notesonmagnetohy01grad_2

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AEC RESEARCH ANDINSTITUTE ^QENCEs r)VELOPM^^^^ ^^^^ REPORT

^^ NEW YORK UNIVERStTYINSTITUTE OF A\ATHEMATiCAL SCIENCESLIBRARY

' [4 WasSinglon Place, New York 3, N. Y

NEW YORK UNIVERSITY

institute of mathematical sciencesc3 /? ^ D

.

MH- IAEC Computing Facility


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UNCLASSIFIED NY0-6i|86

NOTES ON MAGNETO-HYDRODYNAMICS - NTJI4BER I

GENERAL FLUID EQUATIONSby

Harold Grad

AEC Computing FacilityInstitute of Mathematical SciencesNew York University

CONTRACT NO. AT ( 30-1 )-lij.80

August 1, 1956

This report was prepared as an account ofGovernment sponsored work. Neither theUnited States, nor the Commission, nor anyperson acting on behalf of the Commission:A Makes any warranty or representation,express or implied, with respect tothe accuracy, completeness, or use-fulness of the information containedin this report, or that the use ofany Information, apparatus, method,or process disclosed in this reportmay not infringe privately ownedrights; orB# Assumes any liabilities with respectto the use of, or for damages result-ing from the use of any information,apparatus, method, or process dis-closed in this report,

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Contents

Page No,Preface 3Section 1. Notation ... 7

2 Conservation Equations 83 Charge and Current e**..** 11i|. Entropy 13


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Preface

This is the first in a series of notes on varioustopics in magneto-hydrodynamics, A large part of thismaterial has been presented in informal lectures atirregular intervals since 1951;. A set of four mimeo-graphed notes, representing some of the early develop-ments, was issued in July 195^ with the purpose ofacquainting a small group at NYTJ with certain aspectsof the subject. Meanwhile, a certain amount of generalixaterest has been exhibited in these notes, and it wasfelt to be worthwhile to rewrite the original notes ina somewhat more readable form, at the same time extend-ing and elaborating them greatly^ The original fournotes are contained in numbers 1, 3, ^4-, 7 of this series.Although many of these reports are unconnected, an attemptwill be made to supply enough background material to makethe series to some degree self-contained.

The status of the subject of magneto-hydrodynamicsis somewhat paradoxical. It involves in combination,various aspects of fluid dynamics, electromagnetic theory,kinetic theory, and occasionally relativity. Most ofthis was available to Maxwell and he had the material todevelop the subject to a considerable extent, but onlyrecently has it been attacked significantly. Another,and probably related, consideration is that the companion

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subjects of fluid dynamics and electromagnet! sm have awell developed experimental backgroiond, while relativelylittle is known experimentally, in magneto-hydrod3majnics.Consequently, the development of magneto -hydro dynamictheory is rather abstract, principally devoted to thesolution of rather idealized special problems, and thedirections of development are guided to a large extentby what has been done in fluid dynamics and electromag-netics. There do exist important problems in astrophysics,in properties of the earth's atmosphere, and in gaseousdischarges, but these are usually so difficult as to beunapproachable in the framework of a mathematical theory.Controlled experiments, with only a few effects presentat a time, are very scarce.

This first note is largely a matter of bookkeepingand can be motivated only by pointing to the variousapplications made later on or by reference to kinetictheory. Various forms of fluid conservation equationsare listed for future reference. They are presented informs too general to be of specific mathematical use, butthey do serve the useful purpose of correlating and categor-izing the many specific fluid descriptions which will beused.

This note was prepared with the assistance ofIgnace Kolodner and Albert Blank.

