The Behavior of a Vapor-Gas Mixture in theContinuum Limit: Asymptotic Analysis

Based on the Boltzmann Equation

Kazuo Aoki

Department of Aeronautics and Astronautics, Graduate School of Engineering,Kyoto University, Kyoto 606-8501, Japan

Abstract. A binary mixture of a vapor and a noncondensable gas in contact with the condensed phase ofthe vapor, on the surface of which evaporation or condensation of the vapor can take place, is considered.The steady behavior of the mixture in the continuum limit with respect to the vapor is investigated onthe basis of kinetic theory by means of an asymptotic analysis for small Knudsen numbers. First, the casewhere the mixture is confined in the gap between two parallel plane condensed phases, one of which maybe moving in its surface, is considered (two-surface problems), and it is shown that there are two differenttypes of continuum limit depending on the amount of the noncondensable gas, i.e., (a) a local equilibriumstate of the mixture in which no evaporation or condensation takes place but the temperature, numberdensities, and flow velocity along the condensed phases are still affected by the vanishing evaporation andcondensation (ghost effect); (b) a uniform flow of the vapor with the noncondensable gas being confined inthe infinitely thin Knudsen layer on the condensing surface. Then, the mixture around arbitrarily shapedcondensed phases at rest is considered, and the fluid-dynamic type system describing the continuum limitcorresponding to type (a) mentioned above is derived. The cause of the ghost effect in this general case isclarified with the help of the explicit form of the fluid-dynamic type system.

INTRODUCTION

Vapor flows with evaporation or condensation on the boundary have been one of the important subjects inrarefied gas dynamics. For single-component systems consisting of a pure vapor and its condensed phase, manysuccessful results have been obtained. For example, a new type of gas dynamics (i.e., fluid-dynamic equationsand their boundary conditions) describing the vapor flows around arbitrarily shaped condensed phases in thecontinuum limit (the limit where the Knudsen number tends to zero) has been established, together withits higher-order corrections in the Knudsen number, by means of a systematic asymptotic analysis of theBoltzmann equation for small Knudsen numbers [1-5].

In practical situations, however, evaporation and condensation often take place in the presence of other gasesthat do not participate in evaporation or condensation (noncondensable gases). Such two- or multi-componentsystems (vapor-gas mixture) have also been investigated (e.g., [6-9]). But, because of the complexity, the levelof understanding is still unsatisfactory. For instance, the behavior of the mixture in the continuum limit hasnot been clarified yet.

In the mean time, a study based on kinetic theory revealed a serious defect contained in the classical fluiddynamics [10]. That is, contrary to common belief, the Navier-Stokes system fails to describe the behavior ofa gas (a simple gas around solid boundaries without evaporation or condensation) even in the continuum limitin some important situations. This is due to the fact that the gas flows of the order of the Knudsen number,which therefore vanish in the continuum limit, still give a finite effect on the behavior in this limit (ghost effect)(see [11-13]). The effect is expected to manifest itself in a wider class of problems for the vapor-gas mixture.

For the reasons mentioned above, it is an important issue to clarify the behavior of a vapor-gas mixture in thecontinuum limit on the basis of kinetic theory. In this paper, we summarize recent results [14-17] relevant tothis subject obtained by a systematic asymptotic analysis and a numerical analysis of the Boltzmann equation.

CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00

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We consider a binary mixture of a vapor (^4-component) and a noncondensable gas (J5-component), both ofwhich are composed of hard-sphere molecules, in contact with the boundary consisting of the condensed phaseof the vapor, and we investigate the steady behavior of the mixture in the continuum limit with respect to thevapor on the basis of the Boltzmann equation (the results based on model Boltzmann equations will also beused in the following discussions). The conventional complete condensation condition is assumed for the vaporand the diffuse reflection for the noncondensable gas on the boundary. In the former, the vapor moleculesare assumed to be emitted according to the corresponding part of the Maxwellian distribution characterizedby the temperature and velocity of the boundary and by the saturation density of the vapor at the boundarytemperature.

