On the thermodynamics of mixtures with several temperatures

lnt. J. Engrrg Sci. Vol. 8, pp. 63-83.

Pergamon Press 1970.

Printed in Great Britain

ON THE THERMODYNAMICS OF MIXTURES SEVERAL TEMPE~TURES

WITH

R. M. BOWEN and D. J. GARCIA? Rice University, Houston, Texas 77001, U.S.A. AlWract-This work concerns the formulation of a thermomechanical theory of a mixture where each constituent has its own temperature field. The theory also contains the effects of nonlinear elasticity, nonlinear heat conduction, nonlinear viscosity and diffusion. A linearized version of the general theory is also presented. 1. INTRODUCTION

that a distinct temperature can be associated with each constituent in a mixture has long been recognized in general theories of mixtures. The thermomechanical balance equations for a mixture of general materials were first formulated in a major work by Truesdell [l]. Truesdell proposed equations of balance of mass, momentum and energy in a form which is generally accepted today. Kelly [2] followed a suggestion made later by Truesdell and Toupin [3, sect. 2 151 and allowed the partial stresses to have a skew part. This feature is also present in the fo~ulations of Eringen and Ingram [4] and Green and Naghdi [5]. In a sequel to their first paper, Eringen and Ingram[6] investigated the restrictions on the constitutive equations for a mixture of reacting gases resulting from an entropy inequality for each constituent. They allowed each constituent to have its own temperature. Green and Naghdi[5] also proposed an entropy inequality for each constituent which allowed for multiple temperature .effects. Dunwoody and Miiller[7] were the first to investigate a multiple temperature mixture theory in which only an entropy inequality for the mixture was proposed. They considered a mixture of two chemically reacting ideal gases. Their fo~ulation was based on an entropy inequality for the mixture which was proposed previously by Miiller [8] in which the entropy flux is given by a constitutive equation of its own. Steel[9] applied the formulation developed by Green and Naghdi to a mixture of two elastic solids with distinct temperatures. Bowen and Wiese [ lo] adopted Kelly’s generalization of Truesdell’s balance equations and proposed an entropy inequality for multiple temperature mixtures. However, they considered constitutive equations for a mixture of elastic materials with but one temperature. In reference [ 1 l] we adopted the entropy inequality proposed by Bowen and Wiese and investigated theories of heat conduction in a mixture of rigid bodies where each body has its own temperature field. In this paper we continue to explore multiple temperature mixture theories by presenting a thermomechanical theory of a mixture of bodies capable of nonlinear therm~elastic effects, nonlinear heat conduction, nonlinear viscous effects as well as diffusion. The effects of chemical reactions have been omitted here. These effects have been investigated for special mixtures in two recent papers by Bowen [ 12, I 31. In section 2 the kinematics and equations of balance of mass, linear momentum, moment of momentum, energy and the entropy ineq~lity for a mixture of N bodies

THE POSSIBILITY

tPresent Address: Scbhnnberger Well Services, Houston, Texas, U.S.A. 63

R. M. BOWEN and D. J. GARCIA

64

are summarized. In section 3 constitutive equations which define the mixture are stated. In section 4 we deduce the restrictions imposed on the constitutive equations by the entropy inequality. In section 5 the restrictions imposed on the constitutive equations by the axiom of material frame-indifference and symmetry considerations are considered. In section 6 linearized constitutive equations for a mixture of fluids and isotropic solids are presented. NOTATION

Direct tensor notation will be used in preference to component forms. Vectors in the three-dimensional inner product space % and points in the Euclidean 3-space 8 will be denoted by Latin boldface minuscules: x, g,---. No distinction will be made between linear transformations from % into % and second-order tensors; they will both be denoted by Latin boldface majuscules: F, L,---. If A is any linear transformation, AT will denote its transpose, A-’ its inverse, tr A its trace, and det A its determinant. The identity linear transformation will be denoted by I. Component indices will refer to a fixed time-independent rectangular Cartesian coordinate system with basis (e,, q, e3). Spatial coordinate indices will be denoted by Latin minuscules. Material coordinate indices will be denoted by Latin majuscules. The gradient with respect to spatial coordinates will be denoted by grad, and the gradient with respect to material coordinates by V. The divergence with respect to spatial coordinates will be denoted by div. A quantity corresponding to a particular constituent of the mixture will be identified by placing a Latin minuscule directly under the symbol for the quantity: F, r, ---. The summation convention will apply only to summations over coordinate indices. Summations over constituents will always be indicated by the summation symbol. The complete contraction operator will be denoted by the symbol C[lO, section 11. If W is any third-order tensor, and A is any second-order tensor, then C(U @ A @ A) = YIij&jmAknei @ e, @ e,. The symbol Y,, where p is a positive integer greater than or equal to 2, will denote the set of tensors of order p symmetrical in the last p - 1 indices. If W is a tensor of order p and Q, is a tensor of order p + 1, then *[@I is a vector defined by

