-force-free phoretic phenomena [23–25] withthat of electrophoretic particle movement in ionic solutions, the latter driven byelectrical body forces. At the same time this electrophoretic focus enables us to drawa sharp distinction between the purely mathematical Smoluchowski-type vm

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‘‘velocity jump’’ condition existing across the hypothetical Debye ionic boundarylayer [cf. Eq. (8.7)], and its Maxwell-type vm ‘‘slip’’ counterpart (5.2) whichrepresents a true mass slip at the physical surface of the solid rather than amathematical velocity-jump across a thin boundary layer of the type associated withthe parameter G in Eq. (5.4).

8.2.1. Electrophoretic particle motion in incompressible liquids [132– 134]

In contrast with Eqs. (8.3)–(8.4), whose departure from the traditionalformulations thereof is based upon fluid compressibility, explicitly involving thefact that r . vma0; the classical Navier–Stokes/continuity equation formulation forincompressible fluids takes the general form


Dt¼ �rp þ mr2vm þ f ; (8.5)

together with

r . vm ¼ 0 : (8.6)

In this incompressible, nonzero body-force context, consider the application of Eqs.(8.5) and (8.6) to explain the electrophoretically animated movement with velocity U

of a solid particle through an unbounded fluid in the limiting case of a uniformelectric field E1 at infinity, explicitly for the case of an electrically nonconductingparticle possessing a uniform surface charge density characterized by the zetapotential B; and in the Debye thin double-layer limit. The body-force density in suchproblems is given generally by the expression f ¼ reE; where the electric chargedensity re and electric field E are, respectively, given constitutively as re ¼ ��r2fand E ¼ �rf; with � the dielectric permitivity of the fluid and f the electricpotential. Outside of the double layer the electric field is effectively uniform at itsvalue E1; so that f ! fð0Þ

¼ �E1. x as jxj ! 1; whereas the potential satisfies

Laplaces equation, r2f ¼ 0: The potential differs sensibly from its far-field valuefð0Þ only in the immediate neighborhood of the particle surface qVs; where it satisfiesthe boundary condition n .rf ¼ 0 appropriate to a nonconducting body. Moreover,the Helmholtz–Smoluchowski ‘‘slip’’ boundary condition,

vm �U ¼ �Mrf on qV s ; (8.7)

with M ¼ �B=m the so-called electrophoretic mobility, reflects the jump in the massvelocity vm across the Debye boundary layer,Upon neglecting inertial effects in (8.5), the joint solution of this system of

equations and boundary conditions for f and vm; implicitly satisfying the no-penetration condition (1.8), leads eventually to the following expression for theelectrophoretic velocity of the body:

U ¼ ME1 � ��Bmrfð0Þ ; (8.8)

a result which holds independently of both the particle’s size and shape [132, p. 476,133]. As discussed, for example, by Levich [132], the Smoluchowski mass–velocityjump boundary condition (8.7), substitutes mathematically for the electrical body

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forces acting within the Debye layer in the immediate neighborhood of the particle,being equivalent thereto in regard to its physical consequences.We have elaborated on this classical electrophoretic problem because, super-

ficially, it might otherwise appear to have much in common with our priorthermophoretic [23] and diffusiophoretic [25] analyses of particle motion under theinfluence of mass density gradients. However, in contrast with the latter class ofphoretic motions leading to (8.3), such electrophoretic particle motion is, in fact,ultimately caused by the action of electrical body forces, as embodied in f; ratherthan by true mass–velocity slip; that is, the apparent slip, being purely mathematicalin origin, is simply an artifact, rather than the root cause, of the particle’selectrophoretic movement! Moreover, and perhaps most importantly, in contrastwith our nonconducting body-force-free [23,25] phoretic problems, electrophoreticparticle movement is not solely dependent upon the physical properties of the fluid(as embodied in � and m); rather, it also depends upon the particle’s surface zetapotential B; which is a joint fluid � solid property. Additionally, duringelectrophoretic motion, the particle moves through the fluid, rather than beingentrained in the already moving fluid, as evidenced, inter alia, by the appearance ofthe fluid’s viscosity in the denominator of Eq. (8.8). Because of these fundamentaldifferences, body-force-animated electrophoretic motion is very different in itsphysical origins from that of its body-force-free thermophoretic and diffusiophoreticcounterparts, which led to our proposed modification (8.3) of the Navier–Stokesequation.Despite these fundamental differences there nevertheless exists a striking analogy

between electrophoresis and thermophoresis, as discussed in Appendix E. Whenexploited, this analogy enables many of the detailed electrophoretic solutions extantin the literature to be directly and immediately applied to the resolution of theirthermophoretic boundary-value counterparts.

8.3. Thermophoretic particle motion in liquids. The work of Semenov and Schimpf


This subsection resumes the abbreviated discussion of the work of these authorsoffered at the end of Section 3. Their model [135] is based upon the following generalarguments: When the temperature of a liquid (the solvent) varies along the surface ofa solid body, the number-density of solvent molecules will, owing to the dependenceof the solvent density upon temperature, be larger in the colder regions of the surfacethan in its warmer regions. This gradient in molecular solvent number density createsan ‘‘osmotic pressure’’ gradient along the surface, in turn giving rise to an interfacialstress at the particle surface tending to balance this pressure gradient. These osmoticforces are regarded as being short-range body forces f appearing in the traditionalincompressible creeping flow equations emanating from Eqs. (8.5) and (8.6), withsuch forces being sensibly different from zero only in the immediate neighborhood ofthe particle surface. There, similar to body-force-driven theories [30–32] of phoreticmotions deriving from inhomogeneous surface adsorption phenomena, such forcestranslate mathematically into an effective ‘‘slip’’ (strictly, ‘‘velocity jump’’) of the

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fluid’s mass velocity vm near the surface of the particle (more precisely at a pointlying just outside of the ‘‘adsorption boundary layer’’).In physical terms, the situation is not unlike that resulting from the absorption of

surfactant molecules onto a surface, coupled with the fact that any variation in theconcentration of these surfactant molecules along the surface will produce aninterfacial stress [30–32] acting in the tangent plane of the surface. Here, however,the molecules of the solvent play the role of the surfactant molecules, with thesurface temperature gradient serving to produce solvent number-density concentra-tion gradients along the surface. The precise details governing the adsorption ofthese solvent molecules onto the surface depends upon the physicochemicaladsorption forces existing between the solvent molecules and those of the solid.These intermolecular forces are quantified in the Semeneov–Schimpf analysis byLondon–van der Waal’s dipole–dipole intrermolecular forces. Explicitly, thefollowing formula is presented for their thermophoretic particle velocity U [135][which we have re-written in a form deliberately reminiscent of Eq. (3.1)]:

U ¼ �a0b

1þ ðks=2kÞrT ; a0 ¼

ln 3



: (8.9)

In the above, A is the Hamaker constant defining the interaction between the particleand the fluid, m is the fluid’s viscosity, r0 is the radius of a solvent molecule and v0 issolvent volume per molecule, i.e., v0 ¼ vm=N ; where v ¼ 1=r and m are, respectively,the solvent’s specific volume and molecular mass, and N is Avogadro’s number. Forthe liquids with which they are concerned the authors suggest as an approximationthat the molecular radius be estimated from the equation v0 ¼ 8r30; the latter valid foran elementary cubic crystal packing model. They also suggest use of theapproximation A ¼


p; where Ap and As are the respective Hamaker constants

for the particle and solvent. With these approximations, one would have that a0 ¼ðln 3Þ



Despite their very different origins, Eqs. (8.9) and (3.1) obviously have much incommon including the appearance therein of the fluid’s thermal expansivity b; andthe fact that the U is independent of the size of the thermophoretically animatedparticle. However, we note the following major differences in the two models: (i) incontrast with the passive role played by the particle in our analysis, at least in thenonconducting case, the physicochemical properties of the particle play a critical rolein Eq. (8.9) as manifested by the presence therein of the particle’s Hamaker constant;(ii) as evidenced by the appearance of the viscosity in the denominator of (8.9), theparticle moves through the fluid in their model, whereas in our case it moves with thefluid. Of course there is also the fundamental issue of the presence and absence ofbody forces as the animating fluid-mechanical mechanism in the two models.The work of these authors impacts upon the fundamental notion of the tracer

velocity of a fluid, as discussed in Section 6. According to their theory, even innonconducting circumstances, where ks=k51; the particle will not be ‘‘passive’’, inthe sense that its physicochemical properties enter into the particle velocity U in theguise of the particle’s Hamaker constant. As such, the particle would not qualify in

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the role of a tracer in the case of nonisothermal fluids. More general versions of theirtheory [47] suggest that this physicochemical dependence would, in the case of non-spherical particles, also cause the particle velocity U to depend upon its orientationrelative to the direction of the imposed temperature gradient, further comprising itsrole as tracer.A variant of Eq. (8.9) [47] has been applied with some success to explain

Ludwig–Soret thermal diffusion effects in solutions of polymers dispersed innonpolar solvents, with the long-chain macromolecule viewed as a thermophoretic‘‘particle’’. In creating their model these authors use a priori knowledge of the factthat the Ludwig–Soret thermal diffusivities of polymer molecules are known to beindependent of chain length as well as of the presence of side branches [137]. By wayof comparison, our own model [48] of polymer thermal diffusion derived from (3.1)demonstrates this independence, rather than assuming it a priori.While it may appear that a conflict exists between Eqs. (3.1) and (8.9) for the

thermophoretic velocity of a spherical particle, this is not necessarily the case.Rather, it is possible that the two effects may be additive, with Eq. (3.1) referring tothat portion of the total particle velocity constituting motion with the fluid, wherebythe particle is effectively ‘‘entrained’’ (except, perhaps, for the ks=k thermalconductivity factor) in a fluid which is already moving in the absence ofphysicochemical body forces, and with Eq. (8.9) referring to that portion of theparticle’s motion taking place through the fluid, stemming from the action of localphysicochemical body forces causing ‘‘velocity slip’’. To explore this possibility, weadd the two equations in question to obtain

U ¼ �ð1þ �Þab

1þ ðks=2kÞrT ; (8.10)

� ¼ln 3





in which � is a dimensionless parameter (not necessarily small) representing theSemenov–Schimpf ‘‘correction’’ to Eq. (3.1).

8.3.1. Liquids

Semenov and Schimpf provide data enabling � to be calculated in the case of silicaparticles in various liquid solvents. Notationally, with bT their proportionalitycoefficient defined by the formula U ¼ �bTrT ; Eq. (8.11) is equivalent to

� ¼ 1þks


� �bT

ab: (8.12)

Explicitly, Semenov [135] summarizes estimated and experimentally available datafor the parameters bT ; ks; k;b appearing above. Thermometric diffusivity data weretaken from Photothermal Spectroscopy Methods for Chemical Analysis, Wiley, 1996.These data, together with the estimated value of � obtained therefrom, are tabulatedbelow in Table 1.