The main notation used in this paper is summarized here: raa and d^ are the mass and diameter of amolecule of a-component (a = A, B', hereafter, the letter a is used to represent the labels A and B of thecomponents), L is the reference length, TO is the reference temperature, no is the reference molecular numberdensity (a representative number density of the vapor), pQ = fcn0T0 (k: the Boltzmann constant) is the referencepressure, and nB

v is the average number density of B-component in the domain; IQ is the mean free path ofthe molecules of A-component in the equilibrium state at rest with number density no and temperature TO[/o = l/V27r(d£)2no], and Kn= 10/L is the Knudsen number; Xi = Lxi is the space coordinate system,& = (2fcT0/mA)1/2Ci is the molecular velocity, and Fa = n0(2kTQ/mA)-3/2Fa is the velocity distributionfunction of a-component; na — n0na, pa = mAnopa, vf = (2kTQ/mA)l/2vf, Ta = T0Ta, and pa = popa

are the molecular number density, mass density, flow velocity, temperature, and pressure of a-component,respectively; n = n0n, p = mAn0p, vt = (2kTo/mA)l/2Vi, T = T0T, and p = Pop are the molecular numberdensity, mass density, flow velocity, temperature, and pressure of the mixture, respectively. The dimensionlessvariables (xi, C*> and the quantities with a hat) are mainly used in the following discussions.

TWO-SURFACE PROBLEMS

We first investigate the two-surface problem of evaporation and condensation. That is, we consider themixture in the domain 0 < X\ < L (or 0 < x\ < 1) between two parallel plates of the condensed phase. Letus suppose that the surface at X\ = 0 (or x\ — 0) is kept at temperature T/ and is set at rest, whereas thatat Xi = L (or x\ = 1) is kept at temperature T// and may be moving in its surface in the x2 direction with aconstant speed C///. We denote by n/ and n// the saturation number density of the vapor (A-component) attemperature T/ and that at temperature T//, respectively. Here, we take T/ and n/ as our reference quantities,i.e., TO = T/, n0 = n/, and p0 = fcn/T/. This problem is characterized by the following seven parameters:mB/mA, d%/dA, T/j/T/, n///nj, UII/(2kTI/mA)1/2, Kn, and nB

v/nx. The parameter nBv/m (= finBdxi)

controls the amount of the noncondensable gas contained in the domain. Our interest is to clarify the behaviorof the mixture in the continuum limit with respect to the vapor (Kn— 0+).

For this purpose, we carry out a systematic asymptotic analysis of the Boltzmann system for small Kn,following [18,19,3,5,10] as a guideline. First, we seek a moderately varying solution F% (Hilbert solution) inthe form of a power series of Kn: Fg=FgQ-\-FglKn+- - •. Let h be any macroscopic quantity (h=ha, n, pa,p, vf , Vi, Ta, T, pa, or p) and HH be the same quantity corresponding to Fg. Then hH is also expanded ashH=hHo+hHiKn+' • -. By substituting the expansion of F% into the Boltzmann equation, one finds that theleading-order terms Fg0 are the local equilibrium distributions, i.e.,

( -m°[(Ci - vino)2 + (C2 - v2m)2 + ̂ }/fHO ), (1)

where mA = 1 and mB = mB/mA. It should be noted that i)Am = v?HQ = viHQ and f$0 = fjQ = fm

for these distributions. On the other hand, the particle conservation and the noncondensable condition forthe 5-component lead to nBvB = 0, therefore, nf QvB

HO = 0. If nfv/n/ = O(l), then nB 0 does not vanish

identically, and therefore we have v£HO = VBHO = VIHO = 0, that is, the flow velocities normal to the condensed

phases vanish at the leading order; if nfv/n/ < O(Kn), then we have ng0 = 0. We will discuss these two casesseparately. In the latter, we restrict ourselves to the case nB

v/nj = O(Kn).