W@l = scI...kp%I...kp&~.

(1.1)

If A is a second order tensor and W is a tensor of order three, then

The notation currently used in mixture theories is cumbersome. It is for this reason that we have been intentionally careless about confusing functions and their values. Unless some possibility for confusion exists, we have repeatedly used the same symbol for a function and its value and, in some cases, for several functions and their common value. 2. KINEMATICS,

EQUATIONS

OF BALANCE,

AND

ENTROPY

INEQUALITY

In this section the kinematics of motion and the axioms of balance of mass, linear momentum and energy, and the entropy inequality for a multiple temperature diffusing

65

On the thermodynamics of mixtures with several temperatures

, N, are discussed. Each body 99 is considered

mixtureofNbodies9J,a=l,... set with a structure prkcribed A configuration

by Noll[ 141. An element of 9I is deioted

of the body $@is a homeomorphism

to be a

by X.

i of ~8 onto a sibset

Euclidean space 64. A motion of i is a one parameter family of Jonfigurations

of the

g where

t is the time. The position of the particle 5 at the time t is given by

(2.1) A reference

configuration

for f is a fixed configuration

5 The position of the particle

5 in z is given by (2.2)

Jj = T(Z). It follows from (2.1) and (2.2) that

x = gp$ The function

r) = $K(F,?I.

s,, is called the deformation

funcfion

(2.3)

of kl?‘.It will be assumed in this

paper that each function xK has an inverse xi’, and that iach function xI( and xi1 are C3 functions of their a&rents. are

The veloci:y and acceleration

of the pkicle

$ E $

and 2 ‘$

The gradient of the deformation

=z

(2.4)

(?$t).

and the velocity gradient for ?$ E L$?are F = v$q9

t) 9

and f; = grad$(x, As a result of the smoothness follows that

assumptions

r).

made about the functions

(2.5) xK and x;l it D a

(detlJ( > 0.

(2.6)

$‘= E-1.

(2.7)

It can also be shown that

66

R. M. BOWEN

If pa denotes the density off,

and D. J. GARCIA

the density of the mixture is defined by Ph

t> =

E

Pak

(2.8)

t).

a=1

The mean velocity of the mixture is given by

i(x, t) = ;

$ P&X,

t),

(2.9)

O-l

and the di$usion velocity of the particle 5 which at the time t is at x is defined by g(x, t) =

gx, t)

-k(x,

t).

(2.10)

Equations (2.8), (2.9) and (2.10) imply that

Ii1Pa! =0. If q is any function of x and t, the derivatives i and $ are respectively

(2.11)

of W following the motions generated by

@ = !L$ (x, t) +(grad*(x,

t))i

and 2 = g

(x, t) + (grad W(x, t))$.

The equations of balance of mass, linear momentum, energy for the ath constituent are postulated to be %+div

moment of momentum

and

(p,$) = $,

p,x = div T+p,b+$, a

(2.12)

(2

a

(2.13)

a

r$=a-y

and pa&= tr (TX) -divq+p,r+ a a a a

(2a

where 1 denotes the mass supply of the ath constituent due to chemical reactions,

g, per unit volume per unit time

T denotes the partial stress of .$‘, 3 denotes the partial body

force of B, i denotes the diffusive force on L@per unit volume, &l denotes the body lI a a a

On the thermodynamics

61

of mixtures with several temperatures

couple on 5@per unit volume, E denotes the partial internal energy density off,

9 denotes

the partial ieat flux of .G$?, i dznotes the partial heat supply density of $?, and $ denotes the energy supply to f per unit volume per unit time and includes the rate of work of the body couple $I. The equations of balance of mass, linear momentum, energy for the mixture are postulated to be

moment of momentum

and

(2.14) and

Equation (2.14), reflects the assumption that there is no net body couple on the mixture. In order to postulate the entropy inequality of the mixture, the following quantities are introduced as primitive properties of the constituents: (a) a temperature f > 0 for each constituent of the mixture, and (b) a partial entropy density 7 for each constituent

of the mixture.