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Table 1

Semenov–Schimpf correction to Eq. (3.1) for the case of silica particles in various solvents

Solvent bT ð10�7 cm2 s�1 K�1Þ bð10�4 K�1Þ ks=k a ¼ k=rcpð10

�3 cm2 s�1Þ �

Water 0.19 2.1 2.4 1.43 0.14

Tetrahydrofuran 1.39 11 4.4 0.824 0.49

Acetonitrile 2.4 14 3.6 1.08 0.44

Cyclohexane 0.70 12 3.8 0.858 0.20

H. Brenner / Physica A 349 (2005) 60–132108

These data suggest that for liquids the Semenov–Schimpf contribution provides arelatively modest modification of the entrainment model (3.1), a quantitativeconclusion subject, however, to the caveat implicit in the uncertainty surroundinguse of the approximation A �


pfor the Hamaker constant. According to these

authors, their result alone, independently of the further contribution of (3.1),appears to accord with experimental data for silica, at least as regards its order-of-magnitude. However, it needs to be emphasized that the data on which thisconclusion is based are neither single-particle data, nor were the particles studiedsufficiently large to exclude the possibility of Brownian motion. And, when bothfactors are simultaneously present, one is no longer dealing with thermophoresis as aphenomenon in its own right; rather, one is more likely dealing with Soret-effectthermal diffusion phenomena, to which our single-particle, non-Brownian theory isnot directly applicable in the absence of further theoretical considerations [48]. Inany event, despite such reservations and caveats, Eq. (3.1) cannot be ruled out asapplicable to thermophoresis in liquids.

8.3.2. Gases

While Semenov and Schimpf make no claims that their analysis should applyto gaseous continua, there appears to exist no reason in principle while gasesshould be excluded from the domain of applicability of Eq. (8.9). Certainly, theirunderlying notion of fluid-generated osmotic pressure as constituting the drivingforce for thermophoretic motion in fluid continua would appear as applicable togases as it is to liquids. However, upon attempting to apply their equation togases, it becomes evident that major theoretical and experimental conflicts existwith Eq. (3.5) [in which Cs ¼ Oð1Þ], the latter viewed as representing a summary ofexperimental data for thermophoresis in gaseous continua. Even apart fromissues of differences in orders of magnitude, the experimental data support a modelin which the particle’s thermophoretic velocity increases with increasing gas viscositym and decreasing density r; whereas Eq. (8.9) would appear to predict exactly theopposite trends. Moreover, whereas the experimental data (3.5) suggest that thephysicochemical properties of the particle should not be relevant to its thermo-phoretic velocity (apart from particle’s thermal conductivity ks), the presence ofthe particle’s Hamaker constant Ap in the Semenov–Schimpf formula suggestsotherwise.

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8.3.3. The need for experimental liquid-phase data

The preceding discussion reinforces the view expressed in Section 3 of the criticalneed for single-component thermal gradient data in liquids. In subjecting these orany other theories to test, such data need not be limited to thermophoretic of thermaldiffusion data. All such theories involve the notion of slip of one kind or another atthe boundary between a fluid and a solid, whether such slip be of the Maxwellthermal-creep or Semenov–Schimpf velocity-jump types. Steady-state thermaltranspiration experiments [24] offer an interesting alternative to thermophoresis inthis respect, since the presence of slip is manifested macroscopically as a pressuredifference across a capillary tube or across a porous medium, whereas no such‘‘thermolecular pressure’’ difference would arise in the absence of slip. Experimentsof this nature are currently underway at Sandia National Laboratories under thedirection of Lisa A. Mondy.

8.4. Potential applications of the general theory

From a practical point of view, pragmatists could argue that our proposed genericmodifications of the original N–S–F (or N–S–F–F) equation set constitute but a‘‘small correction’’ to those equations in situations where density gradients exist. Inreference to the question of the general viability of this viewpoint, it is useful to firstestablish the circumstances under which the correction will be small. In particular,the corrections do indeed prove to be small in those applications for which the ratio� :¼ jjvj=max jvm; vvjð0p�p1Þ satisfies the inequality �51: Thus, as a practicalmatter, for small density gradients it will often be true that jvmj � jvvj; in whichsituations the disparity between the mass- and volume- or tracer-velocities, vv ¼ vl ;will indeed be small. Conversely, when � ¼ Oð1Þ such corrections will no longer besmall, as occurs, for example, during thermophoretic and diffusiophoretic fluidmotions. Other examples come readily come to mind, such as steady-statethermal diffusion [48], where temperature- and species-gradients coexist. Closelyrelated to the zero mass–velocity condition vm ¼ 0 encountered during thermo-phoretic motions, is the fact that in a number of applications, such as thermaltranspiration, the mass velocity does not vanish locally at each point of the fluid, butrather vanishes only in an integral sense,

RAvm;dA ¼ 0; with A typically a cross-

sectional area. Obviously, in such circumstances the small-correction inequalitycriterion �51 will again be violated, albeit this time in a global rather local pointwisesense. In any event, it is obvious that effects stemming from the existence of adiffusive volume flux, rather than always representing minor corrections toconventional results, may, in fact, be relatively large—indeed, sometimes represent-ing the dominant effect, as in the examples cited above. Microfluidic applications,involving very slow mass motions possibly driven by temperature gradients, come tomind in this context.Apart from practical engineering-type application of the types discussed above,

several other areas of potential interest come to mind. These include use of thepresent continuum theory to rationally re-organize the hierarchical ordering of theBurnett-type noncontinuum terms that arise when solving the Boltzmann equation

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perturbatively at higher order [6,7,57,64,138,139] in the Knudsen or Machnumbers. More rational theories of ‘‘mixtures’’ [140], involving elucidation of theconstitutive and transport properties of so-called ‘‘interpenetrating continua’’,appears to be another area ripe for exploitation. Transport phenomena in porousmedia [141] also offer interesting challenges owing to the fact that the vv-based Eqs.(2.13) and (2.14) together with (1.7) � (1.4) are identical to their classical vm-basedcreeping flow counterparts, often used in deriving Darcy’s law governing flow inporous media.

8.5. Summary

The main results of the present work lie in our proposed modification (8.3) of theNavier–Stokes equation and no-slip boundary condition (5.1) for the case ofcompressible fluid continua, where the mass density r is not uniform. In turn, theselead to the prediction of a Korteweg-like stresses (8.1) in both liquids and gaseswhich, in turn, cause the ‘‘phoretic’’ entrainment of isolated particles towardregions of high fluid density, such that the velocity of the particle through spaceproves to be independent of its size, shape, and physicochemical properties incircumstances where the particle is passive. It is possible in a biological contextthat such density-gradient-induced locomotion may play a role in chemotaxsisor thermotaxis.

Note added in proof

Reconciliation of Maxwell’s slip coefficient Cs ¼ 3=4 for monatomic (Maxwellian)

molecules with our value of C0s ¼ 3=2

Apart from fundamental philosophical continuum vs noncontinuum differencesfor gases, the only quantitative difference between the predictions of Maxwell’stheory and our theory arises from the purely numerical factor of two stemming fromthe different values ascribed to the Oð1Þ Maxwell slip coefficient Cs appearing in Eq.(5.2). Whereas, at least in the case of monatomic gases, Maxwell [8] arrivestheoretically at the value Cs ¼ 3=4 (for Maxwell molecules) on the basis of moleculararguments as per the discussion of section 5, our purely continuum theory givesCs ¼ 3=2 as set forth in Eqs. (5.6) � (3.3) (although without restriction to Maxwellmolecules). In the case of monatomic molecules this difference can, however, bereconciled, as discussed in what follows below.Because the equations of gas dynamics are not valid in the Knudsen layer (whose

thickness is of the order of the mean-free path of the gas molecules) proximate to asolid surface, a rigorous derivation of the continuum boundary conditions to beapplied at the solid-gas boundary involves the solution of the Boltzmann equationinside of the Knudsen layer, and the subsequent matching of this inner solution withthe outer solution of the hydrodynamic equations outside of the Knudsen layer [139].The boundary condition imposed on the inner problem adopts the form of a specified

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law of molecular reflection assumed to exist at the wall. The values of the outer fieldswhen extrapolated to the wall then provide the macroscopic boundary conditionswhich, when enforced, provide the correct solution of the continuum hydrodynamicequations outside of the Knudsen layer. The constitutive form thereby obtained viathis matching scheme for the boundary condition to be imposed upon the relativetangential mass-velocity component vm accords with that of Maxwell’s Eq. (5.2),modulo the explicit numerical value of the Cs slip coefficient appearing therein,which depends on the distribution of gas molecules reflected from the walland, hence, upon the specific physicochemical natures of both the gas and solid [139,p. 367].Various derivations of Cs exist based upon specific models of the gas and wall,

together with use of either the Boltzmann equation, the method of moments, or aMonte Carlo scheme, as recently summarized by Sharipov and Kalempa [142].Explicitly, Sharipov points out in Ref. [143] that: ‘‘According to Refs. [144,145] the

thermal slip coefficient sT [Maxwell’s Cs ] varies from 0.75 up to 1.5, where the first

value corresponds to the specular reflection of molecules on [the] surface, while the

second value corresponds to the opposite situation, i.e. back reflection.’’ (With regardto these values, Sharipov [143] is presumably referring here solely to the case ofmonatomic gases.) Of course, the 3=4 value corresponds to Maxwell’s original 1879model [8] based upon his implicit choice of an accommodation coefficient. On theother hand, the 3=2 value accords with our result for the slip coefficient, derived fromthe assumption that there is no slip of the volume velocity at solid surfaces (togetherwith use of specific heat data for monatomic gases). As such, acceptance of ourtheory would imply that the volume velocity-based no-slip macroscopic boundarycondition is equivalent, at least in the case of monatomic gases, to back reflection ofthe gas molecules at the surface.


I am grateful to Dr. James R. Bielenberg of Los Alamos National Laboratories,formerly a graduate student in the Chemical Engineering Department at MIT. Heshared with me the pleasure of performing the exact phoretic and thermodynamiccalculations, cited throughout this paper, permitting a comparison betweenexperiment and theoretical productions, and resulting in confirmation of thenontraditional constitutive equation (1.4). I am also pleased to acknowledge manyuseful hours engaged in pertinent conversations with Dr. Ehud (‘‘Udi’’) Yarivof the Mechanical Engineering faculty at the Technion, formerly a postdoctoralfellow in the Chemical Engineering Department at MIT. Equally enlighteningconversations were held with Aruna Mohan, currently a graduate student inChemical Engineering at MIT. She was instrumental in helping me to formulate theproof offered in Appendix A. Finally, I am also grateful to Dr. Sangtae Kim ofPurdue University, formerly of Eli Lilly and Company, whose encouragement washelpful in arranging financial support through Lilly for the research embodied in thepresent study.