566

Continuum limit when n^v/71*

By continuing the usual Hilbert procedure with the result VAHQ = v^HO — VIHO = 0, we obtain the equations

of fluid-dynamic type for n#0, n^0, THO, and v2HQ contained in Fg0. Furthermore, thanks to VIHO = 0> theFffQ can be made to satisfy the kinetic boundary conditions (complete condensation for the vapor and diffusereflection for the noncondensable gas) at the order of Kn° by the suitable choice of the boundary values ofn#0, n§0, THO» and V2HO- This choice gives the boundary condition for the fluid-dynamic type equations. Tosummarize, hA, nB , T1, and vi (#3 — 0) in the continuum limit Kn= 0+ (i.e., n#0, n#0, THO, and VIHO) aregoverned by the following equations and boundary conditions [16]:

«t(=tf = «f) = 0, = 0, <ftV)=0, (2)

DAB 2 dxi dXl

(5)

p = pA+pB, pA = f i A f , pB = nBf, (6)

and

hA = l, f = 1, v2 = 0, at xi = 0, (7)nA = nj//n/, f - T///T/, 02 - UII/(2kTI/mA)1/2, at xx - 1- (8)

Here, </>A is an auxiliary unknown function, the physical meaning of which will be clarified below, and /}, A,DAB, and D^B are given functions of the local concentration of A-component X A (or that of jB-componentXB) (Xa = na/h = pa/p with h = nA + nB; XA + XB = 1), depending on mB /mA and d^/dj*, andare related to the conventional transport coefficients, i.e., (^yiT/2)(2kT/mA)l^2l(mAn(l/21 knX, DAB, D^B)are the coefficients of viscosity, thermal conductivity, mutual diffusion, and thermal diffusion, respectively,where T = T/T, n = n/n, and / = l/V^7r(G^)2n (see [16,17] for the details). However, since the explicitfunctional forms of these functions are not known, we have to use approximate formulae [20] or numericalresults in order to make the system (2)-(8) solvable. We should mention here that a database which providesaccurate numerical values of these functions immediately for any XA was constructed recently [21]. Althoughthe auxiliary function (f>A can be eliminated from Eqs. (3)-(6), we retain it for convenience in the followingdiscussions. It is to be noted that the boundary conditions (7) and (8) are valid for more general kineticboundary conditions [16].

The first equation of Eq. (2) indicates that evaporation or condensation does not take place in the continuumlimit. Therefore, in this limit, the problem appears to be identical with the ordinary plane Couette flow withheat transfer between two (non evaporating or non condensing) plates (i.e., the case where both of A— and^-component are noncondensable). However, it is not true. In fact, the solution of the latter problem in thecontinuum limit is given by Eqs. (2)-(6) with (j)A = 0 and boundary conditions (7) and (8) with the conditionfor nA being discarded. (In this case, we have to take an appropriate reference number density no, such as theaverage number density nA

v of A-component.) Hence, the presence of (f)A in Eqs. (2)-(6) makes the behaviorin the present problem different from that of the ordinary plane Couette flow with heat transfer. The physicalmeaning of (f)A is not clear if one considers only the continuum limit. In the framework of the asymptoticanalysis for small Kn, it is identified as (f)A = (2/^)vA

Hl or (pA = lim (2/^)(vA/Kn), which is of 0(1)Kn — >0

(recall that the vapor flow velocity normal to the boundary VAH is expanded as VA

H = vAHlKn + • • •; in the

second expression of <f>A, the correction to vAHl in the Knudsen layers, which degenerate on the surfaces of the

condensed phase in the limit Kn= 0+, is not taken into account). Therefore, in the continuum limit, althoughVA (the flow of the vapor, normal to the boundary, with evaporation and condensation) itself vanishes, its effectremains finite in the form of <j>A. This is one of the examples of the ghost effect found and discussed in [10-13].It is obvious from Eq. (5) that the cause of the ghost effect in the present problem is the mutual diffusion andthermal diffusion [the terms containing DAB and t>T

AB, respectively, in Eq. (5)].

