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A theory of mixtures of nonsimple fluids
hr. 1. EngngSci.Vol. 32, No. 9, pp. 1423-1436.1994 Pergamon
Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved omo-7225/94
A THEORY
OF MIXTURES
OF NONSIMPLE
$7.00 + 0.00
FLUIDS
D. IESAN Department
of Mathematics, University of Iasi, 6600 Ia$, Romania
Abstract-A theory is developed for binary mixtures of nonsimple fluids. The main aim of the paper is to derive the basic equations which govern the mechanical behaviour of a mixture of two compressible Newtonian fluids. The linearized version of the theory is presented. A representation of Galerkin type is established and the fundamental solutions in the case of steady vibrations are derived.
1. INTRODUCTION
The origin of the modern formulation of continuum thermomechanical theories of mixtures goes back to papers of Truesdell and Toupin [l], Kelly [2], Eringen and Ingram [3,4], Green and Naghdi [5,6], Mtiller [7], Dunwoody and Mtiller [8], and Bowen and Wiese [9]. For the history of the problem and the detailed analysis of various results on the subject we refer to the works of Bowen [lo], Atkin and Craine [ll], Bedford and Drumheller [12] and Rajagopal and Wineman [13]. The classical theories of single phase continuum mechanics become inadequate when the wavelength of the imposed motion is comparable to the average dimension of the intrinsic discontinuities in the material. In order to extend the classical theory to include such phenomena, Eringen and his co-workers (see, for example, [14]) have developed the theory of micromorphic materials. In [lS, 161, Twiss and Eringen have established a theory for micromorphic and micropolar mixtures and discussed the dispersion of plane waves. Some authors have proposed that the constitutive functions should include higher gradients of certain state variables, but, as was first pointed out by Demiray and Eringen [17]: “the inclusion of higher gradients leads to long-range forces (i.e. the couple stress theory) which are excluded at the start”. We note that in the context of single phase continuum mechanics Toupin [18], Mindlin [19], Bleustein and Green [20] and Vianello [21] have formulated a theory of nonsimple media, which is characterized by the inclusion of second order gradients of the state variables and hyperstress tensor in the basic postulates. The existence of hyperstresses independent of tractions is essential to this theory. In recent papers devoted to the theories of mixtures (see, e.g. [22-261) the authors consider gradients of state variables of the first and higher order as independent constitutive variables, in the absence of hyperstresses. It is known (cf. [20]) that for fluids and materials with internal variables is not possible to prove from the entropy inequality that hyperstresses must be present when the velocity gradient of second order is included in the constitutive equations. But, as was pointed out recently by Rajagopal, Massoudi and Ekmann [26, p. 2421, “Introducing higher gradients in the theory always introduces the attendant difficulty of specifying additional boundary conditions”. In the absence of the hyperstresses there is no indication as to what the appropriate additional boundary conditions should be. In the present paper, within the spirit of [17], we use the results established in [18-21,27,28] to derive a theory for binary mixtures of fluids, in which the hyperstresses are taken into account and the constitutive functionals depend upon the first and second order gradients of densities and velocities. The existence of hyperstresses leads to new equations of motion. In this theory the counterpart of Navier-Stokes equations form a system of partial differential equations of fourth order. We note that in [27,28] Green and Naghdi have proved that in the case of mixtures with a single temperature there are a number of alternative statements of the field equations which should produce equivalent results. In particular, by using the mixture 1423
D. IE$AN
1424
invariance (cf. [24]) they showed that these other formulations include the equations derived in [5-7,9]. In Section 3, we study the balance laws and derive the equations of motion. Section 4 discusses the constitutive equations. Here we establish the counterpart of Navier-Stokes equations and formulate the boundary-initial-value problems. In Section 5, we derive the linearized version of the theory. Finally, in Section 6 we establish a representation of Galerkin type and derive the fundamental solutions in the case of steady vibrations.
2. NOTATION We consider a mixture of two interacting continua s1 and s2. We assume that each point within the mixture is occupied simultaneously by s1 and s 2, and we refer the motion of the continua to a fixed system of rectangular Cartesian axes. We use vector and Cartesian tensor notation with Latin suffixes having the values 1,2,3. Let X and Y denote the reference positions of typical particles of s1 and s2. The motions of s1 and s2 are represented by x = %(X, t),
y = ?(Y, t),
(2.1)
respectively. These motions are assumed to be sufficiently smooth for the ensuing analysis to be valid. We assume that the particles under consideration occupy the same position at time t, i.e. x = y. The velocity and acceleration vector fields at time t are d(‘)x
dc2’y
u=-&-9
(2.2)
V=y’
where d”)/dt denotes the differentiation with respect to t holding X fixed in continuum sl, and dc2)/dt denotes the differentiation with respect to t holding Y fixed. Let p1 and p2 be the densities at time t of s1 and s2, respectively. We define total density p by p = p1 + p2. The rate of deformation tensors at time t are given by dij
=
i
(Ui,j
+
s U(i,j),
Uj,i)
where a comma stands for partial differentiation define the operator DIDt by
.fij = l)(i.j),
with respect to spatial coordinates.
DQ=aQ+w,aQ
We also
.
rG=Q,
at
Dt
(2.3)
(2.4)
where w=
L(p1u + p2v).
(2.5)
P
3. BALANCE
LAWS
By generalizing the energy balance postulate enclosing a region V, we postulate d dt”I[
given in [27,28] for a fixed closed surface A
pi UlUj +
p,U,+p,U,+2@,u’+p,S)]dV+~[
= (plFl’)Ui +P2Fj2)IJi +pr) I V
In this equation,
(tj’)Ui
dV +
+
p2U2Vi
tj2)Ui
+
+
5 (pIU2Ui
h(,!‘Ui,j
+
+
hjf’Ui,j
p*V%i)
+
4)
1
?li dA
dA.