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Appendix A. Invariance of the basic transport equations under change of reference


From a constitutive viewpoint the respective forms we have adopted for the basicmass, momentum, and energy transport equations, namely Eqs. (1.2), (2.1) and (2.2),follow, inter alia, from the requirement that they remain invariant under translationof the reference frame in which they are written. That they indeed possess thisproperty is formally demonstrated below.In order to keep an open mind on the subject, we begin with the following ‘‘pre-

constitutive’’ forms of this trio of transport equations. By ‘‘pre’’ we refer not only tothe usual notion of constitutive equations for the various diffusive (i.e., ‘‘molecular’’)fluxes such as Newton’s law of viscosity or Fourier’s law of heat conduction; rather,the following equations are equally pre-constitutive with regard to the nonmolecularconstitutive expressions introduced for the specific momentum density m appearingbelow in the momentum equation, as well as for the ‘‘kinetic’’ and ‘‘work’’ velocities,vk and vw; respectively, appearing below in the energy equation. For simplicity wewill suppose that each of the following transport equations are originally written, asbelow, in an inertial reference frame, say x:

ðiÞ Transport of mass : qr=qt þr . ðrvmÞ ¼ 0 ; (A.1)

ðiiÞ Transport of momentum : rDmm

Dt¼ r . Pþ rf ; (A.2)

ðiiÞ Transport of energy : rDme

Dt¼ �r . je þr . ðP . vwÞ : (A.3)

In these equations, f is the specific body force density, assumed (for the time being)to be conservative and hence expressed as the gradient of a time-independentpotential energy function fðxÞ; such that f ¼ �rf in which qf=qt ¼ 0: Additionally,e ¼ u þ v2k=2þ f is the specific total energy; moreover, je is the diffuse total energycurrent which we suppose henceforth to be identical to the diffuse internal energycurrent, ju on the assumption that neither kinetic nor potential energy can betransported diffusively [146].

A.1. Coordinate transformations

Consider two systems of coordinates in space. One of these systems is aninertial coordinate system x; which we regard as being at rest (i.e., fixed inspace). The second system, say x0; moves relative to the former with a (generallytime-dependent) velocity UðtÞ: Let the two coordinate systems coincide at timet ¼ 0: Then,

x� x0 ¼

Z t


UðtÞdt : (A.4)

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The respective velocities v and v0 of an object as measured by the two observers areobviously related by the expression

v� v0 ¼ UðtÞ : (A.5)

With regard to the respective material derivatives in the two reference frames,





þ vm .r andD0




�x0þ v0m .r0 ;

in which r ¼ q=qxÞt and r0 ¼ q=qx0Þt it is readily established since ðqx=qx0Þt ¼ 0 and

ðqx=qtÞx0 ¼ UðtÞ that

r0 ¼ r andD0



Dt: (A.6)

On the constitutive supposition that all diffusive fluxes and other equilibriumphysical quantities such as p and r remain invariant under the coordinatetransformation (so that, for example, P0 ¼ P), and since m� m

0¼ UðtÞ; one finds

from the linear momentum equation (A.2) together with (A.62) that

f0¼ f þ


dt: (A.7)

A.2. Energy equation

Consider the energy equation (A.3) irrespective of any constitutive choices madefor the various quantities appearing therein. Together with use of the linearmomentum equation (A.2) and the definition of the conservative force, inconjunction with several of the above relations one obtains the following expressionupon rearrangement:


Dt¼ �r . ju þ rðvm � vwÞ . f þ rvw .


Dt� r





� �þ PT:rvw :


As an aside, while we have derived the term rvm . f appearing above from time-independent potential energy considerations, this same expression could have beenobtained more generally simply as a rate of working term (per unit volume), rvm . f;added to the right-hand side of Eq. (A.3) while excluding the potential, energycontribution from its left-hand side. That the ‘‘velocity’’ by which f is multiplied is vm

rather than say vw (or even some other velocity) is then justified by reference to thespecific case where the corresponding work term is derived via potential energyconsiderations of the type having led to Eq. (A.8).Our goal in what follows is aimed at demonstrating on purely theoretical grounds

that the following equalities necessarily hold:

vk ¼ vw ¼ m � vm : (A.9)

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We begin by examining the invariance of the internal energy density u under thecoordinate transformation between the inertial and noninertial (i.e., accelerating)frames. Explicitly, we write Eq. (A.8) in the primed (accelerating) system ofcoordinates x0; thereby obtaining



Dt¼ �r0 . ju þ rðv0m � v0wÞ . f

0þ rv0w .







2Þ þ PT:r0v0w :


We have affixed primes to f as well as to each of the velocities, v ¼ ðvk; vm; m; vwÞ;appearing in the above. On the other hand we have suppressed the correspondingaddition of primes to the remaining quantities appearing in (A.10), namelyðr; u; ju;PÞ; based on the recognition that their special physical natures requirethem to remain invariant under the coordinate transformation. Insert into (A.10)the various transformations relating the primed variables to the unprimed ones,including the fact that, in general, r0v0 ¼ rv; and subtract the resulting expressionfrom (A.8). Upon rearrangement and division by r this yields the relation

ðvm þ vk � 2vwÞ .dU

dtþU .


Dtðvk � mÞ ¼ 0 :

Inasmuchas U and dU=dt can each be chosen independently at any given instant oftime, such arbitrariness necessitates that

vm þ vk � 2vw ¼ 0 and vk ¼ m : (A.11)

This is as far as one can go via internal energy invariance arguments alone.In order to complete the theoretical ‘‘proof’’ of Eq. (A.9) we further supposethat the internal energy transport equation (A.8) must not contain any ‘‘kineticenergy’’ terms. This requires that vw .Dmm=Dt ¼ Dm=Dtðv2k=2Þ: Into this equa-tion substitute Eq. (A.112) so as to eliminate m; and rearrange the resultingexpression to obtain ðvw � vkÞ .Dmvk=Dt ¼ 0: Equivalently, ðvw � vkÞ .Dmm=Dt ¼ 0:Hence, with use of (A.2) we obtain ðvw � vkÞ . ðr . Pþ rfÞ ¼ 0: Since this latterrelation has to hold independently of the constitutive equations for either P

and f; this clearly requires that vw ¼ vk: Introduction of the latter into (A.11)yields the further relation, vm ¼ vk: Cumulatively, this completes the proof ofEq. (A.9). Finally, elimination of m; vk and vw from Eqs. (A.2) and (A.3) infavor of vm furnishes the basic momentum and energy transport equations (2.1)and (2.2).Clearly, the pre-constitutive equations (2.1) and (2.2) hold independently of any of

the constitutive relations employed for the fields ðr; u; ju; p;TÞ and jv: Indeed, in thecontext of gas-kinetic theory, Eqs. (2.1) and (2.2) apply even in circumstances wherethe constitutive equations for the physical quantities appearing in the group citedin the preceding sentence, such as T; include noncontinuum terms, as in the caseof (4.3).

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Appendix B. Elementary calculation of the thermophoretic velocity of a nonconducting


Consider a velocity field v satisfying the equations �rp þ mr2v ¼ 0 and r . v ¼ 0;in which v is the field appearing in the deviatoric stress tensor in Eq. (1.1). Accordingto Faxen’s theorem [85] for such incompressible creeping flows satisfying a zero-vector velocity boundary condition, v ¼ 0 on qV s; the hydrodynamic force exertedby the fluid on a solid sphere qVs of radius a translating with velocity U whenimmersed in an undisturbed creeping flow, say fvð0Þ; pð0Þg; from which the sphere isabsent, is given by the expression

F ¼ 6pma ðvð0Þ �UÞ þa2


� �0

: (B.1)

The subscript zero appearing in the above connotes evaluation of the undisturbedvelocity and pressure fields at the center of the fluid space presently occupied by the(center of the) sphere. Accordingly, a force-free sphere will, in the absence of ‘‘walleffects’’, move quasistatically with a velocity

U ¼ vð0Þ0 þ


6mðrpð0ÞÞ0 : (B.2)

While, in the past, Faxen’s law has only been applied to the case where v refers to theusual mass velocity field vm; from a purely mathematical view Faxen’s law may beequally well applied to the volume velocity field vv; since the latter satisfies thevolume–velocity-based creeping flow equations (2.13) and (2.14) and the vectorvelocity boundary condition that vv �U ¼ 0 on qV s: As discussed in Ref. [28], thelatter condition is applicable only to the case where the sphere is nonconducting.To determine the undisturbed volume-velocity fields fvð0Þv ; pð0Þg existing in the

absence of the sphere, we note that since vð0Þm ¼ 0; it follows from Eq. (1.5) thatvð0Þv ¼ jð0Þv : However, from Eq. (1.6) we find upon rearrangement that, as in Section 3,jð0Þv ¼ �abrT ¼ const: so that

vð0Þv ¼ �abrT ¼ const : (B.3)

This undisturbed, pure fluid, sphere-free, volume-velocity field obviously satisfies Eq.(2.13) since r2T ¼ 0; while from Eq. (2.14) we find that rpð0Þ ¼ 0: Thus, Eq. (B.2)becomes

U ¼ �abrT ; (B.4)

in exact agreement with the more detailed result cited in Eq. (3.1) for thenonconducting-sphere case, ks=k ¼ 0:

B.1. Nonspherical particles

While we have demonstrated that Eq. (B.4) applies to the case of nonconductingspherical particles, it is equally applicable to nonconducting particles of arbitraryshape and orientation. This follows from the fact that the generalization [82] of

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Eq. (B.1) for an arbitrarily shaped particle is F ¼ M�1 . ½ðvð0Þ �UÞ þ Oða=LÞ0; whereM is the particle’s mobility dyadic, a is a characteristic particle size, and L is acharacteristic length appearing in the normalization, r� ¼ Lr; of the dimensionlessgradient operator appearing explicitly in the undisturbed nonuniform flow vð0Þ; sothat the Oða=LÞ term represents a wall effect. Accordingly, the velocity of such aforce-free body is U ¼ v

ð0Þ0 þ Oða=LÞ: With use of Eq. (B.3), and in the absence of

wall effects, one thus recovers Eq. (B.4). Thus, remarkably (a fact more formallydemonstrated in Ref. [23]), irrespective of size, shape, and (in the latter nonsphericalcase) orientation relative to the undisturbed temperature gradient rT ; allnonconducting particles will move at the same velocity. Accordingly, provided thatone interprets fluid motion physically as being the (undisturbed) fluid’s volumevelocity rather than its mass velocity, Eq. (B.4) expressed more generally as

U ¼ ðvð0Þv Þ0 þ Oða=LÞ (B.5)

simply states that any passive (i.e., nonconducting) no-slip particle is simplyentrained in the flowing fluid. Alternatively, with use of (1.5) this may be written inthe form

U� ðvð0Þm Þ0 ¼ ðjð0Þv Þ0 þ Oða=LÞ ; (B.6)

a result which holds even when vð0Þm a0: The relative motion represented by the left-hand side of the above then constitiutes the phoretic velocity of the insulatedparticle, as in Eq. (6.4) with ks=k ¼ 0; or in Eq. (7.1).