567

-10

-20 -

\ \ \\ \ \0.2 0.5 l

n*Jn, = 0.1

0.5

(a)Xi/L

FIGURE 1. The behavior in the continuum limit for n%v/m = O(l) in the case Un = 0. (a) The result obtained byEqs. (2)-(8) for mB /mA = 2, d^/d^ = 1, T///T/ = 1.1, and n///n/ = 4. (b) Comparison with the DSMC result forsmall Kn for mB /mA = 5, d^/d^ = 2, T///T/ = 1.3, n///n/ = 5, and nfv/nj = 0.5. In (a) and (b), the solid lineindicates the result by Eqs. (2)-(8), and dotted line the result for the ordinary heat transfer (HT).

Now, we show some examples of the behavior in the continuum limit. We restrict ourselves to the case whereboth plates are at rest (£/// = 0) but their temperatures are different (Tj ^ T//) and focus our attention tothe ghost effect on the temperature field [15].

Figure l(a) shows the profiles of the number densities nA (vapor) and nB (noncondensable gas), the temper-ature T (total mixture), and the auxiliary function <f)A in the continuum limit, obtained from Eqs. (2)-(8), forvarious values of n^v/nj in the case mB /mA = 2, d^/d^ = 1, T///T/ = 1.1, and n///n/ = 4. The dotted lineindicates the corresponding result for the ordinary heat-transfer problem (between ordinary plates), in whichthe parameter n///n/ is discarded, and n/ is a reference number density determined in such a way that nA

v/njis the same as that in the original problem. In spite of the fact that there is no evaporation or condensation inthe present problem, the temperature profile is quite different from that of the ordinary heat- transfer problem.As n^v decreases, J5-component tends to concentrate near the cold wall, and the gradients of nA and nB

become steeper there. Correspondingly, the magnitude of (f>A, which is a measure of the ghost effect and isessentially controlled by the gradient of nA [see Eq. (5)], increases, and thus the deviation of the profile of T

568

from that in the ordinary heat-transfer problem becomes more significant. When n^v is small (n^v/nj = 0.1and 0.2), there appears the region with a negligibly small amount of B-component, where A-component isalmost in a uniform saturated equilibrium state at rest with temperature T// and number density n//. In theordinary heat-transfer problem, on the other hand, the temperature is almost independent of nfv. It shouldbe noted that the temperature profile in Fig. l(a) is not monotonic [see also Fig. l(b)]. This is caused bythe effect of thermal diffusion [15]. On the other hand, Fig. l(b) shows the numerical solution of the originalboundary- value problem of the Boltzmann equation for small Kn obtained by means of Bird's DSMC method[22] for mB/mA = 5, d*/d£ = 2, T///T/ = 1.3, n///n/ = 5, and nfw/n/ = 0.5. Instead of (f>A, the vapor-flowvelocity vf is shown in Fig. l(b). The figure shows that, as Kn — > 0, the vapor flow tends to vanish, and theprofiles of the number densities and of the temperature approach those obtained from Eqs. (2)-(8).

As is evident from the asymptotic analysis, T [= 0(1)] is affected by v\ of O(Kn). Therefore, as Kn decreases,it becomes increasingly difficult to obtain the correct temperature T by the DSMC method, since v\ becomessmall in proportion to Kn. In fact, the result for Kn= 0.02 in Fig. l(b) was obtained by using 2000 (uniform)cells in #1 and 105 simulation particles for the quantity n/L for each component.