(3.1)
I A
U, is internal energy per unit mass of constituent
s,, Ff”’ is the external
1425
Theory of mixtures of nonsimple fluids
body force per unit mass applied to s,, tj”) is the partial stress vector associated with s,, r is the heat supply function, 4 is the flux of heat across A per unit area and unit time, hj:’ is the partial hypertraction associated with s,, and ni is the outward unit normal to A. Without loss of generality we assume that the dipolar body forces are absent (cf. [29]). The method used is essentially the same as that given by Green and Naghdi [5,27]. The equation (3.1) may be written in the form ”
d”‘U --+p2dt =
d”‘U, +p~~~“~~+p2~~2’V~+~~(U~+~~2)+~~(U~+~~)]d~ dt
IV
(plFl’)ui + pzFj’)ui + pr) dV +
IA
(ti’)Ui + ti(‘bi + hj/)Ui,j + h~~‘Ui.j+ 9) dA,
(3.2)
where ml
=
2
+
lll.2
(Plui),i,
=
2
+
(3.3)
(pzUi),i*
We suppose that the body has arrived at a given state at a time t through some prescribed motion. We consider a second motion which differs from the given motion only by a constant superposed rigid body translational velocity, the body occupying the same position at time t. ,, , r and q are unaltered by such superposed rigid We assume that pa, U F@’ - a@‘, t!“) I@) body velocity. If we iie ;he equatidn’(3.2) with Ui and Vi replaced by Ui + ai and Ui+ ai, respectively, where ai is an arbitrary constant, then we obtain Ui
[pl(F!” - a,“‘) + p2(Fi2) - a!“) -
PTZlUi -
+I
WZzUi]
dV (ml+mz)dV=O.
(t!1’+t~2))dA}-~~iai/
(3.4)
V
A
From (3.4) we get (ml + m2) dV = 0, v [pl(Fi’) - al’)) + pz(Fi” - ~1”) - mlUi - LPZzUi] dV +
IA
(ti” + t!“) dA = 0,
(3.5)
for an arbitrary domain V. It follows that
(tji + sji),j + pJ+”
Ml +mz=o, $1) + t!2) = (t.. + s..)n.
(3.6)
i p2FI;) = p:bjl’
(3.8)
(3.7)
1 p:hj2’ + rnlUi + m2ui,
where tji and Sji are the partial Cauchy stress tensors associated with s1 and s2, respectively. Taking into account (3.6)-(3.8), the equation (3.2) reduces to
J-F V
Pr-Pl
d”‘U,
d”‘U, -dt
”
--m*lJl
-m2U2
dt
+ i (PlFf”
-
p2Fi2’)(Ui
-~(p~lZ{1’-p2U~2’)(Ui
-
J-F
Vi) + d (tji + Sji)(Ui
+ A i (ti” -
On applying (3.9) to an arbitrary tetrahedron, T(Ui
-
Vi) + (hJ/’
-
/L,PZ,)Ui,j
t$“)(Ui
-
-
Ui)
+ Ui),j
1
Vi) + hj/‘Ui,j
5[tj”
+ hjf'Ui,j + q] d/4 =
0. (3.9)
we get + (hjf’
-
v,jin,)Ui,j
+ 4 -
where qi is the heat flux vector, p,ji and q,ji are partial hyperstress constituents sl and s2, respectively, and T =
dV
- ti(‘) - (tki - S&)nk]e
qinj
=
0,
(3.10)
tensors associated with the
(3.11)
D. IE$AN
1426
With the help of (3.8) and (3.10), equation (3.9) reduces to (3.12) where PU
= PIG
Q, = (PI Vj”U, + p2V$*‘U*),;,
+ P2Gt
VW I = u.I - w.If
V!” I = u. - w.I>
(3.13)
and
?r;= ; (tji - Sji)*j + ;
pl(Fj” - a,“‘) -
$p*(Fj*’ -
a(*‘).
Let us now consider a motion of the body which differs from the given motion only by a superposed uniform rigid body angular velocity, the continua occupying the same position at time t. We assume that pa, U,, q;, x;, t;j, S;j, pr;i, qr;j and r are unaltered by such motions. in this case Ui and Vi are replaced by Ui + E;jkb+k and vi + E;jkb+k, respectively, where &ii&is the alternating symbol and bi are arbitrary constants. From (3.12) we obtain (Zii + ai;)Ej;pb, = 0, for all arbitrary constants b,. It follows that Zji + Uji
(3.16)
= Zij + aij*
In view of (3.16), the equation (3.12) becomes PIQ + Q, - pr - qi,i = Xi(Ui- Vi) + ‘rc/jjd;j$ U(ij)Jj f Zbi]rij f
&blj;Krji
+ r)rj;Yrj;f
(3.17)
where Kijk
r..
‘I
=
,!!)
11
-
,!?)
=
fJk,ijr
?ijk
1
11 ’
OJi;’ = i (Ui,j
We restrict attention to a single temperature inequality is (cf. [ll, 27,281)
-
=
Uj,;)
vk,ij,
(2) -
= U[i,j]>
Oij
-
u[i,jJ*
(3.18)
1”(>0) for the mixture. The entropy production
where q4 is the entropy per unit mass of the constituent combine to yield
s,. The terms involving qa in (3.19)
r (~4 + W dK
JA
where Y = (PI V”‘7jl C p*Vj2’r/2) Ii * t
(3.20)
If we now apply the entropy production inequality to an arbitrary tetrahedron bounded by coordinate planes at xi and a plane with unit normal nk, we obtain q - q;tZ;3 0. From this, following Green and Steel [30], we get (3.21) 4 = Qini. Thus, the local form of the entropy production
inequality is
pTf/ + T’IJ - pr - q;,; + iq;T;
2 0.
(3.22)
Let A,=
ua - Trlnt
PA = PEAI
+
~42
=
P(U
- 7%).
(3.23)
1427
Theory of mixtures of nonsimpie fluids
From (3.12) and (3.17) we find that - p(A + Pq) - Z + $qtTi
+ Iti(Ui - Vi) + r(ij,U’v+ So,jj f r~~lI’~f
PrjiKrji
+ qaYr/i
20,
(3.24)
where qi* = qi - T(p, Vi”Tj* + p*Vj2)v2).