The entropy density of the mixture is defined by (2.15)

The entropy inequality for the mixture is postulated to be [ 10, sect. 3; 18; 191

pi)+divi(~+pa+g$=0. a=1

a=1

a

(2.16)

a

Equation (2.16) does not represent the most convenient form of the entropy inequality that we may use. The Massieu Function and the coldness for 5 E L$’and at the time t are defined to be respectively (2.17) a and (2.18)

R. M. BOWEN and D. J. GARCIA

68

using equations (2.13),, (2.13),, 12.17)and (2.18), we can rewrite the entropy inequality (2.16)in the more convenient form

5 {h+P,&+tr au a=1 a

[(hI+tPT*)L]+q~g-~u~~-&&2+&?} a

aa

a

a

a

where g = grad 8, and 2 = S + u =@+ 2( E++&). a a aa aa a (I Q

i

aa

a

aaa

aa

3 0,

it foliows from (2.14), that

2=0.

(2.20)

a

a=1

3. CONSTITUTIVE

(2.19)

ASSUMPTIONS

The balance equations (2.13) and (2.14) and the entropy inequality (2.19) must be satisfied by every multiple temperature mixture regardless of material constitution. As we mentioned in the Introduction, we are interested in a theory of a mixture of bodies capable of nonlinear thermoelastic effects, nonlinear heat conduction, nonlinear viscous effects, diffusion and the effects of multiple temperatures. In this section the constitutive equations for this mixture are proposed. Following the nomenclature used by Coleman and Noll[lS], a ?~~~~o~y~~~~c process for the mixture is a set of 13N functions whose values are x=@$.‘)*

Pa = &(X7 a t) 9

f=‘ifc$h 2 = gx, a

$ = $$,

A=lyy, a

$=

c*(X,f), a a

$ = sr’~Y r)*

t=

y~J>,

y = (rr($, f) Y

$ = lg,

t) 3

fa = r;q,

t),

t),

k = f$$, t),

f) 3

for each a = 1,. . ., N, and which satisfy equations (2. 13)1, (2.13),, and (2.13).,. The equations of balance and the entropy inequality suggest that constitutive equations should be prescribed for A, E, T, 6, q, 2, M and & for each Q = 1,. . ., N. Since we are interested in a non-reacti~gQm~xt~re~ it”f;lows :hat our first constitutive assumption mustbec=O,a=l,..., N. It follows that (2.14), is then satisfied identically, and that (2.13), may be integrated to yield p,ldet IJ’I= paor where pa0 is the density

of W in its reference

configuration.

(3.1) We shall assume, for

simplicity, that the densities p:O do not depend on the positions X and that, therefore, they are known constants.

We shail also assume below that the lonstitutive

equations

On the thermodynamics of mixtures with several temperatures

are independent

69

of the positions 5. We shall then postulate that the remaining quantities

A,e,T,G,$,qand2,a=l,...,

N,aredeterminedbyzP

b,~,I$V~andt,b=

l,...,

N.