Continuum limit when n^/n/ = O(Kn)

Let us put n^v/nj = AKn, where A is a given constant, and investigate the behavior in the continuum limitwith respect to the vapor (Kn= 0+). It should be noted that n^v/nj vanishes in this limit. Since njjQ = 0or FjJQ = 0 in this case, we only need to deal with F^0. If we proceed following the Hilbert procedure, wecan easily show that n#0, #mo> ^2HQ, and THQ in F^0 are all constant, and furthermore FA

rn=FBrn— 0 for

m > 1. At this point, one might conclude that the present case is trivial because no noncondensable gas iscontained in the vapor. However, one should notice that, unlike the previous case, F^0 with VIHQ / 0 cannotbe made to satisfy the boundary condition on the condensed phases. In order to obtain the solution satisfyingthe boundary condition, we have to seek the solution in the form Fa=F§Q-\-F^Q (FHQ = 0), where F£0 are thecorrection terms appreciable only in the thin layers (Knudsen layers) with thickness of the order of the meanfree path [or 0(Kn) in xi] adjacent to the condensed phases. In other words, the noncondensable gas can bepresent inside the Knudsen layers. However, it cannot stay in the Knudsen layer at the evaporating surface[23], i.e., FKQ = 0 there, and can be present only in the Knudsen layer adjacent to the condensing surface. Thesituation that n^0 (i.e., nB corresponding to FB

0) is of O(l) in the Knudsen layer with thickness of O(Kn)(in #1) is consistent with the condition nfv/n/ = 0(Kn).

The analysis of these Knudsen layers are essentially the same as that of the evaporating or condensing flowin a half space (the so-called half-space problems). That is, F^0 for the evaporating surface is given by thesolution for the steady flow of a pure vapor evaporating from a plane condensed phase [24-26], while F^0 andFKO f°r the condensing surface are given by the solution for the steady flow of a vapor condensing onto aplane condensed phase in the presence of the noncondensable gas [27,28]. The conditions on the condensedphases that determine the constants ?V#0, VIHQ, ^2HO> and THQ are obtained together with the solution for theKnudsen-layer corrections. We summarize the conditions below.

Let us suppose that evaporation is taking place on the surface at X\ = L, and condensation on the surfaceat Xi — 0, namely, T/ < T//, n/ < n//, and VIHQ < 0- Then, the boundary conditions for n"4, T, and vi(1)3 — 0) in the continuum limit (i.e., n^0, THQ, and Vino) are given as follows.

On the evaporating surface (at xi = 1):

t>2 - (2kTI/mArl/2Un, nA = (n///n/)[MMi)/MMi)], f = (T///T/)/i2(M1), (9)f-1/2, Mi<l , (10)

where MI is the Mach number based on the normal component of the flow velocity, and hi and h^ are givenfunctions of MI. Accurate numerical values of these functions were obtained by using the BGK model [29-31]by Sone and Sugimoto [24].

On the condensing surface (at x\ = 0):

n^f =Fs(Mi,M2,f,r), ( f o r M i < l ) , nAf > Fs(l,M2,f,r), (for MI - 1), (lla)uAT> F6(Mi,M2,f,r), (for Ml > 1), (lib)

569

M2 = (12)

where MI is the same as Eq. (10), M% is the Mach number based on the tangential component of the flowvelocity, Fs and F^ are given functions of four variables, and Fn/ is a quantity of the order of the averagenumber density of ^-component in the Knudsen layer; in Eq. (12), e — 0 for hard-sphere molecules and e = 1for the model Boltzmann equation proposed by Garzo et al. [32]. The functions Fs and Ft, were constructednumerically in [27] on the basis of the model of [32] for the case where the molecule of the vapor and that of thenoncondensable gas are mechanically identical. (In this case, the problem is essentially decomposed into twoproblems: one is for the total mixture, which is equivalent to the half-space problem for a pure vapor [33-35],and the other is for the noncondensable gas. In [27,28], this property being fully exploited, the F-dependenceof Fs and Fb is obtained explicitly.) In [27,28], however, only the case of M% = 0 was considered. The analysisof [27,28] for Fs was extended to the case of general M% by T. Taguchi recently (private communication), andit was shown that the dependence of Fs on M<2 is rather weak, as in the pure vapor case (F = 0) [35] . It shouldalso be mentioned that some result of Fs(Mi,0,T, F) for hard-sphere molecules was obtained recently by theDSMC method in the case where the molecules of the two components are not identical [36].