2 = (pi V{‘)At + pzV$2)Az),i,
(3.25)
Following [27,31] we introduce the notations t/j =
S@= $j - $&ij*
7Ei= ifi + p,ip
(3.26)
where (3.27)
rp= ?(A,-A,)=p,(A,-A). The inequality (3.24) then becomes 1 -P(A + fn) + ,qrTi
f #i(Ui - Ui) + Ztij,dii + Ci;,ij,$j+ rtijlI?ij+ FijkKijk + q[jkyijk 30,
(3.28)
where (3.29)
ZQ= xj + p$cL,ij,s, ~ij = &j + ?$f$j,re It follows from (3.8) and (3.15) that tiw-ri+p,F:*)-~
’ (m*Ui + m2Ui) = plaf’),
Sji,j+ ni + p2FIZ’- ’ mlUi + mZUi)= p2Uj2)* 2( Moreover,
(3.30)
using (3.23), (3.25), (3.26) and (3.29), the energy equation (3.17) becomes
p(A + ?n + T4) - pr - q?i = $(&i - vi) + ?{ij,dij + @,ij,Aj + r[ji)r@ + /kijkKijk+ qijkyijks (3.31) We observe that the parts of tji, Sji and Ei in (3.26) which arise from 9 do not contribute equations of motion (3.30), the energy equation (3.31) and the total stress.
4, CONSTITUTIVE
We assume that the mass of each constituent
to the
EQUATIONS
is conserved, so that
ml =m2=0,
(4.1)
and that the constituents of the mixture are Newtonian compressible fluids. We exclude the heat conduction phenomena, and assume that A,, qa, ii;:, $j+ Ei,i,/hijk, %jgkand qT are nonlinear functions of p,, p2, pl,i, p2.i and T, and linear functions in the variables pl,ij, pz,ij, & = ui - ui, dij, $1, I,, Kijk and yijk* The axiom of objectivity reqUireS that the constitutive functionals must be hemitropic. In what follows we restrict our attention to isotropic fluids. It follows from (3.28) that A =A(P~, ~21 p~,it P2,i, 7’), and
r)=-2,
aA
qp=o,
(4.2)
D. lE$AN
1428
In view of (2.4), (3.2) and (4.1) we obtain PA2 = -PPZLr
PPI = -PPldrr - p2pl,i&, PP1.i = -P(P~,kuk,i
-I pt,drr
-P(P2,kUk,i
+ P2,i.L
Pb2.i
=
+ PlP2,iViF
PI Kiss) - P2Pl,ijy,
+
(4.4)
+ P*Yiss) + PI PZ,ijV;:-
From (4.3) and (4.4) we conclude that A =&PI,
PZ, ~1, K2, T),
KI = Pl,iPt.i,
(4.5)
K2 = P2,iP2,it
and Qj, = -pq + z&,,
G(jjj = -4ij + ffQ>,
&=Qi+n:, Pij =
9ij =
&jk =
PPI g+
(
-sijk
2PK12
PP2 g
>
+ 2PK2 2
21
+
b$k,
4j + 2P e
=
PI e
lijk
=
P2
2
@l,iajk
+
Pl,jaik),
(P2,iajk
+
P2,jsik),
=
-rijk
-t
q$k,
Pl,iPl,j, 1
8, + 2P g
Vi=P,Pl,i~-PIP,,i~-P1~Kl,i+P,~K2,i,
%jk
%jk
2
p2,ip2,jy
1
2
(4.6)
2
where the functions 7fij),a&,, p$k, q$;, I@ and Tuilare given by z&j = h,&rSjj + 2pldij + A,f,sij + 2/++$j, afij, = h,.IZ$, + 2pzf;j.+ h&,&j + 2Pddij, Z[ji]
@ = d& + bl K,,j+ b2Kirr+ dl ‘Yrri + dzyirr,
= Z[ ji] = CT,
1 F$k
=
i
@I(Kmiajk
+
Krr$ik)
KjrrSik)
$
i
$
?jrraik)
f
k,SijVk + kz&ikF + k,6jkq,
+
yjrraik)
+
pl(?rriajk
+
+
+
6YI(Krrtajk
Kjrraik)
LZiY3K,rk6ij
’
+
-6 i
f
2Kkrrhj
+
2Ykdij
@3yrrkhj
2v3Yrrkaij
+
+
263KrrkSij
2ff~Kijk
’
+
+
Yrrj&k)
zv4Yijk
+
@2(Kdjk
f
+
v5(Ykji
+
+
f
Kkij)
+
ykij)
PZ(yidjk
&(ykji
+
Krrjaik)
&LKijk
(Tg(Kkji
+
2e4yijk
+
2Kkrraii
+
f
Ytcij)
f;(Kirrajk
fCS(Kkji
+
Kkij)
(4.7)
4h,Sijvkk+2~ik~++h3~jkV;..
Here 8, is the Kronecker delta, and the constitutive coefficients are functions of ~1, PZ, K3
=
Pt,iP2,i
and
r
KI,
~2,
1429
Theory of mixtures of nonsimple fluids
The entropy inequality (4.3) yields the following restriction 1rF6 + Z$j,dij + ob,fij +
CI’ijTv
+ /L&Kijk
+ qJky/jk 2
0.
(4.8)
In view of (3.27) we find that Q = b + 456
+
6fir
63~1.m +
+
where the coefficients are prescribed functions of pl, p2, (3.14), (3.26), (3.29) (4.6) (4.7) and (4.9) that
K~,
K2,
~~
QS,
+
A,drrSij + 2pldij + h,f,6,
+ 2p3&j +
Q6,
+
A2f,6,
h,d,r6,
+ 2pddij -
rij = -pij + -
(4.9)
54~2,ss,
Uij
=
-qij
G
=
Qi + & +
+
2p2Jj
+
and T. It follows from
CT,, CT,,
(4.10)
Q,i + bl Krri + b2Kirr + dl ‘Yrri + d,Yirr.