~o~rn~ll; wi iave aisumed that (3.2) In order to simplify an already cumbersome notation, all the entries in (3.2) are not shown explicitly. For example, the appearance of 4 denotes (+ $, . . ., PL), and the appearance

of Q denotes

(f, 9,. . ., 8). Also the component

functions associated

f will be assigned the same symbol as the value in question. write from (3.2)

with

For example, we shall

(3.3) For the arguments presented in this paper, it will suffice for the response function f, and therefore its component functions, to be of class C. The choice of independent variables in (3.2) is a natural extension of previous mixture theories. where a single temperature was allowed. (See, for example, Bowen and Wiese[lO, sect. 43.) It was Miiller [8] who stressed for the first time the importance of a dependence of the constitutive functions on gradients of the deformation. Without this dependence, expressions for the partial stresses are obtained which are known to be too special. In this context see Bowen and Wiese[lO, sect. 51 and Bowen[ 13, sect. 61. An admissible thermodynamic process for the mixture defined by (3.2) is a thermodynamic process which is consistent with the postulate (3.2) and the postulate c = 0, a=1 , . . ., N. By an obvious generalization of the theorem by Coleman and No11 [ 15, sect. 31, it can be shown that for every choice of the N deformation functions xK, a= l,..., N, and the N coldness distributions 9, a = 1, , . ., N, there exists a kzique admissible thermodynamic process. Further generalizing the logic of Coleman and Noll[15, sect. 41, we shall now require that equations (2.13)s, (2.14),, (2. 14)3 and (2.20), and the inequality (2.19) be satisfied for every admissible thermodynamic process. We can easily include as part of the defining constitutive assumptions the restrictions imposed by (2.1 3)3, (2.14),, (2.14), and (2.20). The interesting restrictions will arise from the inequality (2.19), and these restrictions will be examined in the next section.

4. CONSEQUENCES

OF THE SECOND

AXIOM

OF THERMODYNAMICS

Following the logic outlined in the previous section, it is now necessary to investigate the restrictions that must be imposed on the component functions of the response function f in order that the constitutive assumption (3.2) be compatible with the entropy inequality (2.19). We shall obtain this restriction by using a standard technique originally formulated by Coleman and Noll[15, sect. 41. Substituting (3.2) in (2.19), using equations (2.7), (2.1 2)2, and rearranging the terms, we find that

70

R. M. BOWEN and D. J. GARCIA

where the Massieu function for the mixture A is defined by

A=$+, a=1 a

(4.2)

and we have used the identity +

(ya = z +(grad

y)($ - $1

(4.3)

which follows from (2.12),. Equation (4.1) must be satisfied for every choice of the N functions xK, a = 1, . . . , N, and the N functions

9, a = 1, . . ., N. By an argument similar to that Lf Coleman and

Noll[15,sect.4],(4.1)impliesthatfora=

l,...,N,

(4.4)

On the the~od~n~cs

of mixtures with several temperatures

71

for all 5 in 9’s and

for all y in Y*. Equations (4.4) follow from the fact that each t, 4, grad 111t, V$, VV!, and $, a = 1, - * *, N, can be assigned arbitrary values. Equations (4.4),, (4.4), and (4.4), show that A=A(lp),

(4.5)

so that equation (4.4), implies that

-

~-~(jj a

F) a

(4.4)

b’b

The results (4.5) and (4.6) generalize those obtained for the mixture of rigid bodies by Bowen and Garcia[ 11, sect. 31.As a result of (4.4), the inequality (4.1) reduces to

(4.7) where (4.8) and

(4.9) If equations (4.4)@,(4.4), and (4.4), are differentiated with respect to f and then evaluatedat#=O,b= l,..., N,wefindfora= l,..., N

q,p$,$VT,O, agi

fi

= 0,

(4.10)

72

R. M. BOWEN and D. J. GARCIA

(4.11) and (4.12) Therefore Vp,b=l,...,

when the velocities

vanish $, a = 1,. . ., N, does not depend on f, $ and

N. Further conditions on A may be obtained from equations (4.4),, (4.4),

and (4.4), by computing higher derivativis theresultsat$=O,b= I,..., N.

with respect to the velocities and evaluating

The mixture is said to be in an equilibrium stare if $=gb=...=~=6

1

N

2

..=g=&

f=f=’

N

f=$

=...=

$=o

(4.13)

and i=

x=** 2

*=8=O,

where I!+is any coldness distribution in the domain of the constitutive function $ It would be sufficient for our purposes to require that for each a = 1, . . ., N g = v and $ = W$‘, where v is a constant

vector,

and W is a constant

skew-symmetric

linear

transformation. We shah see in the next section that the restrictions imposed by the axiom of material frame-indifference imply that no loss in generality is incurred by adopting the definition (4.13) in the form stated. It follows then from (4.7) and (4.13) that at the equilibrium state 1(6,0, f, 0, VS’ 0) = 0. Since I( Q, p %, $‘, VT, f) has a minimum at equilibrium, vanish, and its second differential that is