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General Fluid Equations

Any macroscopic picture of a fluid, electricallyconducting or not, is based on conservation laws, usuallyof mass, raoinentum, and energy, and occasionally of others.It is Important to realize several things. First, theseare "lack of conservation" more than "conservation" equa-tions. For example, the energy of an element of fluid(or of the whole fluid) is usually not a constant, but isassumed to vary in certain definite ways. Second, theform taken by these equations is, macroscoplcally speak-ing, pure postulation. One is governed by certain generalconsiderations of what has been found to be useful, butno question of derivation can arise iinless referred to amore primitive basis such as motion of particles. Thisremark applied also to the the rmo -dynamic formalism whichis microscopically derivable but macroscoplcally postu-lated. Finally, in a strictly formal sense, many of theterms included below are redundant. For example, in themomentum equations, we find terms X and ^P ^/ix^%Clearly, it would be completely general to omit ^P "^/3x"'and claim that the force term X sometimes takes the formof space derivatives of auxiliary quantities, and, con-versely, it is easy to see that a force term X can alwaysbe put into the form ^P 'V^x by suitable choice of P '.

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This separation into terms is significant only becausethe conservation equations are Incomplete, mathematicallyspeaking, and the supplementary relations which willeventually have to be introduced will be of several kindsfor which the various forms such as ^p'V^^ and X"*" aremore convenient. It may be remarked that, even from amolecular standpoint, the separation into JP ^/d7i, andX turns out to be somewhat arbitrary. Insofar as theequations listed below are sufficiently general to covermost cases of normal occurrence, they cannot be completebut must be supplemented by relations which specify theparticular nature of the fluid under study; these rela-tions will range from functional relations (e.g., anequation of state) to differential equations (e.g., rela-ting stresses to velocity gradients, or relating a forceterm to a magnetic field vector which in turn is coupledto other of the fluid variables through a set of auxili-ary electromagnetic differential equations).

By the inclusion of general terms like X , we obtaincomplete and universal validity for these equations (e.g.,by defining X as the force required to make the momentumequation correctl), but, by the same token, they cannotserve more than a bookkeeping piJrpose. This use isImportant, however, because of the great variety of fluidmodels which may be used to describe a conducting medium.

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1 NotationWe consider a fluid consisting of n homogeneous

components, each with its ovm Intrinsic properties (suchas molecular mass, charge, etc.) and each with its ownthermodynamic state variables.Molecular properties:

m^ = mass , .re = charge (signed) r = l, , n'r r r

Molecular structure is referred to only to establish theconnection between mass density and charge density below.State variables:

V = number densityp = m V = mass densityq = e V = charge density (signed)q /p = Y - constant for a given fluidu = macroscopic flow velocityp u = momenttim per vmit volume^r r ^e = Internal energy per unit volume

12= total energy per imit volume ~ v, "*" p P-n^-rT = temperaturep = pressure^i, = chemical potential

Supplementary variables:P^'' = stress tensor, pj'' = Pj'' - p^..^^*^

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Q^ = heat flowX = force per unit volumer ^C = energy source per unit voliitne(T = mass source (chemical creation) per unit volumerCw= momentum source (chemical) per unit volumer r

Internal energy, stresses, etc, are defined to be thoseobserved moving with the individual fluid, i.e., at thevelocity u2 Conservation Equations

The follovdng "conservation" equations are postulated:-"-

-W "-^^ (Pr"r) = ^r ^^^"^^^^^ { it ^Pr^r^ " O ^^v'V^i "- ^r^^ = ^J "" ^r^r ("'Omentum)

ae^ ax"" -"^ (^r^r "" ^r^r'^' + Q^) = ^ (energy)Using V = u - u, Jl Pt.% = we define the following

gross (single-fluid) variables:V = I v^p = Z pj,q = I Oppu = I Pr^r

-;> The summation convention is used for repeated tensorialindices, but not for indices distinguishing the fluids.- 8 -


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e = I (e^ + 1/2 p u^) - 1/2 pu^ = I (e^ + l/2 p v^e = 51 e =s e + 1/2 pu^rX = I X r

-r

The V are the relative or diffusion velocities. Thersingle fluid internal energy, stresses, etc., are definedwith respect to the mass flow velocity, u, and are notmerely sums of the corresponding quantities of theconstituent fluids.