Thus, the constitutive equations are given by (4.2), (4.5)-(4.7), (4.9) and (4.10). We note that in the present theory the free energy can depend also on the gradient densities. Taking into account (4.2) and (4.6), the energy equation (3.31) reduces to pT7j = Z$dij + U$fij + ZI$;j + ?$V, +
CL~~Kij~
+
TlJkyijk
+
pr.
(4.11)
In view of (3.14) and (4.1), the equations of motion (3.30) may be written in the form rji,j - /.Llji,rj- Ici + ~1FI” = plUj’), Uji,j- qlji,rj + Xi + p2Fi2) = p2aj2).
(4.12)
From (3.3) and (4.1) we get $
+ (Pl”i),i
2
= Or
+ (pzUi),j
= 0.
(4.13)
To the above equations we must add boundary conditions and initial conditions. Let B be a regular region of three-dimensional Euclidean space bounded by the smooth surface LIB. It follows from (3.1) (3.7), (3.10) and (3.21) that the total rate of work over the surface tIB is P = I,,(f!‘)Ui
+ ti(2)Ui +
hjj)Ui,j
+
h~f'Ui,j)dA
=
l,B(ljiU;
+SjiUi
+
/aLjrsU*,,
+
qjrsVs,r)nj
dA.
(4.14)
Following Toupin [18] we can write P =
I aB
(Sjl)Ui + Sj2)Ui+ R$“DUi + Rj2)Dq) &I,
(4.15)
where Of =fini and $I’
= (rji - Prji.r)nj
- nrDsllsri
+ /&si@rs - Pmmnrns)~
Sl” = (flji - l)rji,r)nj - Osrlsri + 7)&m R!”, = CLi-s,.n I n s, Rt2’ = 77 .n I IS, r n S’
- PmdVs)~
(4.16)
form of the surface aB and Dj/.tsjriis the surface gradient
In (4.16), Pij is the second fundamental of the hyperstress, i.e.
DjPjri = Pjri,j - njD/-+. It follows from (4.15) that P = i I,
[(Si” + S$“)(ui +
Vi)
+
(Si”
+
Many types of boundary
-
Si(2’)(Ui
-
Vi)
(RI” + Ri2’)D(ui + Vi) + (RI” - R$2’)D(ui - Vi)] dA.
conditions have been suggested (cf. [ll, 121). The first boundaryby the following mechanical boundary conditions
value problem is characterized Ui
=
fii,
(4.17)
Vi
=
Oi,
Dui = A,
Dui=gi
on
aBXZ,
1430
D. IE$AN
where 1zi, 6(, A, gi are prescribed functions and I = [0, a). In the second boundary-value problem the mechanical boundary conditions are $1’ Rj” + Ri2’ = &, & = Vi, D&=& on MXI, I + $2’ = p.It where pi, cl/, pi and di are given. The thermal boundary conditions have the classical form (see f321). We consider the following initial conditions P,(xt 0) = p?(x), Ui(X,
0) z
P2@,
Up(X),
0)
r)(X,
=
Ui(X,
PW,
0)
0) =
U:(X),
x E B,
= TO(X),
where pf, pf, up, up and 71’are prescribed functions. Let us assume that the constitutive coefficients Ai, cli (i = 1, 2,3,4), c, CY,,&, v~,f; {s = 1,2,3,4,5) are constants. Then, in view of (2.3), (3.18), (4.7) and (4.10), the equations of motion (4.12) can be written in the form (7, - PZ~A)AU,+ (22 - m2 A)U,ri f (~3- m,A) Au) + (24 - m4A)u,, - d(ui - Ui)+ Ai + PIFir’ = praf’)* (xl - rlA) Aui + Cxz- rzA)u,ri + 013- r3A)Aui f (~4 - rdA)u,,ri + d(ui - Vi)+ I’i + p2F12)= PZU~~‘,(4.18) where A is the Laplacian, Ai = skji,kj
-
Pji,j-
Qi,
ri = rkji,kj
-
4ji.j
Qit
+
(4.19)
and
z3 =
y3
+
;c-
dl + kl,
Zq=A3+p3-!c-dd2+k2+k3, 2
1
X,=pa+;c+bl-hl,
xz=h4+C14
-2cCbZ-hZ-h3,
X3=p++dl+hir
X4=&+p2+;c+dp+h2+h3, m2 = 2(al f a2 + as),
ml = 2(a3 -+-a4),
m4 = 2@1+ P2 + Ps), rl = 2(6x + fbh r4 = 2(v, + v2 + vs). r3 = 2(Y3 -t “4)~
5. LINEARIZED
m3 = 2(& + &a), 5
=
WI
+
32
+
Sd
(4.20)
THEORY
In this section we consider the linearized version of the theory developed in Section 4. We assume that there exists an equilibrium state of the mixture in which the constituent densities pl, p2 and the temperature T have the uniform values py, p! and To, respectively. In what follows we assume that the motion takes place under isothermal conditions. We introduce the notations th=Pc-Pt
Jtz=pz-P%
(5.1)
and assume that 3/i, I@~, II and v are small, i.e. JIr = &I&,e2 = E&, u = EU’and v = EV’,where E is a constant small enough for squares and higher powers to be neglected, and ++I;,JI;, u‘, vf are independent of E. To the second order, A is taken in the form A =A,+ YI& + KI
=
@l.iJll,i,
~292+$71J;:+
K2
=
?#‘2,i’!‘2,i,
fl2&+hh3/2+
~4~1
+GK2),
(5.2)
1431
Theory of mixtures of nonsimple fluids
where Ao, yI, y2, us (s = 1,2, . . . , 5) are prescribed constants. It follows from (4.6), (4.19) and (5.2) that Pij = (a10 + all+l +
9ij = (a2o + a214+1
a12+2cz)sij9 1
Qi = (c2+2Cjk
=
i
ri
=
B
Cl +l).i,
&5($2.i~jk
W2.i
sijk
+
-
(b21
i
=
pya,(Jll,iajk
Ai
I(I2,jhk)t
J/l
+
=
a
+
$l,jsik),
A+l,i
-
(bll
JIl
+
bl2+22),i,
(5.3)
b22J12),i,
where a11 = Yl(PO
a10 = P0Ph9 a12 = YIP:
+ P”Ph
Y2(P0 + Pi)
a22
=
hl
= a11 -cl, a
=
+ Pa
020
=
P0Ph
=
a21
Cl = Ph
+ P0P%2> bl2
+ Pod%
a12
+
b2l
c21
=
a21
=
Y2P!: + P0P%39
c2
=
+
Cl7
PO=P(:+P;,
PYY2, b22
=
a22
-
~2,
P = Ph.