Appendix C. Fortuitously-correct thermal solutions based upon Maxwell slip

The energy equation (2.9) pre-constitutively incorporates both possible choices forv; namely Eqs. (1.3) and (1.4). In the case where vm is chosen, it can be shown that forboth the unsteady-state thermophoretic [23] and steady-state thermal transpiration[24] cases that, to leading order in the appropriate small parameter, ajr ln T j51;asymptotically characterizing the heat-transfer process, the energy equation reducesto the form rvm .rT ¼ kr2T : This same equation is obtained using v ¼ vm; albeitonly on the proviso that the fluid is supposed ‘‘incompressible,’’ so that r . vm ¼ 0(see below), as it done in the literature for the classical thermophoretic [35] andthermal transpiration [8] analyses. Now, in circumstances where a prescribedtemperature gradient, quantified by a characteristic magnitude jrT j; is the onlymechanism animating the fluid motion vm (as in the thermophoretic and thermaltranspiration cases to be discussed), it is found in the leading-order linearapproximation that vm ¼ Oðajr ln T jÞ: As such, to this degree of approximationthe convective term, vm .rT ; in the energy equation is of OðjrT j2Þ; whereas in thissame linear approximation the diffusive term, kr2T ; is of OðjrT jÞ: Consequently,when jrT j is small, convection may be neglected compared with conduction. In suchcircumstances, the energy equation reduces simply to r2T ¼ 0; which is applicable toboth cases, v ¼ vm and v ¼ vv: Similar arguments apply with respect to the

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convective/inertial and diffusive/viscous terms in the momentum equation, so that,only the viscous term remains relevant.The following table shows the resulting energy, momentum, and mass transport

equations describing both the Maxwell-based vm scheme, Eq. (1.3), and the modifiedvv scheme, Eq. (1.4).In the Maxwell scheme the mass transport or continuity equation is taken to be

r . vm ¼ 0 despite the fact that the fluid density is not constant as a consequence ofthe fact that the equation of state is of the form r ¼ rðT ; pÞ: The scheme based uponvv correctly takes the continuity equation to be r . ðrvmÞ ¼ 0: It is, however,unnecessary to explicitly employ the latter equation in the calculations. Rather, asthe ‘‘third’’ modified equation we instead use the accurate volume transport equation[cf. Eq. (2.13)], r . vv ¼ 0; expressing the fluid’s ‘‘quasi-incompressibility’’ [27].The trio of Maxwell equations appearing in Table 2 is to be solved for vm using

Maxwell’s slip boundary condition (5.2), whereas the modified equations are to besolved for vv; using the no-slip boundary condition (5.1). These two velocities arerelated by Eq. (1.5), in which the required diffusional volume current jv; Eq. (1.6), isgiven explicitly for the present heat-transfer case by Eq. (5.9). As such, as aconsequence of the fact that r2T ¼ 0; it follows from the latter relation that r . jv ¼

0 and r2jv ¼ 0: Accordingly, from Eq. (1.5) we find that to the order of the presentapproximation, r . vm ¼ r . vv ¼ 0 and r2vm ¼ r2vv: Comparison of the latterexpression with that in Table 2, leads to the surprising conclusion that both theMaxwell and modified schemes are each governed by exactly the same genericequations, namely r2T ¼ 0; r . v ¼ 0; and rp ¼ mr2v: Nevertheless, their respectivesolutions will not coincide, as each is to be solved subject to ‘‘different’’ boundaryconditions on qVs; namely Is . ðvv �UÞ ¼ 0 in the modified vv case and Is . ðvm �UÞ ¼

Csvrs ln T in the Maxwell vm case, all other boundary conditions [such as theimpenetrability condition (1.8)] being the same for both. Note that despite thevelocities being different in the two cases, the respective pressure gradients rp willeach be the same owing to the fact that r2vm ¼ r2vv [147].The calculation of some overall force F usually constitutes the main item of

interest in connection with the class of problems under discussion, either the force ona thermophoretic particle [23] or the force on the walls of the capillary tube in thethermal transpiration problem [24], each for the case of a prescribed temperaturedifference at the two ends of the system confining the fluid. The force on animpermeable solid surface can be calculated from the expression F ¼


dS .P;with the pressure tensor given by Eq. (2.12) in conjunction with (1.1). Since, as we

Table 2

Energy, momentum, and continuity equations

Maxwell scheme, v ¼ vm Modified scheme, v ¼ vv

Energy equation r2T ¼ 0 r2T ¼ 0

Momentum equation 0 ¼ �rp þ mr2vm 0 ¼ �rp þ mr2vv

Continuity equation r . vm ¼ 0 r . vv ¼ 0

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have indicated, the calculated pressure p will be the same for both the Maxwell andmodified schemes, one can form the difference, Fv � Fm ¼


dS . ðTv � TmÞ;between the respective forces. In view of Eq. (1.1) we find upon using Eq. (1.5)together with the relation r . jv ¼ 0 that Fv � Fm ¼ 2m


dS .rjv: In conjunc-tion with Eq. (5.9) considered jointly with the constancy of the multiplier of rT ;this yields

Fv � Fm ¼2mk




� �p


dS .rrT :

The surface integral appearing in the latter relation can be converted into a volumeintegral, and the relation r2T ¼ 0 employed in the resulting expression to concludethat Fv ¼ Fm: This shows, at least for the present class of gaseous thermal problems,that the Maxwell scheme does, in fact, furnish the same overall global results as themodified scheme, despite the inconsistency of the set of Maxwell equations basedupon the constitutive relation, v ¼ vm: However, such agreement, rather thanembodying some fundamental principle, is seen to be merely fortuitous, being limitedto a very special set of circumstances. More general circumstances, wherein, forexample, nonlinear terms are retained in the calculations, would almost certainlypoint up a difference between the two. Note also that in this limited class ofproblems, not only will the global forces be the same for both schemes, but so toowill be the respective local pressures p; densities r; temperatures T ; and massvelocities vm: As such, it is perhaps not surprising that the true, no-slip, nature of theMaxwell slip condition has gone undiscovered for so long.

Appendix D. First-law violation occasioned by Maxwell slip in rigid, fluid-filled,

immobile containers

Following up the closing remarks of Section 5, we demonstrate here that theMaxwell slip condition (5.2) imposed upon vm violates the First law ofthermodynamics when used in conjunction with the following (single-component)standard incompressible creeping flow equations used in the literature [8,35] toanalyze phoretic phenomena:

r . vm ¼ 0 ; (D.1)

r . Pm ¼ 0; Pm ¼ �Ip þ Tm ; (D.2)

Tm ¼ m½rvm þ ðrvmÞT : (D.3)

According to the First law, the temporal rate at which the energy E (internal pluskinetic) of a closed system increases is given by the expression dE=dt ¼ _Q þ _W ;where _Q ¼


dS . q is the rate of heat flow into the system and _W is the rate atwhich the surroundings are doing work on the system. Here, qV s denotes the rigid,solid, impermeable, and immobile boundaries of the apparatus confining the fluidwithin its interior.

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Based upon the above constitutive form (D:22) for the pressure tensor P; the rateof working is thus given in present circumstances by the expression

_W ¼


dS .Pm . vm (D.4)

for the case where the boundaries are immobile, so U ¼ 0 in Eqs. (1.7) � (1.3) and(1.8). For definiteness we suppose that the container, within which some transientheat transfer process is occurring, is suddenly insulated at time t ¼ 0; so that _Q ¼ 0for all subsequent times t40: As no mechanism exists by which work can beperformed by the surroundings on the contents of the fluid-filled container V s whoseimmobile boundaries are qV s; this requires that

_W ¼ 0 ð8t40Þ : (D.5)

This condition is consistent with the fact that the total energy E; internal pluskinetic, of the isolated system will necessarily remain fixed for all time follow-ing placement of the insulation on the boundaries. As such, when the tran-sients have decayed the fluid will eventually attain a homogeneous equilibriumstate in which the pressure, temperature and density are each uniform throughoutthe fluid.Under conventional non-Maxwellian no-slip conditions, one would have that

vm ¼ 0 on qVs; whence Eq. (D.5) would be satisfied automatically irrespectiveof the constitutive form of the stress tensor appearing in (D.4). However, in thepresence of initial temperature gradients imposed along qVs; the Maxwell slipcondition (5.2) generally obviates the possibility that vm ¼ 0 on qV s; since theintegral (D.4) will not generally vanish owing to the generally nonzero tangentialmass velocity component, Is . vma0: As a consequence, in contrast with the classicalcase (1.3) where v ¼ vm; the velocity boundary condition alone no longerautomatically assures satisfaction of (D.5). It is possible, however, that the integralmay nevertheless vanish in some integral or other sense. To explore this possibility,we convert (D.4) into a volume integral and use (D.2) in conjunction with (D.1)to obtain

_W ¼


2mrvm:rvm dV : (D.6)

Obviously, the nonnegative nature of the above integrand at each point of the fluidprecludes the possibility of Eq. (D.5) being satisfied. This demonstrates thethermodynamic inconsistency of the set of mass-based velocity equations(D.1)–(D.3) when used in conjunction with the Maxwell slip condition (5.2). Viewedalternatively, insistence upon the applicability of this equation set together with (5.2)would violate the first law of thermodynamics.The escape from this dilemma involves replacing the set of equations (D.1)–(D.3)

by our more physically appropriate set, namely with vm throughout replaced by vv

[see Eqs. (2.13)–(2.14) and (1.1), in which v ¼ vv]. Thus, in place of (D.4), one would

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now write

_W ¼


dS .Pv . vm ; (D.7)


Pv ¼ �Ip þ Tv; Tv ¼ m½rvv þ ðrvvÞT : (D.8)