(4.14) its first differential

must be positive semi-definite

must

at (8,0, H, 0, VT, 0),

(4.15) and

forall (a,a,A,m) bbb

in (-co,03)NX~NX_EP(~!;~)NX~N. b

On the thermodynamics of mixtures with several temperatures

73

A simple calculation shows that (4.15) implies for each a = 1, . . . , N

q( 8,0, F, 0, VF,0) = 0, a

p,

b

0,

I,

$29,0,$0,

(4.17)

b 0,

VI,

0)

=

0,

(4.18)

V$O) = 0,

(4.19)

S( 6,0, H, 0, VT, 0) = 0. (1

(4.20)

and

The most general implications of (4.16) are too complicated to reproduce here. However, in section 6 we shall illustrate the results implied by (4.16) for a special linearized theory. Equations (4.4),, (4.4),, (4.4), and (4.5-4.7) represent necessary conditions that the component functions of the response function f must satisfy in order that the entropy inequality (4.1) be satisfied for every admissible thermodynamic process. It is easy to show that these conditions are also sufficient. We may therefore summarize the main results as a theorem. Theorem. In order that the entropy inequality (2.19) be satisfied for every admissible thermodynamic process of the mixture defined by (3.2), it is necessary and suficient thatfor each a = 1,. .., N

(4.21) (b) x=tj@$H)

(4.22)

=-+&rL a a

(4.23)

(4.24) for all K in Y3, 0 (e)

(4.25)

for all 9 in Yd. and (4.26)

for all admissible

thermodynamic

processes,

where each q and each e^ vanish in D D

74

R. M. BOWEN

equilibrium, each T takes the value - f

and D. J. GARCIA

4'1 + ?$

a

zT in equilibrium, and each fi takes n >

the value

in equilibrium. The symbol + indicates that the function is evaluated at the equilibrium state. 5. MATERIAL

FRAME-INDIFFERENCE

AND

SYMMETRY

In this section the restrictions imposed upon the response function (3.2) by the axiom of material frame-indifference? and by material symmetry are considered. The axiom of material frame-indifference requires that an admissible thermodynamical process remain admissible after a change of frame defined by

x* = c+Q(x-O),

(5.1)

where c is a time dependent element of 8,O is the origin of 8, and Q is a time dependent orthogonal linear transformation of S onto @. It follows from (2.1 3)4 and (3.2) that the axiom of material frame-indifference is satisfied if and only if equation (3.2) and the following equation are valid ($, 5, QTQT, Q$QT9 QP, Qt, $-- tr (T’Q’Q>

a

)

=f(~,~,Q~,~~+Q~,c(Q~q,,c+Q~+i2(~-0)), for all orthogonal

tensors Q, all Q such that @

F, $‘,VT, 5) in the domain of the response

(5.2) = - (w)

T, all i: in S, and all (f, $,

function J? Of course, the domain off is

assumed to be large enough so that the right side of (5.2) has meaning. The trace term associated with the component function { is a direct consequence of requiring that 9 in equation (2.1 3)4 include the rate of work of the body couple 9. Equations (5.2) (4.2) and (4.5) require that

for all ($, T) and all orthogonal Q. Equation (5.3) implies that (5.4)

The formal proof that (5.3) implies (5.4) is an elementary generalization of an argument which was first given by No11 in order to prove that the stress was necessarily symmetric tSee reference [16], sects. 17-19.