Since hA, T, v\ (or MI), and t)2 (or M2) are all constant, they are determined by Eqs. (9)-(lla) and (12)(note that MI < 1). Some examples of the numerical result, based on the functions hi, h^ for the BGK modeland Fs for the model of [32], for the case where both of the condensed phases are at rest (i.e., t/// = 0) are givenin Table 1 [14]. On the other hand, Fig. 2 shows the result by the DSMC method for hard-sphere moleculesfor small Kn (i.e., the numerical solution of the original Boltzmann system) in the case where T///T/ = 1,nii/ni — 2, UH = 0, and the molecules of both components are the same (mA = mB and d^ = d% forhard-sphere molecules): the result for n^v/nj = Kn (A = 1) and that for n^v/nj — IKn (A = 2) are shownin Figs. 2(a) and 2(b), respectively, and that for a pure vapor (n^v = 0 or A = 0) is shown in Fig. 2(c). Thecorresponding result of Table 1 for the continuum limit (VA = vi in this limit) is also shown by the dotted linein the figure. As shown by Figs. 2(a) and 2(b), the noncondensable gas, which is distributed over the wholeregion at Kn = 0.1, is confined near the condensing surface at Kn = 0.01, and except for this region and for thevicinity of the evaporating surface, the flow field is uniform. At Kn = 0.005, the nonuniform regions shrink,but the uniform flow of the vapor does not change. Such behavior agrees with the result of the asymptoticanalysis described above.

In general, the way of comparison of the results based on different molecular models is not unique. If we sup-pose that the mutual diffusion coefficient is a basic and common quantity, we have the relation (/o) model =7 Go) hs

TABLE 1. The constants nA, T, vi, and Mi for given values of the parameters T/j/Tj,nn /m, and A (Un = 0) when the molecule of the noncondensable gas is mechanically thesame as that of the vapor. The values in the parentheses are those obtained by the use of theconversion of A explained in the main text.

Turniiiiiiii1.11.11.11.11.11.11.11.1

nn/m1.21.21.21.22222555510101010

A0

0.5120

0.5120

0.5120

0.512

nA

1.1181.128 (1.131)1.137 (1.142)1.149 (1.154)1.5431.606 (1.622)1.655 (1.679)1.720 (1.747)2.7813.158 (3.245)3.411 (3.533)3.743 (3.881)4.6145.686 (5.913)6.336 (6.632)7.141 (7.470)

f0.9810.984 (0.984)0.985 (0.986)0.987 (0.988)0.9300.941 (0.944)0.949 (0.953)0.960 (0.964)0.9190.960 (0.969)0.985 (0.996)1.014 (1.024)0.8540.926 (0.939)0.961 (0.976)0.999 (1.013)

-Vi0.04230.0367 (0.0351)0.0321 (0.0298)0.0259 (0.0233)0.15640.1318 (0.1257)0.1136 (0.1048)0.0902 (0.0810)0.37890.2942 (0.2761)0.2436 (0.2206)0.1833 (0.1601)0.50690.3641 (0.3378)0.2921 (0.2620)0.2138 (0.1847)

Mi0.04680.0405 (0.0387)0.0354 (0.0329)0.0285 (0.0257)0.17770.1489 (0.1417)0.1278 (0.1176)0.1009 (0.0904)0.43310.3289 (0.3073)0.2689 (0.2422)0.1994 (0.1733)0.60090.4145 (0.3819)0.3265 (0.2906)0.2344 (0.2010)

570

z

1.6

1.2

0.4

0

1

0.96

0.15

0.05

_ Kn = 0.005/ ^MC^O ft^

\ /0.005i7*^W.? nB/ru

10 AAA

:° o.oos *^^

"'"'oof -"-0.005

5 0.5 Xl/L 1(a)

1.6

1.2

0.4

0

1

0.96

0.15

0 05

Kn = 0.005

-.v^o.oi aA^^iia"7

SA/Kn = 0.1•° A&*4

^0.005

- Kn = 0.1

0.005i i i ......