Prom the conservation of mass and an appropriatedefinition of CJ ,r

I (y^(^ = 0. ;;::, ,Adding the individual conservation equations: _. ,,

^ +-^ (pu^) = ,. ,( ax-(2) 4r (pu^) + -K (pu-^uJ + ?^h = X^

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It is sometimes convenient to consider the following(Lagrangian) form of the momentiain equations:

(3)

ih)

- + w^ )t ^x dpij

^:c J= x^ + (T (coi - ui)

and also the difference momentvun equations^v û >P.^ ^ ^ ^ Pr )x

1.11 apijp JJ

X-^ î (5- . .= -^ -2L + -I (u,=^ - u^)Pp P Pp I' rEquations (3) and (U) are obtained by combining themomentiAm and mass equations of (1) and (2) respectively

The conservation equations (2) are equivalent(assuming differentiability of all quantities) to theintegral relations

1^ J pdV + i pu^dS . =D S

(6) 1^ j pu^dV + 6 (puû^ + P^'^)dS, = J X^dV

1^ J edV + a (eu^ + u^P^^ + Q^ )dS = /(fdVdV +D S

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The surface integrals are taken over the boundary,S, of the fixed domain, D. The Lagrangian form of (6)for a domain D(t) which is carried along by the fluidat the velocity u is ,

4t / pdv = oD(t)

it J p-'^^ V ^'^^sj = J ^'^D(t) S(t) D(t)^ / 5dV + / (u^P^^ + Q^')dSj = y dV

V< D(t) S(t) D(t)Similar integral forms hold for the indii^idual conserva-tion equations (1).3. Charge and Current

We note the conservation of charge

^ 'r rand define the current densities

J =5'qv,J=yqu =J + qu;u ^ ^r r' ^ ^r r u ^ 'J is the current observed in the fixed system and J"moving with the fluid" J the convection current is qu.Prom (1), the individual "conservation" of charge is

^X

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whence, adding,(9) 32 ^ î^ J ^ 0,

Similarly, from (1) we can obtain individual currentequations(10) Jr (q u^) + -K (q uû^' + y ^^ h = Y (X^ + (ft/).^ ' di ^r r ^J^rrr 'rr 'rr rrand the total current equation(11) ^* I 7^ ^%44 * Â''' = I Yr"r *


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U EntropyA local the ririodynamic structure can be introduced by

making certain assinnptions. First (this has already beendone), extensive quantities such as mass, momentum, andenergy are introduced as densities. We know from equili-brium thermodynamics that all thermodynamic parameterscan be computed from the single function "entropy density"as a function of the other densities. We need only adoptthe same fvinctional relation in nonequilibrium and computeall other quantities, e.g., temperat\are and pressiore, bydifferentiation.

We define the entropy per unit volume, s, by(li;) Ts = e + ^ ^^p^ + pand write the second law(15) Tds = de + j; tipdp^and the Cxibbs-Duhem relation

= ed(^)+Ip^d(^) + d().We have expressed the intensive thermodynamic

coordinates per unit volume rather than per unit masssince this provides a simpler (additive) transition fromindividual fluid properties to total fluid properties.Expressed in terms of unit mass, (li|.) and (l5) take theform(16) < p p ^ ^^r p p

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Prom the above we compute that(17) If -^ div (su + I + i X t^rPr^r) = ^ ( - u-X + Iix^cr )

+ Q . grad(i) + Ip^v^ . grad (^) -J p^J ^ .This thermodynamic formalism is appropriate in case

it is sufficient to deal with the total momentvun andenergy equations alone, although taking n individual massequations. In case we wish to treat all n velocities andall n temperatures as primitive variables, it is necessaryto introduce the generalized n fluid fon7ialism:->

r r r ^r'^r r(18) "S T ds = de + u dpr r r "^r ^r

^ = e d(i-) + p d(^) + d(^)r r rThe total entropy per volume is(19) 3^ = I rWe compute that(20) ^^+ dlv H s^u^ ^l^) =l^it^. u^.X^)

o-

T+ Z f^ (t^r "" i^r ~ ""r* '^r ^ " ^ ^ * ^rad (i-)V 1 niJ ^^

^ This is significant only for perfect gases, i.e.,when there is no significant intercomponent energy.- Ik -