Ph
By (4.9) and (5.1), Q = b + 51&r+
52.h
+
~3’7h.s~
+
(5.4)
64@2,ss,
where the coefficients are prescribed constants. Here and in what follows we denote the constitutive coefficients and their values in the equilibrium state by the same letters. From (3.14), (3.26), (3.29), (4.6) and (4.7) we get rij =
-(UIO+~II$‘I
+a12$2
Q)aij +A~drraij +2~1dij+h3f,6ij +2/~3f;, +Crijt
-
aij=-(a20+UZI+I+a~$2 + Q)Sij + h,f,aij +~/Jz& +b&r6ij +2/~4dij -Crijt ni= (czICIZ - CI'!'I + Q),i + d% + blKrri + b2Kirr + d,Yrri + &3/irr, hjk
=
-sijk
+
qijk
p;k,
=
-rijk
+
(5.5)
q$k,
where the constitutive coefficients are constants. The entropy is determined energy and initial conditions. It follows from (4.13) that a*, at +
Ph,i
a*2
at
= 07
+ P:Ui,i =
by the equation of
0.
(5.6)
The equations of motion reduce to r1i.j-
ffjisj
-
~kji,kj
?)kji,kj
-
Ei
+
+
Zi
+
p~FI” = p’: ~, p!Fi2’ = pi:.
(5.7)
From (2.3), (3.18), (5.5) and (5.7) we obtain the following form of the equations of motion (ri -
mi A) Aui + (22 - mzA)u,,ri + (~3 - m3A)Aui +
(~4
(~2
-
r2Ab,n
-
+
(x3
-
r3A)
Avi
+
m4A)u,Vi
A+i -bll$1
-d(u,-ui)+((~ (XI - M) AUi+
-
(x4
-
-b12+2),i +p~F~“=p:~p
r4A)vr,ri
d(ui - vi) + (p A$+ - b21#1- b22t,&),i+ p;Fj2)= pzz.
(5.8)
Thus,we have obtained the system of field equations (5.6) and (5.8) for the unknown functions #i, &, UI and Vi*To this system we must add boundary and initial conditions.
D. IESAN
1432
6. FUNDAMENTAL
SOLUTIONS
In the first part of this section we establish a solution of Galerkin type of the field equations within the linearized theory. Then, we use the Galerkin representation to determine the fundamental solutions of the field equations in the case of steady vibrations. Galerkin representations and fundamental solutions in the context of a single phase polar fluid were presented in various papers (see, e.g. [33,34]). In what follows, for convenience, we consider the system Dllu + D12 grad div u + D13v + D14 grad div v + D,5 grad JI1 + D,6 grad & = -p’$(“, Dzlu + Dz2 grad div u + Dz3v + Dz4 grad div v + D,, grad J/l + Dz6 grad J12= -p;F(*),
a*, at + py div u = -M(l),
a** at + p; div v = -MC*),
(6.1)
where D,,=(r,-m,A)A-p:;-d,
D12 = r2 - m2A,
D,, = (q - m,A) A + d,
D1,+=r4-m4A,
D21 = (xl - r,A) A + d,
022 = x2 - r2A,
D23=(X3-r3A)A-pgi-d,
024
=
016
=
-h*,
D15 = CYA - bl,,
X4
025
-
=
-b21,
026
r4A,
=
0
A - b2*.
Clearly, if M(i) = M(*) =O, then the system (6.1) reduces to (5.6), (5.8). We introduce notations Aa2
=
Da2
;
P:&s,
-
Au4
=
Dcr4
+
023
-
+
013
a
-9
212
at
Z2, = A2*A + D2, d at’ T,, = D Cm,
THEOREM6.1.
Z11&(4+a)
-
the
P;&,
f
a &I = &A
(6.2)
=
Al4A
LA,D23
-
Dd21,
=
Ao2Z11
-
&4Z21>
at
>
Z~Z=&A+DI,$. Ga
z12D2(4+a),
=
-
I-I =
=
z22z11
ca2
=
-
‘%x4222
z22D2(4+a)
-
Z21D1(4+a)
z12z211 -
(a
&2&2,
=
1,2).
(6.3)
Let
u = -llD23G + IID13H + (Cl1 023
-
Czl D,,)grad div G af
+ (C12D23- C22D,3)grad div H + T,, grad at + T12grad $, v = llD21G - IIDllH + (C21Dll - C,,D,,)grad
+
(C22Dll
-
div G
C 12 D 21 ) g rad div H + T,i grad $+
I,$ = pyZll D div G - P?Z~~D div H - (II + p?T,,A)f
T2*grad 2, - pyT12 Ag,
J12= -P~Z~~D div G + p!DZ2* div H - piT2, Af - (II + p!T,,A)g,
(6.4)
where G, H, f and g satisfy the equations DIIG = p:F(‘),
D IIH = p;F’*‘,
Then II, v, JI1 and x2 satisfy the equations (6.1).
II $= M(l),
II $ = M(2).