0.5 Xl/L 1(b)

1.6

1.4

0.92

0.2

0.16

0.12

Kn = 0.005

/Kn = 0.1

Kn = 0.

0.01 0.005

0.5

(c)

FIGURE 2. The behavior for small Kn in the case nfw/n/ = O(Kn) [T///T/ = 1, n///n/ = 2, t/// = 0, and themolecules of both components are mechanically identical (mB/mA = 1 and d^/d^ — 1 for hard-sphere molecules)], (a)riav/ni = Kn (A = 1), (b) nfv/nj = 2Kn (A = 2), (c) pure vapor case, i.e., nfv = 0 (A = 0). The symbols •, o, and Aindicate the DSMC result for hard-sphere molecules for small Kn, and the dotted line indicates the corresponding resultfor the continuum limit given by Eqs. (9)-(lla) and (12) (the values in the parentheses in Table 1; see the main text).

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(^ = 0.764215) when the molecules of A— and B— component are identical. Here, the subscripts model andhs indicate, respectively, the quantity for the model of [32] and that for hard-sphere molecules. But, sincenav/ni is a giyen quantity irrespective of the molecular model, we should have (AKn)mode/=:(AKn)^s, or(A)morfe/=(A)/ls/7. This gives a conversion formula between the model of [32] and hard-sphere molecules. Ifwe regard the A in Table 1 as (A)^s and use the resulting (A)mode/ (= A/7) in Eqs. (9)-(lla) and (12) (e — 1),we obtain the values in the parentheses in Table 1. These values are shown by the dotted line in Figs. 2(a)and 2(b).

To summarize, in the continuum limit, all the noncondensable gas is confined in the Knudsen layer with avanishingly small thickness (compared with L) at the condensing surface, and the vapor flow becomes uniform.The uniform state depends on A. This means the following striking fact. Since we are considering the continuumlimit Kn= 0+ under the condition nfv/n/ = AKn, the average number density nfv of the noncondensable gasis vanishingly small compared with the reference number density n/. Nevertheless, the vapor flow is affectedby the presence of the noncondensable gas through A and is different from that of the pure vapor case.

GENERAL GEOMETRY

As we have seen in the preceding section, even in the simplest two-surface problems, the continuum limitwith respect to the vapor is not trivial. In one of the limits where n^v/ni = 0(1), an infinitesimal flow ofthe vapor gives a finite effect on the state at rest of the mixture (ghost effect), whereas in the other limitwhere n^v/nj = O(Kn), an infinitesimal amount of the noncondensable gas gives a finite effect on the uniformvapor flow. The corresponding limits are also present for the case of a general geometry, though the situationbecomes more complicated (the two types of the limit may coexist). In this section, we discuss the limit of thefirst type for the general geometry [17].

Let us consider the case where the shape of the boundary (the condensed phase of the vapor) is arbitrarybut smooth. We investigate the behavior of the mixture in the continuum limit with respect to the vapor. Forsimplicity, we consider the following situation: (i) the boundary is at rest; (ii) the noncondensable gas is presenteverywhere in the domain; and (iii) the mixture is in a state at rest with a uniform pressure at infinity whenan infinite domain is considered. In this case, an asymptotic analysis similar to that in the previous sectioncan be carried out. Here, we give only the result for the fluid-dynamic system that describes the behavior inthe continuum limit.The fluid-dynamic type equations are:

«,(=«* = t»f ) = 0 , |̂ = 0, ^-(na<pf)=0, (13)

p_

dxj 2 dxj n 4 dxt dxj

-where

pa = naf, pa = (m°'/mA)na, n = nA + nB, P=pA+pB, p = pA+pB, (17)Xa = ha/h, p<t,i=pA<t>A + pB<t,f, (Aij) = Aij+Aji-(2/3)Aee6i:i. (18)

Here, TI, T2, TS, T^, and T^ are given functions of the local concentration XA of A-component, and 0^,<f)f , fa, and II are auxiliary unknown functions (note that </>^, </>f , and fa are vectors).The boundary conditions are:

nA = nw/n0, T = TW/TQ, (19)

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on the boundary. In Eqs. (19) and (20), Tw is the temperature of the boundary, nw is the saturation numberdensity of the vapor at temperature Tw (Tw and nw may vary along the boundary), TO and no are, respectively,the reference temperature and number density (see the last paragraph in Introduction), 67 and b£ are givenfunctions of XA on the boundary, and ef' and e^ are, respectively, the tangential and normal unit vectorsto the boundary.