It is confirmed in what follows that the state of motion of the confined fluid iscorrectly described by the fact that vv ¼ 0 and p ¼ const throughout the fluid andfor all t40: This proposed solution, for which Tv ¼ 0 everywhere, clearly satisfiesEqs. (2.14) and (2.13) as well as the no-slip boundary condition (5.1). Moreover, thecondition that the container be insulated requires that n .rT ¼ 0 on qVs; from whichit follows from Eq. (5.9) that n . jv ¼ 0 on qV s: In turn, from Eq. (1.5) this yieldsn . vv ¼ n . vm; thus assuring that the no-penetration condition (1.8) is satisfied. Inturn, it follows that dS .Pv ¼ 0 at each point on qV s: Satisfaction of Eq. (D.5) is thusassured despite the slip of vm along qV s:To complete the problem, it remains to demonstrate that the proposed solution,

vv ¼ 0; is consistent with the continuity equation (1.2) and the energy equation(2.11). In the course of the demonstration, we use these equations to determine theremaining fields: r; T and vm: Consider first the density field. Substitution of (1.5)and (1.6) together with use of vv ¼ 0 yields

vm ¼ �k

cpr2rr : (D.9)

Introduction of the latter into the continuity equation (1.2) furnishes the followingequation governing the density field rðx; tÞ:




r2 ln r : (D.10)

The insulation boundary condition n .rT ¼ 0 on qVs together with the fluid’sequation of state furnishes the boundary condition n .rr ¼ 0 on qV s imposed uponr: This same condition arises from the impermeability of the boundary, n . vm ¼ 0 onqV s; considered in conjunction with Eq. (D.9). The initial condition rðx; 0Þ imposedupon r is obtained from the fluid’s (assumed pressure-independent) equation of stater ¼ rðTÞ together with knowledge of the initial temperature distribution, Tðx; 0Þwithin the fluid. Solution of Eq. (D.10) subject to these boundary and initialconditions serves to determine rðx; tÞ: In turn, such knowledge immediately furnishesvmðx; tÞ from Eq. (D.9), and Tðx; tÞ from the fluid’s equation of state, therebydemonstrating the validity of the assumed vanishing volume velocity condition, vv ¼

0: As an aside, it should be noted that this latter condition is consistent with thespeculations of Onsager [114] and Haase [15,115], cited in Section 7, albeit inconnection with their isothermal binary diffusion problems rather than our presentsimilarly-structured single-component heat transfer problem.

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Appendix E. Mathematical analogy between electrophoretic and thermophoretic


Upon identifying the pair of electrophoretic fields ðvf; jfÞ; defined below, with theirthermophoretic analogs ðvv; jvÞ; defined earlier in Eqs. (1.5) and (1.6), a completemathematical analogy will be seen to exist between steady-state electrophoretic andthermophoretic particle motions in liquids for the respective nonconducting particlecases, at least in the Debye thin double-layer limit. Explicitly, define the followingelectrophoretic quantities:

vf :¼ vm þ jf where jf :¼ � Mfrf ; (E.1)

in which Mf :¼ �z=m is the electrophoretic mobility. Inasmuch as r2f ¼ 0 outside ofthe Debye layer it follows that

r . jf ¼ 0 and r2jf ¼ 0 : (E.2)

Equations (E.1) and (E.2) represent the analogs of the corresponding thermophoreticquantities,

vv :¼ vm þ jv where jv:¼� MvrT ; (E.3)

in which Mv :¼ ab: Since r2T ¼ 0 in the nonconducting thermophoretic case (seeTable 2) it follows that

r . jv ¼ 0 and r2jv ¼ 0 : (E.4)

Owing to the respective analogs, Eqs. (E.2) and (E.4), one has for the respectivechoices of f or v; both denoted below by the common symbol g; that

r . vm ¼ r . vg ¼ 0 and r2vm ¼ r2vg ¼ 0 : (E.5)

Both the electrophoretic boundary-value problem outlined in Section 8 (with thesubscript f implicitly appearing therein replaced throughout by g) and thenonconducting thermophoretic boundary value problem outlined in Section 3 (withthe subscript v replaced therein by g) satisfy exactly the same set of equations:

rp ¼ mr2vg; r . vg ¼ 0; r . jg ¼ 0; vg ¼ vm þ jg ; (E.6)

and boundary conditions

n . jg ¼ 0 on qVs; vm �U ¼ �jg on qV s; vg ! jð0Þg =Mg as jxj ! 1 ; (E.7)

where jð0Þg ¼ const refers to the undisturbed uniform value at infinity.The two classes of problems are thus seen to be mathematically identical, which is

why both lead to the ‘‘same’’ size-, shape-, and orientation-independent result for thevelocity of the respective phoretic particles, namely

U ¼ jð0Þg � �Mfrf


MvrT ð0Þ;


in the electrophoretic and thermophoretic cases. This analogy obtains despite thevery different physics underlying the sources of the respective particle motions—one

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animated by externally-imposed electrical body forces, the other by an externallyimposed temperature gradient in the absence of body forces! [148].While we have pointed out the analogy only for the elementary case of an isolated

particle in an effectively unbounded fluid, the analogy can be shown to persistirrespective of the presence of other (nonconducting) boundaries as, for example, incases where a plane wall or a circular cylinder bounds the particle-containing fluidexternally. Existing solutions of the electrophoretic problems posed for suchconfigurations [149] thereby automatically provide solutions for their thermophore-tic counterparts.


[1] G.G. Stokes, Trans. Cambridge Philos. Soc. 8 (1845) 287 (see also Mathematical and Physical

Papers, vol. 1, Cambridge University Press, Cambridge, 1901, p. 75).

[2] An account of pre-1845 work by others on the Navier–Stokes equations can be found in G.G.

Stokes, Report on recent researches in hydrodynamics, British Assoc. Advance. Sci., 1846, pp. 1–20.

Reprinted in Mathematical and Physical Papers, vol. 1, Cambridge University Press, Cambridge,

1901, p. 157. A concise history of the conceptual foundations of fluid mechanics from the time of

Newton’s Principia in 1687 up to the definitive work of Stokes in 1845, can be found in the following

articles: C. Truesdell, Am. Math. Monthly 60 (1953) 445;

O. Darrigol, Arch. Hist. Exact Sci. 56 (2002) 95.

[3] Here and throughout, a question mark surmounting an equality sign serves to suggest that the stated

equality is, at this point in the manuscript, an as yet unresolved issue, one which will, however, later

be resolved against the equality!.[4] H. Brenner, Kinematics of volume transport, Physica A (2005), in press [doi:10.1016/


[5] The volume velocity vv is, in fact, identical with the volume flux density or volume current nv [4],

defined such that with dS a directed element of surface area fixed in space, the scalar dS . nv gives the

volume flowing across dS per unit time. This is the analog of the fact that with dS . nm the flux of

mass across dS; in which nm is the mass current, the mass velocity, vm:¼nm=r; represents the

convective portion of the volume flux and jv the diffusive portion.

[6] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, third ed.,

Cambridge University Press, Cambridge, 1970.

[7] D. Burnett, Proc. London Math. Soc. 39 (1935) 385;

D. Burnett, Proc. London Math. Soc. 40 (1936) 382.

[8] J.C. Maxwell, Philos. Trans. Roy. Soc. (London) A 170 (1879) 231. Reprinted in: W.D. Niven (Ed.),

The Scientific Papers of James Clerk Maxwell, vol. 2, Cambridge University Press, Cambridge, 1890,

p. 681.

[9] R.J. Adrian, Ann. Rev. Fluid Mech. 23 (1991) 261;

R.J. Goldstein (Ed.), Fluid Mechanics Measurements, second ed., Taylor & Francis, Washington

DC, 1996;

Th. Dracos, Three-Dimensional Velocity and Vorticity Measuring and Image Analysis Techniques,

Kluwer, Dordrecht, 1996;

F.T.M. Nieuwstadt (Ed.), Flow visualization and image analysis, Fluid Mechanics and its

Application, vol. 14, Kluwer, Dordrecht, 1992;

M. Raffel, C. Willert, J. Kompenhans, Particle Image Velocimetry. A Practical Guide, Springer,

New York, 1998;

M. Stanislas, J. Kompenhans, J. Westerweel (Eds.), Particle Image Velocimetry, Kluwer, Dordrecht,


[10] Presumably, vm can be independently measured experimentally at a point of the fluid by some

colorimetric method, involving the addition of dye to the fluid or, even better, instead of adding a

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foreign coloring agent (and thereby obfuscating the notion of a single-component fluid) by

performing an optical experiment with a single-component fluid whose molecules are photochromic

or fluorescent. These latter techniques involve so-called ‘‘molecular tagging velocimetry’’ (MTV)

[11], as opposed to ‘‘particle-image velocimetry’’ (PIV) [9], which involves monitoring tracer

particles, namely foreign objects deliberately introduced into the fluid.

[11] K.G. Roesner, Mol. Cryst. Liq. Cryst. 298 (1997) 243;

C.P. Gendrich, M.M. Koochesfahani, D.G. Nocera, Exp. Fluids 23 (1997) 361;

W.R. Lempert, in: A.J. Smits, T.T. Lim (Eds.), Flow Visualization: Techniques and Examples,

Imperial College Press, London, 2000;

P. Mavros, Trans. Inst. Chem. Eng. 79 (2001) 113;

S.J. Muller, Korea-Australia Rheol. J. 14 (2002) 93.

[12] I. Muller, T. Ruggeri, Extended Thermodynamics, Springer, New York, 1993;

K. Wilmanski, Thermomechanics of Continua, Springer, Berlin, 1998.

[13] S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962.

[14] D.D. Fitts, Nonequilibrium Thermodynamics, McGraw-Hill, New York, 1962.

[15] R. Haase, Thermodynamics of Irreversible Processes, Dover reprint, New York, 1990.

[16] G.D.C. Kuiken, Thermodynamics of Irreversible Processes: Applications to Diffusion and

Rheology, Wiley, New York, 1994.

[17] L. Onsager, Phys. Rev. 37 (1931) 405;

L. Onsager, Phys. Rev. 38 (1931) 2265;

H.B.G. Casimir, Rev. Mod. Phys. 17 (1945) 343;

I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, second ed., Interscience,

New York, 1961.

[18] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, second ed., Wiley, New York,


[19] With the apparent exception of a few authors (see, for example, Refs. [20]) it has not been clearly

recognized in the literature that a need exists for a formal proof that the specific momentum, say m;of a fluid is equal to its mass velocity vm: Rather, as judged by accounts found in fluid mechanics

textbooks, which implicitly assume it a priori without discussion, the constitutive relation in m ¼ vm

is regarded as an identity.

[20] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed., Butterworth-Heinemann, Oxford, 1987,

p. 196;

P. Kostadt, M. Liu, Phys. Rev. E 58 (1998) 5535.

[21] J.R. Bielenberg, H. Brenner, Continuum thermodynamics in the presence of the diffusive transport

of volume, Contin. Thermo. Mech., 2005, to be submitted.