(6.5)
Theory PROOF.
of
1433
mixtures of nonsimple fluids
It follows from (6.4) that
D, 1~ + D,2 grad div u + D,sv + D14grad div u + Q5 grad ++I+46 grad +CIZ
= -DI-IG + {(D,4D21+
(Cz,D,l -
+ #42013
&2023)n
+ (cl1023
+ 04)
+ pZ1445
cl,DZl)(D,3 -
&&,)n
+ (c22&
-
+ {T,,(D~~
+ QUA)
+ { T2(D11
+
+ (c12O23
-
+ ~~~A)
-
c,2~2,)(~13
$
+ 7i4h3
-
c21~13)(011 -
~22°13)(011
i
DlzA) ; + q2(D13 + D,J)
-
i -
~Z2~~k&wddivG + 0124
&%2D&5
+ 4A)
+ D12A)
+ diQd22WgraddivH
&(n
+ dW)
p:D,s
TA
-
- Ddn
d%J&Gwdf
+ p%ANwd
g.
If we take into account (6.3), then we obtain D,,u + Or2 grad div u + Di3v + D14grad div v + D15grad I+Q~ + D16grad Q2 = -DIIG. Similarly, we find that D2iu + D22grad div u +
D23~ + D24
grad div v + 025 grad @I+ 026 grad $2 = -DJ%
af
z+pydivu=--II--,
w2
ag
T+p;divv=
With the aid of (6.5) we obtain the desired result.
-II%.
0
We now consider a mixture which occupies the entire three-dimensional Euclidean space. In what follows we use the solution (6.4) to determine the fundamental solutions of the field equations for the case of steady vibrations. We suppose that F(“) = Re[F*@(x)exp(
-ior)],
M’“’ = Re[M*@(x)exp(
of vibration and i = (-l)ln.
where w is the frequency form
u = Re[u*(x; w)exp( -iot)],
-it&)],
If we seek for solutions to (6.1) of the
v = Re[v*(x; o)exp( -it&)],
+m= Re[$,*(x; w)exp( -hot)], then the field equations introduce the notations
reduce to a differential
D& = (q - mlA)A + iopy - d, AZ2 = -iwD a2 Z;2 = A&A -
-
p:D,s,
u*, o*, I,$,*.We
Dz3 = (x3 - r3A)A + iop; - d,
AZ4 = -hoDa - p:Dp6, Z,*, = A&A - iwD&,
1oD13,
TL = Z&Dw+a) -
system for the amplitudes
Zif2D2(4+aJ,
T%
Z?, = A&A - ~oD;~,
Z,* = AT2A- ioDT,,
= Z2D2c4+,,
-
D* = D:ID;3 - D,3D2,,
l-I*= z,*,z:1 - ZT2Z&,
C,*, = Az2Z5 - A:,Z;,,
C:2 = A:4Z;2 - Az2ZT2.
Z&4~~+n~,
If we assume that G = Re[G*(x; o)exp( -iwt)],
f = Re[f *(x; w)exp(-iot)],
H = Re[H*(x; w)exp( -it.&)], g = Re[g*(x; w)exp( -iot)],
(64
D. IE$AN
1434
then (6.4) leads to the following representation u = -II*D&G*
+ II*&H*
of Galerkin type for amplitudes
+ (C:,D,*, - C&&)grad
div G*
+ (CWT3 - C%&)g rad div H* - ioTT, grad f * - iwT& grad g*, v* = II*D,,G* - II*DTIH* + (C$,Dh - C:,&)grad
div G*
div H* - iwT& grad f * - ioTz2 grad g*,
+ (C&DTI - CT&)grad
$7 = pyZT,D* div G* - pyZ&D* div H* - (II* + p’lTf,A)f* - p?TT2 Ag*, $,T = -p;ZT,D*
div G* + piD*Z,*, div H* - piT& Af * - (II* + p!T&A)g*.
(6.7)
The functions G*, H*, f * and g* satisfy the equations D*II*G* = pyF*“‘,
D*l-I*H* = p!$*‘2’,
wfl*f * = iM*(‘),
on*g
= iM*t2).
(6.8)
From (6.3) and (6.6) we get D* = mIdA + m,,A3 + m12A2+ m,,A + ml,,
(6.9)
where m 13 -
ml4 = mlr3 - m3r1,
d(m,+m3+h+r3)+
m 12=
m3xl
+
rlz3
-
rlr3
-+x3,
zlX3-X123-iiw(mlp~+r3~~),
m 11= -(z, +x3 + 23 + xl)d + io(rlp;
+ x3p3,
m 1o= -pyp!w2 - iwd(p? + pi).
(6.10)
The operator II* can be expressed as II* = mz4A4+ m23A3+ mz2A2+ m2,A + mzo,
(6.11)
where m24 m 23 --
m 22
=
=
-m14w2
-m13w2
-
lZl20
-m12w2 - izllo +
m20
--w
=
2
+
ml4 +
-
m1n24
m3G2
+
nl2n54 -
+
(nl2n24 +
n24tIy2
ny4ni2,
-
-
-
n 14n;2
-
-m1102 -
=
n 24
=
’
lor4
-
+
rd4
P%
z12 = -
n24 zln24
zll
+
x3n~2 + n12(iwp!j- d) - z3ni2 - dn22 -
xlfh4,
-
Ob
~2
mld4 =
22 -
-
zlnfii4
D* = m14(A + &(A + &)(A + &)(A + E:),
and ~f+~ (i = 1,2,3,4)
are the roots of the equation -
m23Y3
+
m22Y2
-
mzl
Y +
m20
=
0,
are the roots of the equation m14y4
+
(iop? -
+ iw(pyng, + p$ny2). + nS4 + ny2 + r102~)
II* = m24(A + &:)(A + &$)(A+ &)(A + E:),
-
ml3y
3
+
m12y2
-
mlly
+ml0=
0.
iw4,
r3nY2
xl& - w&
It follows from (6.8) and (6.10) that
m2,Y4
ZZ~OW,
ny2 = pyb,, - iwz2, n22 = iwr2,
x3n12
where ~3 (S = 1,2,3,4)
nLn22,
0 _
+
zlo= -d(ny,
n14n22),
n?, = p;b12 - iw4,
r3n12,
23n22
-
rn,,
n12 = iwm2 - pyff,
mlo,
nX2= pyb2, - iwx2 , m3n22
izl3w
ny2ni4
q4 = iwm,,
z13 =
-
d)nt4
Theory of mixtures of nonsimple fluids
1435
In what follows we denote by sS (S = 1,2, . . . ,8) the roots with positive real parts and assume that e, # &2# - - - # tz8# E]. If F:(I) = QS, (j fixed), Fi*(2) = 0, M*(a) = 0, then we take GF = Y6,, Ht = 0, f* = g* = 0. It follows from (6.7) that Y satisfies the equation D*rI*Y = j@. If the functions q (j = 1,2, . . . ,8) satisfy the equations (no sum, j = 1,2, . . . ,8) then the function Y can be expressed as
where z;’
=
(S = 1,2, . . . , 8).