On the thermodynamics of mixtures with several temperatures

for a hyperelastic

material [ 16, sect. 841. The importance of (5.4) here is that

$ W--T) a a

a=la

as

15

=

$l$(!‘-“f’>

(5.5)

follows from (5.4) and (4.9). Also, in obtaining (5.5), we have the fact that N

an

&f=o. a=1

Equation (5.6) is another implication

of (5.2) which requires that A depends upon the b

N velocity vectors through N - 1 velocity differences (5.5) in equilibrium

(5.6)

a

(see (5.7) below). If we evaluate

and use (4.20) we see that the total stress a& T is automatically

symmetric in equilibrium. Another way of stating this result is that (2.14), is automatically satisfied in equilibrium. A typical solution of equation (5.2) is

(5.7) where

u=i-ir,

bd

b

d

(5.8)

E=vpW, c

W = +(La a

LT) is the spin tensor, a

$ = CT! is the right Cauchy-Green

tensor

and c, d, and e are fixed integers satisfying 1 s c s N, 1 s d s N, 1 s e s N. We shall now consider the restrictions imposed by material symmetry on the component functions of the response function f in equation (3.2). The approach taken here is a generalization of one formulated by Bowen[ 171. The isotropy group Y for the dth constituent in the mixture is the set of all unimodular linear transformations & such that f(g, f, T, $9 VI, f) =f($+,$

,...,

p

,...) E,$ )...) $I,..

.,$VT

)....,

(5.9)

(?(V;@H@H),...,Vf,i) for all ( f, 9, T, $, Vf, t) in the domain of the response

function J Again we assume

the domain offis sufficiently large to make the right side of (5.9) defined.

R. M. BOWEN and D. J. GARCIA

76

As an illustration

of a special type of material symmetry,

the case for $ = @, for

all d= 1,. ..,M,where~istheunimodulargroup,and~=0,foralld=M+l,...,N, where 0 is the orthogonal group of M fluids, and N-M isotropic solids or fluids exclusivelyt. It that for a mixture of M fluids and

will be considered. This case corresponds to a mixture solids, and can be specialized to a mixture of either can be shown from equations (3.2), (5.7) and (5.9) N-M isotropic solids

(5.10) where a,b= l,..., N, c= l,..., 1 s e s N, 1 6 h =SN,

. ., N, e and h are integers such that

M, d=M+l,.

B = FFT is the left Cauchy-Green tensor, (I aa D = 4(L+ LT) is the stretching tensor, &&&-Wp’,

(5.11)

and, where if (5.6) is valid the following equation is also valid

(4, z, Q'fQT9 Q$QTyQh % < +

tr Cry ) 1

a

QT@ QThQ$

=f(t,$,~,Q~QT,Q~T,Q~QT,Q~(~s for all orthogonal

tensors Q, and all values of

$ f, i,

7, y, r,

y, ;

the response function $ Equation (5.12) implies that the component response function fare isotropic functions. 6. LINEAR

ISOTROPIC

>

(5.12)

in the domain of functions of the

THEORY

In this section we shall present linearized expressions for the internal energy, stress, heat flux, energy supply and diffusive force for a mixture of M fluids and N -A4 isotropic solids undergoing small coldness changes, small deformations and small departures from equilibrium. The displacement of X E S relative to the configuration K is D

a

0

(6.1) The gradient of the displacement respectively

and the injinitesimal strain tensor at X E 9j are (I (I 5 = VT
(6.2)

tFor a precise statement of what is meant when we refer to ‘kids’ and ‘solids’ see reference [ 163, sects. 32 and 33.

On the thermodynamics of mixtures with several temperatures

77

and (6.3) It follows from (2.5) and (6.1) that T=I+H.

(6.4)

a

It follows from (5.1 1)1, (5.1 1)3, (6.3) and (6.4) that ~=1+2~+IIIv

aa

and $I= C(Vy8

(I+y-%

(l+Iy-1).

(6.5)

We wish to expand the response functions about the reference state

8, g, ‘, B, b b PC d

..,N, c= 1,. ..,M, d=M+l,..., = 6.,0,+-,I,O,O,O,O , b=l,. N. > > ( In order to measure departures from the reference state we shall use the parameter E, defined by D,W,M,u b

bN

e2=

b

bN

& {(+-o)2+ a

g*g+tr(HTH)+trD2+tr(~%J+C(VH@VH)+~.~}. 0 D 0 a a a

a (6.6)

The temperature changes, the deformations, and the departures from equilibrium are said to be small if E < 1. A quantity of order 8 is any scalar, vector, or tensor, denoted by O(@), with the property that there exists a real number K such that