[22] J.G. Kirkwood, I. Oppenheim, Chemical Thermodynamics, McGraw-Hill, New York, 1961.

[23] H. Brenner, J.R. Bielenberg, A continuum theory of phoretic phenomena: thermophoresis, Physica

A (2004) submitted.

[24] J.R. Bielenberg, H. Brenner, A continuum model of thermal transpiration, J. Fluid Mech., 2004,


[25] J.R. Bielenberg, H. Brenner, A continuum theory of phoretic phenomena: diffusiophoresis, Phys.

Fluids, 2004, submitted.

[26] Indeed, at the hands of D.D. Joseph, his co-workers, and others (see Ref. [127] as well as the

extensive references cited in Ref. [4]), Eq. (2.13) is often used in applications to ‘compressible’ fluids,

at least in the case of isothermal binary diffusion problems, where our single-component

adiabatically additive volume ‘law’ based on (qv=qTÞp is replaced by its better known (cf. [4])

multicomponent species additive volume ‘law’ counterpart based on ðqv=qwiÞp;T ; where wi is the mass

fraction of species i: In the latter context, Eq. (2.13) is referred to as expressing a condition of ‘‘quasi-incompressibility’’ [27] in circumstances where r is not constant throughout the fluid.

[27] J. Lowengrub, L. Truskinovsky, Proc. Roy. Soc. (London) A 454 (1998) 2617.

[28] This elementary equivalence is true only in circumstances where the no-penetration boundary

condition (1.8) imposed upon vm at solid surfaces can be replaced by a comparable condition

imposed upon vv; for in such circumstances Eq. (1.7), in conjunction with the latter condition, then

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leads to the single vector velocity boundary condition, vv ¼ 0 on qVs: This no-penetration

equivalency will obviously obtain in circumstances where n . ðvm � vvÞ ¼ 0 on qV s: Equivalently,from Eq. (1.5) this necessitates that n . jv ¼ 0 on qVs: From (1.6), this latter condition will prevail

whenever n .rr ¼ 0 on qV s or, equivalently, when n .rv ¼ 0 on qVs: In the present single-

component case, and for the case where the law of adiabatically additive volumes prevails, this

requires that n .rT ¼ 0 on qV s and, hence, from Eq. (2.5) that n . q ¼ 0 on qVs: In turn, from

Eq. (2.7) this is equivalent to the condition that n . ju ¼ 0 on qVs; which, because it is also true that inthese same circumstances that n . jv ¼ 0 on qVs; leads to the observation that in such circumstances itis immaterial whether ju is given constitutively by the classic expression (2.4) or by its nontraditional

counterpart (2.6). In summary, the complete vector velocity boundary condition, vv ¼ 0 on qVs; willobtain whenever no diffusive transport of internal energy occurs across the solid-fluid interface,

corresponding to the ’insulation’ boundary condition, n . ju ¼ 0 on qVs: For nonconducting cases,

Eqs. (2.13) and (2.14) together with the boundary conditions (1.7) and (1.8) are indistinguishable

from those governing vm in the classical creeping flow case.

[29] With regard to use of the term ‘‘phoretic’’ forces to describe particle motion in the presence of

gradients, Anderson [30] has inadvertantly sowed some degree confusion owing to his use of terms

like thermophoresis and diffusiophoresis, normally reserved for gases [23–25], to describe

phenomena that are actually driven by surface-gradient forces in liquids [31,32], see also [132].

The latter category is typified by Marangoni forces resulting from interfacial tension gradient rsg;caused by a surface temperature gradient rsT along the particle surface, owing to the functional

dependence of interfacial tension g upon T : The resulting Marangoni surface stress causes the

particle to move against the temperature gradient. However, the forces associated therewith give rise

to a particle velocity U generally dependent upon the size of the particle [32], whereas in non-

Brownian thermophoretic experiments [23] U is observed to be independent of particle size, ruling

out Marangoni forces as possibly responsible for the observed, size-independent, thermophoretic


[30] J.L. Anderson, Ann. Rev. Fluid Mech. 21 (1989) 61.

[31] D.A. Edwards, H. Brenner, D.T. Wasan, Interfacial Transport Processes and Rheology,

Butterworth-Heinemann, Boston, 1991.

[32] E. Ruckenstein, J. Colloid Interf. Sci. 83 (1981) 77;

T. Keyes, J. Stat. Phys. 33 (1983) 287;

V.G. Levich, V.S. Krylov, Ann. Rev. Fluid Mech. 1 (1969) 293.

[33] This occurred when Tyndall observed dust-free regions in proximity to heated surfaces and wires in a

chamber filled with dust-laden air; J. Tyndall, Proc. R. Inst. 6 (1870) 1. (For further historical

references, see Ref. [34]).

[34] F. Zheng, Adv. Colloid Interf. Sci. 97 (2002) 255.

[35] P.S. Epstein, Z. Phys. 54 (1929) 537.

[36] J.R. Brock, J. Colloid Sci. 17 (1962) 768;

G.M. Hidy, J.R. Brock, The Dynamics of Aerocolloidal Systems, Pergamon Press, Oxford,


[37] L. Waldmann, K.H. Schmitt, Thermophoresis and diffusiophoresis of aerosols, in: C.N. Davies

(Ed.), Aerosol Science, Academic Press, London, 1966, p. 137.

[38] B.V. Derjaguin, Yu.I. Yalamov, J. Colloid Sci. 20 (1965) 555;

B.V. Derjaguin, A.I. Storozhilova, Ya.I. Rabinovich, J. Colloid Interf. Sci. 21 (1966) 35;

B.V. Derjaguin, Yu.I. Yalamov, The theory of thermophoresis and diffusiophoresis of aerosol

particles and their experimental testing, in: G.M. Hidy, J.R. Brock (Eds.), Topics in Current Aerosol

Research, Pergamon Press, Oxford, 1972, p. 2;

B.V. Derjaguin, Ya.I. Rabinovich, A.I. Storozhilova, G.I. Shcherbina, J. Colloid Interf. Sci. 57

(1976) 451.

[39] L. Talbot, R.K. Cheng, R.W. Schefer, D.R. Willis, J. Fluid Mech. 101 (1980) 737;

L. Talbot, Thermophoresis—a review, in: S.S. Fisher (Ed.), Rarefied Gas Dynamics, Part 1, AIAA,

New York, 1981, p. 467.

[40] W. Oostra, J.C.M. Marijnissen, B. Scarlett, Space Forum 3 (1998) 251.

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H. Brenner / Physica A 349 (2005) 60–132 125

[41] Knudsen’s work on noncontinuum effects in thermal transpiration flows did not even appear

until 1910 [42], whence it is unlikely that the concept of ‘‘noncontinuum’’ behavior would

have even arisen in Maxwell’s mind in 1879. [Indeed, the fact that Maxwell applied his slip

condition to the strictly continuum N–S–F equations supports our belief that he regarded

his so-called slip condition to be a continuum effect arising from the surface temperature gra-

dient. In this context it is noteworthy that the adherence of the fluid to a solid surface—so

widely accepted today in the case of continua, irrespective of whether or not the surface is

isothermal—would, in the case of nonisothermal continua, not likely to have been regarded as

sacrosanct during Maxwell’s era. After all, very little data pertinent to the issue existed at that

time.] Concomitantly, the standard explanation found in textbooks [43] to the effect that

the thermophoretic particle motion observed in gases is ‘‘molecular’’ (i.e., noncontinuum) in

origin, arising from more energetic particles striking the hotter side of the particle and overcom-

ing the opposing effects of the less energetic particles on the colder side, is untenable in the

continuum limit.

[42] M. Knudsen, Ann. Phys. (Leipzig) 31 (1910) 205;

M. Knudsen, Ann. Phys. (Leipzig) 33 (1910) 1435.

[43] E.H. Kennard, Kinetic Theory of Gases, McGraw-Hill, New York, 1938;

L.B. Loeb, The Kinetic Theory of Gases, Dover reprint, New York, 1961. For a discussion of

Maxwell’s role in explaining Crookes’s radiometer, see Ref. [44].

[44] S.G. Brush, The Kind of Motion that we call Heat, North-Holland, Amsterdam, 1976.

[45] G.S. McNab, A. Meisen, J. Colloid Interf. Sci. 44 (1973) 339.

[46] Others [32] have suggested that ‘‘phoretic motion’’ in liquids may actually be due to Marangoni-like

surface effects [31], wherein the surface is not ‘‘passive’’, as in our model, but rather interacts

physicochemically with the fluid. However, as discussed in Section 8 such particle motion requires

the action of body forces, which are absent as the animating mechanism underlying Eq. (3.1) for

liquids and (3.4) for gases.

[47] S. Semenov, M. Schimpf, Phys. Rev. E 69 (2004) 011201.

[48] J.R. Bielenberg, H. Brenner, A hydrodynamic/Brownian motion model of thermal diffusion in

liquids, Phys. Rev. E (2004), to be submitted.

[49] The absence of bulk viscosity effects in (4.1) derives from the fact that the volume velocity appearing

in Eq. (1.1) is assumed to obey Eq. (2.13), a conclusion consistent with the choice of the constitutive

equation (1.6) and valid, for example, in the case of ideal gases.

[50] To describe these as being ‘‘the’’ noncontinuum terms, without including the first-order ‘‘near-

continuum’’ OðKnÞ N–S–F terms in the appellation, is surely confusing, certainly to fluid

mechanicians who regard the OðKnÞ N–S–F terms, and not the OðKn0Þ Euler terms, as the

equations of continuum fluid mechanics; that is, owing to their apparent Knudsen number

dependence, the latter classical ‘‘near-continuum’’ first-order N–S–F terms should, for consistency,

also be classified as noncontinuum terms, despite their being regarded by fluid mechanicians as

strictly continuum-level terms.

[51] Even higher-order, OðKn3Þ; so-called super-Burnett terms [52] exist. For a contextual evaluation of

the Burnett, super-Burnett, and generally higher-order contributions to the linear momentum

equation, see Ref. [53].

[52] A.V. Bobylev, Sov. Phys. Doklady 27 (1982) 29;

F.J. Uribe, R.M. Velasco, L.S. Garcia-Colin, Phys. Rev. E 62 (2000) 5835;

M. Slemrod, Arch. Rational Mech. Anal. 150 (1999) 1.

[53] R.K. Agarwal, K.Y. Yun, R. Balakrishnan, Phys. Fluids 13 (2001) 3061.