(E; - E:), fi j=l(jZs)
Let us assume that 6 = S(x - y) where S is the Dirac delta and y is a fixed point. Then we get r ti
exp(kr),
(S = 1, 2, . . . , 8)
where r = Jx - y[. In this case we have Gf = ES,, Hi* = 0, f* = 0, g* = 0, where 8 r P(: X ZjexP(i&jr).
E=-
j=l
From (6.7) we obtain + (CTlD,*, - C,*,D,,)E,jk,
u:(j) = -II*D,*,Eajk
u:(‘) = lYI*Ql ES,, + (C~~D~~ - Cf1Dz1)E,jk, ~,2*“’= -p$Z,*,D*E,j
Icr:“’ = p~Z;r,D*E,j,
If we assume that F)(l)= I
0, FFc2) = 6(x - y)S,,
M*(p)=
(6.12)
0, then we find the solutions
uzc3+‘) = lI*D13ESjk + (CTzD,j - C,*,D,,)E,jk, vzf3+j) = -II*DflEGjk
+ (C&Of1 - Cy2D21)E,jk,
#fc3+j) = -p;Zf2D*Evj,
I,#~+” = p;Z,:D*E,.
(6.13)
= 0.Then we have G* = 0, H* = 0, Let us assume now that F*(p)=O, M*(l)=6(x-y),M *(2) g* = 0 and f* = e, where
e;l = fi
(E;- E:),
j=l(jZs)
(S = 1, 2, 3,4).
It follows from (6.7) that &7) = -itoTf,e,k, $ft7’ = -(n* Finally,
if F*(O) = 0, M*(l)
IJE(~)= -ioT$e,k,
+ pyT:,A)e,
= 0, m*t2) = 6(x -
+zt7) = -p!T&Ae.
y) then we obtain the solution
*(‘I = -iuT&e,k, Uk
vf8) = -rwT&e.k, *
JI?“‘=
+j@’ = -(II*
-p?T&Ae,
(6.14)
+ p!Tz2A)e.
(6.15)
1436
D. IE$AN
The functions z&‘, I#‘), I,@“‘, I,@“’ (s = 1, 2, . . . ,8) given by (6.12)-(6.15) fundamental solutions of the system of field equations for steady vibrations. Acknowledgement--I
represent
the
express my gratitude to the referees for their helpful suggestions.
REFERENCES C. TRUESDELL and R. TOUPIN, The classical field theories. In Handbuch der Physik (Edited by S. FLUGGE), Vol. III/3. Springer, Berlin (1960). P. KELLY, Int. J. Engng Sci. 2,129 (1964). A. C. ERINGEN and J. D. INGRAM, Int. J. Engng Sci. 3,197 (1965). J. D. INGRAM and A. C. ERINGEN, Int. J. Engng Sci. 5,289 (1%7). A. E. GREEN and P. M. NAGHDI, Int. J. Engng Sci. 3,231 (1965). A. E, GREEN and P. M. NAGHDI, Int. J. Engng Sci. 6,631 (1968). I. MULLER, Arch. Rationa!,Mech. Anal. 28, 1 (1968). N. DUNWOODY and I. MULLER, Arch Rational Mech. Anal. 29,344 (1968). R. M. BOWEN and J. C. WIESE, Znt. J. Engng Sci. 7,689 (1%9). R. M. BOWEN, Theory of mixtures. In Continuum Physics (Edited by A. C. ERINGEN), Vol. III. Academic Press, New York (1967). R. J. ATKIN and R. E. CRAINE, Q. J. Mech. Appl. Math. 29,209 (1%7). A. BEDFORD and D. S. DRUMHELLER, Int. J. Engng Sci. 21,863 (1983). K. R. RAJAGOPAL and A. S. WINEMAN, Developments in the mechanics of interactions between a fluid and a highly elastic solid. In Recent Developments in Structured Continua (Edited by D. DE KEE and P. N. KALONI), Vol. II. Longman Scientific & Technical, Essex (1990). A. C. ERINGEN and C. B. KAFADAR, Polar .fieId theories. In Continuum Physics (Edited by A. C. ERINGEN). Vol. IV. Academic Press. New York (1976). R. J. TWISSand A. C. ERINGEN, Ini. J. Engng ici. 9,‘1019 (1971). R. J. TWISS and A. C. ERINGEN, hat. J. Engng Sci. 10,437 (1972). H. DEMIRAY and A. C. ERINGEN, Plasma Phys. 15,903 (1973). R. A. TOUPIN, Arch. Rational Mech. Anal. 17,85 (1964). R. D. MINDLIN, Arch. Rational Mech. Anal. 16,51 (1964). J. L. BLEUSTEIN and A. E. GREEN, Znt. J. Engng Sci. 5,323 (1967). M. VIANELLO, Arch. Mech. 41,641 (1989). M. SILHAVY, Arch. Rational Mech. Anal. 88, 135 (1985). H. GOUIN, C. R. Acud. Sci. Paris, 305,833 (1987). I. SAMOHYL and M. SILHAVY, Arch. Rational Mech. Anal. 109,299 (1990). H. GOUIN, Eur. J. Mech., B/Fluids 9,469 (1990). K. R. RAJAGOPAL, M. MASSOUDI and J. M. EKMANN, Mathematical modeling of fluid-solid mixtures. In Recent Developments in Structured Continua (Edited by D. DE KEE and P. N. KALONI), Vol. II. Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Essex (1990). A. E. GREEN and P. M. NAGHDI, Q. J. Mech. Appl. Math. 22,427 (1%9). A. E. GREEN and P. M. NAGHDI, Arch. Rational Mech. Anal. 48,352 (1972). R. D. MINDLIN and N. N. ESHEL, Int. J. Solids Struct. 8,1407 (1972). A. E. GREEN and T. R. STEEL, Int. J. Engng Sci. 4,483 (1966). A. E. GREEN and P. M. NAGHDI, Acta Mechanica 9,329 (1970). A. C. ERINGEN, Mechanics of Continua. Wiley, New York (1967). H. RAMKISSON and S. R. MAJUMDAR, ZAMM 56,197 (1976). C. V. EASWARN and S. R. MAJUMDAR, Int. J. Engng Sci. 28,843 (1990). (Second revision received and accepted 17 August 1993)
Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved omo-7225/94
A THEORY
OF MIXTURES
OF NONSIMPLE
$7.00 + 0.00
FLUIDS
D. IESAN Department
of Mathematics, University of Iasi, 6600 Ia$, Romania
Abstract-A theory is developed for binary mixtures of nonsimple fluids. The main aim of the paper is to derive the basic equations which govern the mechanical behaviour of a mixture of two compressible Newtonian fluids. The linearized version of the theory is presented. A representation of Galerkin type is established and the fundamental solutions in the case of steady vibrations are derived.