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1 2It is interesting to note that the combination u, + :? u*^r 2 rwould have appeared in such a fashion as to warrant thename chemical potential if we had chosen as thermodynamiccoordinates mass, total energy, and momentum instead ofmass and internal energy, (see Grad, J. Phys. Chem,, 56,1039-101+8, 1952).

A few words should be said about the differencebetween (1?) and (20). First of all, the "entropy" s-"-is not the same as sj specifically, it is not a functionof the same variables. In the one fluid formulation, (17)there is an entropy source related to diffusion velocitiesand gradients of the chemical potentials,. This does notappear in (20 )j in other words, the individual fluidvelocities are here treated as primitive variables, conse-quently reversible. If the energy sources are related tothe forces by the relation o = u X , then (20) considers./ r r rthis as reversible, while (17) has an entropy source ofmagnitude

g - u . X = 5: v^ . X^ ;i.e., slipping is irreversible. On the other hand, if we

have energy sources, C' in the absence of forces, X ,and have = ^^ - 0 then (1?) considers this state ofaffairs as reversible, while (20) does not if the variousfluids are at different temperatures.

The choice of the entropy function s-"- can be- 15 -


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justified on the basis of statistical mechanics; however,its most important macroscopic justification is that wemay keep the concept of increasing entropy together witha fluid description involving several momentum and energyequations.

As an example, choose for the force X and energy (>the expressions

r X^ = qÊ + qû^ x BP^ = u ' X^ = q u^ . E

^ ut r r r r(E and B are the electromagnetic field vectors, ) Thereis no entropy production from this source in the secondCase, (20), while in (17) we have(22) C - u.X = J .E = ( J - qu) (E + uxBThe entropy source is the product of the current and electricfield each measured in a system moving with the instantaneouslocal fluid velocity, u.

Another possibility isf X^ = qÊ + (qû^ + J^) X B(23) ) ^( C = E . (q u + J )\ ^r ^r r r

The extra currents, J , could be introduced, for example,to take into account magnetic polarization. In this case,we have

^X = qE + JxB(24) i pO s E J

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by redefining

(Instead of just ^ ^r^j,)* whence- u X = E .J

as before, but also(26) ^^ - ^r - y^^ = i^ + n^ X B) J^ .

It will be observed that we have not Included electricor magnetic polarization terms in the thermodynamics.Nevertheless, we are not restricted to the case of unitdielectric constant and permeability. To achieve this weneed merely adopt the viewpoint ahat J and q include polari-zation current and charge respectively (cf, Stratton,equation (7), p. 12). Moreover, this Interpretation isalmost forced upon us in any consideration involving energy.The usual formulation in which electric energy is representedby jE'dD rather than f^ E^ (Identical remarks hold for Hand B ) is a useful expedient only when the internal energyof the medium is disregarded. In certain very simple situa-tions, it is convenient to distinguish electric chargeswhich are \mder the direct control of the experimentorfrom other "polarization" charges. In the process of polari-zation, there is a transfer of energy between the electricfield and the dielectric (Internal thermal energy). Ifone is not interested in this transfer of energy (which

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is frequently reversible), one can maintain an artificialconservation of those energies which one chooses toacknowledge by using / EdD instead of / -i E In oomplexproblems of fluid flow, retention of D and H would seemto be a hindrance rather than a help. It should beremarked that equations (8) and (10) - (13) are validonly in the absence of polarization.

In succeeding notes chemical terms will be ignoredunless the context plainly demands their inclusion.

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