IIW”) II < Kek as E + 0. It will be assumed from this point on that E < 1. Since the internal energy, the stress at equilibrium and the diffusive force at equilibrium are expressed as derivatives of the Massieu function, it is convenient to expand the Massieu function first to obtain the common coefficients. If q is any function of *O = 9 aoo,0, &, I,O, 0, 0,O . > state, and recalling that since 4 is an isotropic

we shall adopt the notation Expanding

4 about the reference

function, the coefficients in its expansion must be isotropic tensors, we get

R. M. BOWEN

78

b=l

-f

and D. J. GARCIA

d=M+l

$j 2 b=M+t d=M+l

b=M+l d=M+l

P tr(~~)+~kj,~,~,M,

b

abd

The fact that 4 is an isotropic

u) +o(Ey.

bN

function insures that (4.10) and (4.12) are satisfied in

the reference state. We have used (4.11) in order to exclude the occurrence of a term linear in D in (6.7). The quantity T(g, D, W, M, u ) in (6.7) is O(@) and, as the notation a

b

b

b

eN

b

bN

indicates, depends only on g, D, W, M and u. The coefficients abc, v ,tC, & and CLsatisfy bN b b bN b nbc the symmetry conditions V = u,

abc

acb

K = abc

K, acb

h=h abc

acb

(6.8)

and P =

abc

P.

acb

If we now compute the expression for the internal energy from (4.6), and (6.7) we find

(6.9) b=l

where

ab

ba

d=l dab

N

and

r=9

ab

The stress ‘JFis obtained

d=l

(6.10)

7.

dab

from (5.10) by expanding

directly

and comparing

the

On the the~odyn~ics

of mixtures with several temperatures

79

result of the expansion with equations (4.9), (4.20) and (6.7). By following this procedure we find

(6.11) fora=l,...,M,and

Yp+++;

(p&+-)1 b=l

(6.12) b=l

fora=M+l,...,N, where

(6,13)

By use of equations (4.17) and (4.18), the linearized expressions for the heat flux q and the energy supply g are easily shown to be a a

80

R. M. BOWEN

and D. J. GARCIA

(6.14) and (6.15) In order to insure that equations (4.18) and (4.20) are satisfied for all 6, not just 9, equations (4.9), (6.1 l), (6.12) and (6.15) yield 0 ili$=O,

a=

l,...,

&a,=Ov a = M+

M,

(6.16)

1,. . ., N,

(6.17)

and iI_=0, bXlab

a=1

,...,

N.

(6.18)

Equation (2.20), when applied to (6. IS), requires that i I’=O, a=1ab

b=l,...,

N,

(6.19)

i R=O, az:1ab

b=l,...,

N.

(6.20)

and

Equations (6.18) and (6.19) are generalizations of equations (5.15) of reference [ 111. If we compute I$ from (2.13), (6.11) and (6.12) and then force (2.14), to be satisfied we find that

jlz=O forb= l,...,N-I. The diffusive force 6 can be obtained

by expanding

(6.21)

directly

and comparing

the

result with (4.8), (4.19) aid (6.7) to obtain

(6.22)

fora=

l,...,M,and

On the thermodynamics of mixtures with several temperatures

+

$

{Egrad

(trE)--grad D ab

b-M+1

81

(tr?))

(6.23)

for a=M+l,.

. ., N, where in order to insure that equation

(2.14), (with c= 0) is

satisfied, the coefficients of equations (6.22) and (6.23) must satisfy the conditions

and

(6.24) jJ=O

forb= l,...,N. Certain of the coefficients in equations (6.7), (6.1 l), (6.12), (6.14), (6.22) and (6.23) are restricted by the inequality (4.7). If we substitute these equations into (4.7) it is possible to write, after considerable algebra, the result in the form

N-l

N-l

a=1

bxl

N-l

abm

bN

(6.25)

N-l

Equation (6.25) is equivalent to the following four inequalities

a=1

N-l

b=l

ab

N-l

(6.26)

~l~~~tr(~-t(tr~)I)(~-t(tr~)I)+O(ES)

IJ.Es.Vd.8

No 1-P

5

0,

(6.27)

R. M. BOWEN N-l

and D. J. GARCIA

N-l

(6.28)