[54] The comparable Burnett terms for the heat flux do not impact upon whether or not Eq. (2.6) is or is

not correct, since gas kinetic theory [6] draws no clear-cut distinction between the heat flux q and the

diffuse internal energy current ju:[55] M.N. Kogan, V.S. Galkin, O.G. Fridlander, Sov. Phys. Usp. 19 (1976) 420;

M.N. Kogan, Ann. Rev. Fluid Mech. 5 (1973) 383;

M.N. Kogan, Progr. Aerospace Sci. 29 (1992) 271.

[56] A.V. Bobylev, J. Stat. Phys. 80 (1995) 1063.

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[57] E. Yariv. H. Brenner, A continuum alternative to the ‘‘ghost effect’’ of gas-kinetic theory, Phys.

Fluids, 2004, submitted.

[58] C. Truesdell, R.G. Muncaster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic

Gas, Academic Press, New York, 1980.

[59] Although not required for the subsequent calculations, as an aside we note that m ¼ ðconst:Þ T [6] for

Maxwell molecules, from which it follows that K2 ¼ 3 for such molecules.

[60] In the latter context, note that Eq. (4.8) is consistent with the fact that k is known [6] to be identicallyzero for monatomic ideal gases owing to the assumed spherically symmetric nature of such


[61] A.D. Kovalenko, Thermoelasticity, Wolters-Noordhoff, Groningen, 1969;

H. Parkus, Thermoelasticity, Springer, Wien, New York, 1976;

D. lesan, A. Scalia, Thermoelastic Deformations, Kluwer, Dordrecht, 1996;

G.A. Maugin, A. Berezovski, J. Thermal Stresses 22 (1999) 421;

N. Noda, R.B. Hetnarski, Y. Tanigawa, Thermal Streses, Taylor and Francis, London, 2002.

[62] Indeed, in the case of solids, the notion of a ‘‘noncontinuum solid’’ does not even appear to exist,

except perhaps in the case of granular materials, although fractures and dislocations, representing

isolated singularities, may exist within the solid.

[63] D.J. Korteweg, Arch. Neerl. Sci. Exactes Naturelles II 6 (1901) 1.

[64] C. Cercignani, Mathematical Methods in Kinetic Theory, second ed., Plenum Press, New York,


[65] H.A. Kramers, J. Kistemaker, Physica 10 (1943) 699.

[66] D.A. Noever, Phys. Fluids A 2 (1990) 858;

D.A. Noever, Phys. Lett. A 144 (1990) 253;

D.A. Noever, Phys. Rev. Lett. 65 (1990) 1587;

D.A. Noever, Phys. Rev. A 45 (1992) 7302.

[67] W. Crookes, Philos. Trans. Roy. Soc. (London) 166 (1876) 325.

[68] A detailed and historical discussion of attempts to explain the principles underlying the windmill-like

rotation undergone by the rotor in Crookes’s radiometer based upon noncontinuum concepts is

given in Ref. [44].

[69] J.C. Maxwell, Philos. Mag. 19 (1860) 19;

J.C. Maxwell, Philos. Mag. 20 (1860) 21.

[70] Some of the historical context, chronology, and acrimony in the matter of priority surrounding the

competition between Maxwell and Osborne Reynolds [71] to use their respective thermal

transpiration models to explain the physical mechanism underlying the working of Crookes’s

radiometer [67] can be found in the biography by I. Tolstoy, James Clerk Maxwell, University of

Chicago Press, Chicago, 1981, pp. 150–151, 166–167; see also Ref. [44].

[71] O. Reynolds, Proc. Roy. Soc. London 38 (1879–1880) 300. This paper is only a preliminary abstract

of the lengthier paper published some time afterwards as O. Reynolds, Philos. Trans. Roy. Soc.

(London) 170 (1879) 727.

[72] The importance of understanding the mechanism behind Crookes’s radiometer [67] played a vital,

and under-appreciated, role in the history of gas-kinetic theory, in particular in regard to the

boundary conditions to be applied to the Boltzmann equation at solid surfaces. After all, an

important part of the verification of the validity of the Boltzmann equation necessarily lies in the

agreement of its predictions with experiment, for which circumstances the solution of boundary-

value problems (either imposed upon the Boltzmann equation itself or upon the coarser-scale

transport equations derived therefrom, such as the N–S–F equations) plays a pre-eminent role.

[73] For a modem version of the slip boundary condition involving G for gases, see F. Sharpov,

D. Kalempa, Phys. Fluids 15 (2003) 1800.

[74] Given the interpretation of Maxwell slip as a noncontinuum OðKn2Þ effect owing to its origin in

connection with the Burnett terms, Epstein [35] and those who followed should, for mathematical

consistency as regards the hierarchical ordering of the Knudsen number terms appearing in their

transport equations, have then solved the corresponding noncontinuum OðKn2Þ-level transport

equations, rather than the near-continuum OðKnÞN–S–F equations. At a minimum, this would have

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resulted in adding the Maxwell thermal stress term (5.3) to the OðKnÞ viscous Newtonian term (4.2)

appearing in the momentum equation. Additionally, because the gas is ‘‘compressible’’ owing to its

density varying with temperature, the continuity equation used by Epstein, namely r . vm ¼ 0; isvalid only to OðKnÞ: At OðKn2Þ another term should have appeared in his continuity equation in

order that the latter be correct. However, as discussed in Appendix C, owing to a fortuitous

combination of circumstances in the present class of phoretic thermal problems [23,24], these

additions do not affect the calculation of U:[75] As discussed in connection with Eqs. (5.4) and (5.5), the notion of noncontinuum slip is associated

with the parameter G appearing therein, rather than with the last term of Eq. (5.4), which alone

governs Maxwell’s ‘‘slip coefficient’’, Cs: In the literature [73], G is associated with the notion of

‘‘velocity slip’’, a truly noncontinuum phenomenon occurring even in isothermal fluids.

[76] L. Euler, Mem. Acad. Sci. Berlin 11 (1755) 274. Reproduced in: Leonhardi Euleri Opera Omnia.

Series II, vol. 12, Fussli, Zurich, 1954, p. 54. Additional historical information can be found in the

‘‘Editor’s Introduction’’ to the latter volume by C. Truesdell, Rational fluid mechanics, 1687–1765,

pp. VII–CXXV; see also L. Euler, Hist. Acad. Berlin 1755 (1757) 316–361.

[77] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge,


[78] By ‘‘small’’ is meant the following: If a is the maximum linear dimension of the particle, it is required

that ajjrvl jj=jvl j51; with the modulus bars denoting appropriate norms.

[79] C.C. Truesdell, R.A. Toupin, The classical field theories, in: S. Flugge (Ed.), Handbuch der Physik,

vol. IIII/1, Principles of Classical Mechanics and Field Theory, Springer, Berlin, 1960, p. 226;

C. Truesdell, W. Noll, The Nonlinear Field Theories of Mechanics, in: S. Flugge (Ed.), Handbuch

der Physik, vol. III/3, Springer, Berlin, 1965;

W. Noll, R.A. Toupin, C.C. Wang, Continuum Theory of Inhomogeneities in Simple Bodies,

Springer, Berlin, 1968.

[80] Of course, in the case of unsteady flows, the necessity of performing repetitive experiments with

different size particles, all at the same instant of time, would, no doubt, pose a daunting challenge to

the experimentalist!.

[81] A perhaps equally remarkable fact about Eq. (3.2), applicable to gases, is that it reveals a totally

counter-intuitive fluid-mechanical phenomenon—namely, the larger the viscosity of the gas the

faster does the particle move! This fact alone signals the extraordinarily unique nature of

thermophoretic motion, since viscosity generally retards rather than enhances relative particle

motion through fluids, a fact well known to every low Reynolds number fluid mechanician [82].

[82] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs,

NJ, 1965.

[83] P. Goldsmith, F.G. May, Diffusiophoresis and thermophoresis in water vapour systems, in: C.N.

Davies (Ed.), Aerosol Science, Academic Press, London, 1966, p. 163 (see also Ref. [38]).

[84] In order for an investigator be able to objectively identify his velocity measurements as

representative of those of the fluid itself, and not an artifact of the properties of the tracer

particle, he needs to assure himself that his experimental tracer particles do not possess any

physical attributes that, in the zero-size limit, would distinguish the particle’s velocity from

that of the fluid itself. It was in order to fulfill this requirement of ‘‘passivity’’ that only

(effectively) thermally insulated thermophoretic spheres were selected by us in order to identify

the velocity vl of the undisturbed fluid. As revealed by Eq. (3.1), thermophoretically

animated spheres possessing a nonzero ks=k ratio move with a velocity that depends

significantly upon the magnitude of this conductivity ratio, even in the limit of effectively

zero size. As such, (effectively) noninsulated particles may not serve as fluid velocity tracers.

It is only to this extent that the experimental fluid mechanician, in deciding upon the choice

of appropriate tracer particles with which to conduct his velocity experiments, would have

to contemplate the possible complicating effects of temperature gradients. Even were

he insufficiently insightful to recognize a priori the need for insulated particles, were he to

next perform a sequence of replicate size-varying experiments using a series of particles

possessing different thermal conductivities (just as he might do with a series of particles of

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different densities, so as to assure himself of their zero-size ‘‘passivity’’), he would presumably

soon come to recognize that all low conductivity particles yielded identical extrapolated zero-size

velocities. Accordingly, he would presumably then reject all zero-size particle data obtained with

his high conductivity particles as failing to fulfill the requirement of ‘‘passivity’’ (even were he

unable to identify thermal conductivity as the source of the observed differences in the zero-size

velocity measurements).

[85] Even were external forces such as gravity to act on the fluid, enabling the particle to sediment

relative to the surrounding fluid if its density differed from that of the fluid, such relative motion

would vanish in the pointsize tracer-particle limit, thereby having no effect upon the ability of the

tracer particle to monitor the fluid velocity that exists in its absence.

[86] By the phrase ‘‘gas-kinetic molecular interpretation’’ is meant that the property cannot be derived

directly simply by summing each of the three elemental extensive properties of the individual

molecules in some small domain of volume V (namely the mass m; kinetic energy mc2=2; andmomentum mc of the molecules, with c the molecular velocity) and subsequently dividing by the

volume of that domain in order to obtain the corresponding intensive volumetric pointwise mass,

kinetic energy, and momentum densities at a point of the continuum.

[87] The reason for separating these two items stems from the fact (noted in connection with Table 2

appearing in Appendix C) that it is possible under certain well-defined circumstances for the

traditional and modified N–S–F equation set to fortuitously yield identical results, both of which

accord with experiment, albeit on the proviso that the correct velocity boundary condition be used

(either that of no-slip imposed upon vl or the equivalent Maxwell slip condition imposed upon vm).