To give the physical meaning of the auxiliary functions cj)f, $f, (f>i, and II, we mention that, in the asymptoticanalysis, any physical quantity g (g =Fa, na, n, vf, Vi, etc.) is expressed as a sum g=gn+9K with gHbeing the moderately varying (Hilbert) part and QK the Knudsen-layer correction. Each part is expanded asgH=gHQ+gHiKn-i- - • and #K=g*:iKn-|- - •. In particular, vim=vfm=vfHQ=Q in this case. [As in the precedingsection, na, T, £«, etc. in Eqs. (13)-(20) correspond, respectively, to n#0, THQ, ^HO, etc.] Then, the auxiliaryfunctions are identified as <J>f=(2/^/K)v?Hl, 4>i=(2/-\/n)viHi, and II=(4/7r)pH2» and they are of O(l) (PHI,which does not occur in the above system, is also a constant). Incidentally, to derive the boundary condition(20), we need the analysis of the Knudsen layer F^l.

As in the two-surface problem with nfv/n/ = O(l) (in the case £/// = 0), the flow of each componentvanishes in the continuum limit Kn = 0+. Nevertheless, as seen from the system (13)-(20), the numberdensities, temperature, etc. in this limit are affected by the vanishing (or nonexisting) flow through c/)f, fa,and II. That is, the ghost effect manifests itself. The cause of the ghost effect is the cause of the flow of O(Kn),which will be discussed in the following. Let us first examine the boundary condition (20). If the temperatureof the boundary varies along it, a flow of 0(Kn) is induced along the boundary because of the second conditionof Eq. (19) and the first term on RHS of Eq. (20) [recall that fa = (2/A/5r)viHi]- This flow is known as thethermal creep (see [37-39] for a single-component gas and [40] for a binary mixture). On the other hand, if theconcentration XA varies along the boundary, a flow of O(Kn) is induced because of the second term on RHSof Eq. (20). This flow is called the diffusion slip (see [41-43,40,44]). Next, let us look into Eqs. (13)-(16). Theterms containing TI and T2 in Eq. (14), which originate from the thermal stress, can be the cause of a gas flowof O(Kn) [45,46,10], and the terms containing TS and T^, which come from the concentration stress, can alsocause a flow of O(Kn) [47,46]. On the other hand, Eq. (16) indicates that the gradient of the concentration andthat of the temperature cause a relative flow of each component, which is also a possible cause of the flow ofO(Kn). These are known as the mutual diffusion and the thermal diffusion. To summarize, the ghost effect iscaused by (i) thermal creep, (ii) diffusion slip, (iii) thermal stress, (iv) concentration stress, and (v) diffusion.The causes (i) and (iii) have been clarified in the case of a single-component gas [10]. The cause (v) has alreadyappeared in the two-surface problem. The rest is peculiar to the present case of general geometry.

The system (13)-(20) is still formal because accurate numerical values of TI, T2, T3, TA, and T^ are notavailable at present. A database for these functions similar to that for /}, A, DAB, and DT

AB [21] is underconstruction. Concerning the coefficient of thermal creep 67 and that of diffusion slip b^, though approximateresults are available [40], we stick to accurate numerical results by a finite-difference analysis of the Knudsenlayer problem. Such a result has been obtained for b$ in [44].

ACKNOWLEDGMENT

The author's attendance at the 22nd International Symposium on Rarefied Gas Dynamics was supported bythe Ministry of Education, Science, Sports and Culture in Japan.

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