These inequalities obviously imply that certain symmetric matrices be positive semidefinite. For example (6.27) requires that the matrix whose components are #s+z) be positive semi-de~nite. These conditions are necessary but not sufhcient to satisfy (6.26) through (6.29) uniess we make sufhcient assumptions so as to drop the order terms. The linear theory we have presented here must be regarded as an approximation which holds for those values of the independent variables sufficiently close to the reference state so as to make the order terms in (6.26) through (6.29) negligible. It should be noted at this point that we could have used (4.16) to obtain the necessary conditions mentioned above. The resulting equations are just (6.25) through (6.29) with the order terms omitted. Acknowfedgment-The research reported here was supported under Grant Nos. GU-1153 and GP-9492.

by the U.S. National Science Foundation

REFERENCES [l] C. TRUESDELL,Rend. Lincei22,33,158 (1957). [2] P. D. KELLY, Int. J. Engng Sci. 2,129 (1964). [3] C. TRUESDELL and R. A. TOUPIN, The Classical Field Theories, in Fliigge’s Handblrch der Physik, Band III~I. Springer (1960). I41 A. C. ERINGEN and J. D. INGRAM, Int. .I. Engng Sci. 3,197 (1965). [S] A. E. GREEN and P. M. NAGHDI,Arch. r&on. Mech.A~~ysis~, 213 (1967). [61 J. D. INGRAM and A. C. ERINGEN, Int. J. EngngSci. 5,289 (1967). [7] N. T,.DUNWOODY and I. MULLER,Arch. ration. Mech.Analysis 29,344 (1968). [81 1. MULLER,Arch. ration. Mech. Analysis 26,118 (1967). [91 T. R. STEEL, Determination of the Constitutive CoejtFcients for a Mixture of Two Solids, Ofice of Naval Research, Rep. No. AM-68-4. (1968). [ 101 R. M. BOWEN and J. C. WIESE, Int. .I. Engng Sci. 7,689 (1969). [ 111 R. M. BOWEN and D. J. GARCIA, Q. uppl. Math. (Submitted for publication). [12] R. M. BOWEN,J. them. Phys.49,1625 (1968). [I33 R. M. BOWEN,Arch. ration. Mech.A~Iysis~, 97 (1969). [141 W. NOLL, The Axiomatic Method, With Special Reference to Geometry and Physics. North Holland (1959). [151 B. D. COLEMAN andW. NOLL,Arch. ration. Mech.Analysis 13,167 (1963). [161 C. TRUESDELL and W. NOLL, The Non-Linear Field Theories of Mechanics, in Fliigge’s Handbuch der Physik, Band Ill/jr. Springer (1965). [ 171 R. M. BOWEN, Arch. ration. Me&. Analysis 24,370 (1967). [IS] C. TRUESDELL, Rend. Lincei44,381(1968). El91 C. TRUESDELL, Rational Thermodynamics, A Course of Lectures on Selected Topics. McGrawHill ( 1969). (Received 2 Muy 1969)

On the thermodynamics

83

of mixtures with several temperatures

R&urn&La presente etude conceme l’elaboration dune theotie thermomCcanique d’un m&urge de constituants ayant chacun sa propre distribution thermique. La theotie s’etend aussi aux effets de I’blasticite non fin&ire, a la conduction de chaleur suivant des lois non lin&ires, de la viscosite et de la diffusion non lidaires. Une version hnCarisCe de la theotie &&ale est egalement presentte. Znsammenfassung-Diese Arbeit betritTt die Formuherung einer thermodynamischen Theorie einer Mischung, deren jeder Bestandteil sein eigenes Temperaturfeld hat. Die Theorie enthiilt such die Effekte nichtlinearer Elastizitlit, nichtlinearer W%meleitung, nichtlinearer Viskositat und Diffusion. Eine linearisierte Version der allgemeinen Theorie wird such vorgelegt. Sommario- Nell’articolo si tratta la formulazione di una teoria termomeccanica di una miscela in cui ciascun componente ha un proprio campio di temperatura. La teoria presenta anche gli effetti dell’elasticita non hneare, della conduttivid termica non lineare, della diffusione e viscosit& non lineare. Si presenta pero anche una versione linearizzata della teoria generale. HaCTOIlUleM

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