[88] The isothermal assumption is needed in order to avoid complications associated with thermal

diffusion species fluxes, while the isobaric assumption is similarly required to avoid pressure

diffusion contributions to the species flux density ji [18].

[89] This has the effect of enabling the right-hand side of (7.1) [and, equivalently, that of Eq. (1.6) for the

isothermal, isobaric, binary diffusion case] to be re-written in the form jv ¼ Dr ln r ¼ �D�rv ¼

�D�ðqv=qw1Þp;Trw1 ¼ ðqv=qw1Þp;T j1:[90] We use the word ‘‘semi-empirical’’ here because there does not appear to exist in the literature a

theoretical proof of the concentration-slip boundary condition, derived along the lines laid out by

Maxwell [8] in the thermal gradient case, wherein the concentration analog of Eq. (5.4) is derived

from the analog of the Maxwell–Burnett thermal stress term (5.3). Rather, owing to this lack,

Kramers and Kistemaker [65] adopted their widely-used concentration-slip velocity condition on a

different basis, namely a molar rather than mass basis. Explicitly, we are not aware of the existence

in the literature of the Burnett extra stress concentration analog of Eq. (4.5), although if our theory

is correct it should be given by Eq. (4.6), in which jv ¼ Dr ln r (see Ref. [89]). According to our

theory, the generic no-slip boundary condition should be given by Eq. (5.8), where Is . jv ¼ Drs ln rin the present binary mixture case.

[91] Reynolds’ [71] experiments were actually performed with porous plugs rather than with well-defined

capillary tubes.

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June 2003;

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[93] In regard to these experiments, note that according to Eq. (3.3) the slip coefficient is different for

monatomic and diatomic gases.

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Rev. E (under revision; originally submitted under the title ‘‘Molecular, Brownian motion

confirmation of the Euler/Lagrange velocity disparity’’).

[95] E. Nelson, Phys. Rev. 150 (1966) 1079;

E. Nelson, Dynamical Theories of Brownian Motion, Princeton Univ. Press, Princeton,


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E. Nelson, Connection between Brownian motion and quantum mechanics, in: H. Nelkowski, A.

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[96] K. Ito, Stochastic Calculus, Springer Lecture Notes in Physics, vol. 39, Springer, New York, 1975,

p. 218.

[97] In probabilistic terminology, vl constitutes Nelson’s ‘‘drift velocity’’ (more precisely, his ‘‘forward’’

drift velocity, since a hypothetical mathematically-defined ‘‘backwards’’ drift velocity also appears in

Nelson’s theory, a fact that need not concern us here). In his notation, Nelson’s symbol v [no relation

to our v in either Eqs. (1.3) and (1.4)] is equivalent to our mass velocity vm; as is apparent from its

appearance in Nelson’s continuity equation, analogous to our Eq. (1.2).

[98] In our interpretation of Nelson’s work, D might better be termed the fluid’s ‘‘self-diffusion’’

coefficient, since his analysis appears limited to single-component fluids within which

mass density gradients exist; that is, the symbol D appearing in (7.3) is regarded as being a

‘‘self-diffusion’’ coefficient, intrinsic to the single-component fluid itself, rather than arising from the

presence a foreign object, namely a colloidal particle, present in the fluid. In this sense, D should be

regarded as an isotropic correlation coefficient, ID ¼ ð1=2ÞhDxDxi=Dt; in which the position vector

x ¼ xðx0; tÞ represents the statistical location at time t of a ‘‘fluid particle’’ that at time t ¼ 0 was

located at the position x0: The phrase ‘‘fluid particle’’ here refers not to a ‘‘material particle’’ (which isan extensive entity) but rather to a fluid particle (an intensive entity) in the sense implicitly understood

in connection with Eq. (1.9), where the tracer fluid field, vl ðx0; tÞ; is regarding as describing the (mean)tracer motion, the so-called ‘‘forward motion,’’ of such a hypothetical fluid particle. Mathematically,

the symbol D is that appearing in the Markoff process stochastic relation [99] dxðtÞ ¼ vl ½xðtÞ; tdt þffiffiffiffiffiffiffi2D

pdwðtÞ; in which dxðtÞ � x� x0; with dwðtÞ a normalized Wiener process [100].

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Brownian recoil-like effect.

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[109] The essentially kinematic analyses of Einstein and Smoluchowski preceded the later dynamical

theories of Brownian motion phenomena, such as those due to Langevin, Ornstein-Uhlenbeck,

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[111] Youri L’vovich Klimontovich, late Professor Emeritus in the Physics Department at Moscow State

University, died of cancer on November 27, 2002, approximately three months before I learned of

his contribution to the subject under discussion.

[112] Yu.L. Klimontovich, Statistical Theory of Open Systems, vol. 1: A Unified Approach to Kinetic

Descriptions of Processes in Active Systems, Kluwer, Dordrecht, 1995 (Chapters 13 and 14);

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Yu.L. Klimontovich, Theor. Math. Phys. 96 (1993) 1035.

[113] It is interesting to note that an identical term appears in the well-known book of de Groot

and Mazur [13], but only in the context of a class of applications involving what they

term ‘‘discontinuous’’ systems [cf. Eqs. (69) and (72) of their Chapter XV]. Indeed, they

explicitly identify the term jv appearing in our subsequent Eq. (7.4), which they term the

‘‘volume flow’’.

[114] L. Onsager, Ann. Trans. NY Acad. Sci. 46 (1945) 241. It is interesting to note in the present context

that Onsager comments as follows when discussing problems of ‘‘pure’’ multicomponent diffusion in

liquids, involving what appears to us to be thermodynamically ideal solutions: ‘‘..provided only that

the volume change due to mixing may be neglected, it is possible to arrange matters such that v ¼ 0

everywhere’’ (where Onsager’s ‘‘hydrodynamic’’ velocity, v; is understood by us to be the volume

velocity). Moreover, he goes on later to further state that: ‘‘Viscous flow is a relative motion of

adjacent portions of a liquid. Diffusion is a relative motion of its different constituents. Strictly

speaking, the two are inseparable; for the ‘‘hydrodynamic’’ velocity in a diffusing mixture is merely an

average determined by some arbitrary convention’’.

[115] As pointed out by Haase [15, p. 221] and others [116], experimentalists who measure molecular

diffusivities usually choose a volume- rather than mass-based reference frame (so as to avoid having

to explicitly address what Haase terms ‘‘convective velocities’’, namely nonzero mass-average

velocities, vmÞ: In this frame of reference it is supposed: (i) that the volume velocity vanishes

everywhere, vv ¼ 0 (corresponding here to vl ¼ 0), despite the fact that vma0; and (ii) that the

diffusional process is unidirectional, However, to the best of our knowledge, it appears never to have

been pointed in this connection that the assumption of requiring that vv ¼ 0 everywhere, including on

the boundary, is incompatible with the traditional no-slip tangential boundary condition, Is . ðvm �

UÞ ¼ 0; imposed upon vm [117]. Equally, the Kramers–Kistemaker [65] species concentration

boundary condition, analogous to (5.2), would also be violated unless it was true that, when

expressed in appropriate binary diffusion terminology, vv ¼ vm � Csurs ln T ; for, in that case,

Eq. (3.5) becomes identical with Eq. (5.1), corresponding to no slip of the volume velocity (and hence

of the Lagrangian velocity vl ).

[116] E.L. Cussler, Diffusion, second ed., Cambridge University Press, Cambridge, 1997.

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[118] J.C. Maxwell, Proc. Lond. Math. Soc. 3 (1870) 82. Reprinted in: W.D. Niven (Ed.), The Scientific

Papers of James Clerk Maxwell, vol. 2, Cambridge University Press, Cambridge, 1890, p. 208. This

paper discusses Maxwell’s view of the relation between the Eulerian and Lagrangian velocities of a

fluid. While it might appear from Maxwell’s remarks that he is literally referring to a ‘‘molecule’’, it

is clear from his Lagrangian pathline example that his paper addresses, as well as from his later use

of the word ‘‘molecule’’, that he is actually referring to a ‘‘particle’’ of fluid, what today would be

referred to as a ‘‘material fluid particle’’.

[119] R.F. Streater, J. Math. Phys. 38 (1997) 4570;

R.F. Streater, Rep. Math. Phys. 40 (1997) 557;

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[124] R. Balescu, Statistical Dynamics, Imperial College Press, London, 1997.

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[129] In the case of gases, kinetic theory [18] suggests that a better assumption would be Dv ¼ D�v=r; where

D�v is a constant, independent of density. This would result in different expressions for the four

Korteweg coefficients than those given in Eq. (8.2).

[130] In the interest of greater generality we could have added a body force per unit volume, say f; to the

right-hand side of (8.3). However, we have refrained from doing so in order to clarify subsequent

arguments regarding other classes of phoretic particle motion [30], which, in contrast with the thrust

of our work involving circumstances where f ¼ 0; are driven by nonzero body forces.

[131] It is important to distinguish between the phenomenon of thermophoresis, which is a single-particle

theory applicable to non-Brownian particles, and that of thermal diffusion, which involves multiple

Brownian particles in a fluid [48], collectively forming a second species, so that the solute and solvent

together constitute a nonisothermal binary mixture We bring this up because the Semenov–Schimpf

interpretation of experimental data does not appear to strictly constitute a test of single-particle


[132] V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, New Jersey, 1962.

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D.A. Drew, S.L. Passman, Theory of Multicomponent Fluids, Springer, New York, 1998;

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[146] In a generic context, the diffuse current jc; of some extensive propertyC is defined as the flux density

of the property over and above the corresponding convective contribution nmc thereto carried by the

mass current nm ¼ rvm: Stated more explicitly, the total current nc of the extensive property under

discussion in a Eulerian space-fixed reference frame is regarded as being of the form nc ¼ nmcþ jc;with c is the amount of the property per unit mass, i.e., the specific density of the property C: Thelatter density appears in the generic Eulerian transport equation qc=qt þ r . nc ¼ pc in which c ¼

rc and pc are, respectively, the amount of the property and temporal rate of production of the

property, both on a per unit volume basis. This generic Eulerian transport is formally equivalent to

the generic material derivative form, rDmc=Dt þr . jc ¼ pc:[147] The latter fact shows, for example, why the thermomolecular pressure difference in the thermal

transpiration problem [24] is correctly given by Maxwell’s scheme despite the fact that Maxwell’s

transport equations are inappropriate.

[148] Note that in terms of the fundamental ‘‘Newtonian’’ stress issue (1.1), the ‘‘extra’’ deviatoric stress

[cf. (4.2)], Tþg ¼ 2mrjg; makes no contribution to the present problems, just as was true in Appendix

C, owing to the fact that since r2jg ¼ 0; it follows that r . Tþg ¼ 0:

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