1. INTRODUCTION
The origin of the modern formulation of continuum thermomechanical theories of mixtures goes back to papers of Truesdell and Toupin [l], Kelly [2], Eringen and Ingram [3,4], Green and Naghdi [5,6], Mtiller [7], Dunwoody and Mtiller [8], and Bowen and Wiese [9]. For the history of the problem and the detailed analysis of various results on the subject we refer to the works of Bowen [lo], Atkin and Craine [ll], Bedford and Drumheller [12] and Rajagopal and Wineman [13]. The classical theories of single phase continuum mechanics become inadequate when the wavelength of the imposed motion is comparable to the average dimension of the intrinsic discontinuities in the material. In order to extend the classical theory to include such phenomena, Eringen and his co-workers (see, for example, [14]) have developed the theory of micromorphic materials. In [lS, 161, Twiss and Eringen have established a theory for micromorphic and micropolar mixtures and discussed the dispersion of plane waves. Some authors have proposed that the constitutive functions should include higher gradients of certain state variables, but, as was first pointed out by Demiray and Eringen [17]: “the inclusion of higher gradients leads to long-range forces (i.e. the couple stress theory) which are excluded at the start”. We note that in the context of single phase continuum mechanics Toupin [18], Mindlin [19], Bleustein and Green [20] and Vianello [21] have formulated a theory of nonsimple media, which is characterized by the inclusion of second order gradients of the state variables and hyperstress tensor in the basic postulates. The existence of hyperstresses independent of tractions is essential to this theory. In recent papers devoted to the theories of mixtures (see, e.g. [22-261) the authors consider gradients of state variables of the first and higher order as independent constitutive variables, in the absence of hyperstresses. It is known (cf. [20]) that for fluids and materials with internal variables is not possible to prove from the entropy inequality that hyperstresses must be present when the velocity gradient of second order is included in the constitutive equations. But, as was pointed out recently by Rajagopal, Massoudi and Ekmann [26, p. 2421, “Introducing higher gradients in the theory always introduces the attendant difficulty of specifying additional boundary conditions”. In the absence of the hyperstresses there is no indication as to what the appropriate additional boundary conditions should be. In the present paper, within the spirit of [17], we use the results established in [18-21,27,28] to derive a theory for binary mixtures of fluids, in which the hyperstresses are taken into account and the constitutive functionals depend upon the first and second order gradients of densities and velocities. The existence of hyperstresses leads to new equations of motion. In this theory the counterpart of Navier-Stokes equations form a system of partial differential equations of fourth order. We note that in [27,28] Green and Naghdi have proved that in the case of mixtures with a single temperature there are a number of alternative statements of the field equations which should produce equivalent results. In particular, by using the mixture 1423
D. IE$AN
1424
invariance (cf. [24]) they showed that these other formulations include the equations derived in [5-7,9]. In Section 3, we study the balance laws and derive the equations of motion. Section 4 discusses the constitutive equations. Here we establish the counterpart of Navier-Stokes equations and formulate the boundary-initial-value problems. In Section 5, we derive the linearized version of the theory. Finally, in Section 6 we establish a representation of Galerkin type and derive the fundamental solutions in the case of steady vibrations.
2. NOTATION We consider a mixture of two interacting continua s1 and s2. We assume that each point within the mixture is occupied simultaneously by s1 and s 2, and we refer the motion of the continua to a fixed system of rectangular Cartesian axes. We use vector and Cartesian tensor notation with Latin suffixes having the values 1,2,3. Let X and Y denote the reference positions of typical particles of s1 and s2. The motions of s1 and s2 are represented by x = %(X, t),
y = ?(Y, t),
(2.1)
respectively. These motions are assumed to be sufficiently smooth for the ensuing analysis to be valid. We assume that the particles under consideration occupy the same position at time t, i.e. x = y. The velocity and acceleration vector fields at time t are d(‘)x
dc2’y
u=-&-9
(2.2)
V=y’
where d”)/dt denotes the differentiation with respect to t holding X fixed in continuum sl, and dc2)/dt denotes the differentiation with respect to t holding Y fixed. Let p1 and p2 be the densities at time t of s1 and s2, respectively. We define total density p by p = p1 + p2. The rate of deformation tensors at time t are given by dij
=
i
(Ui,j
+
s U(i,j),
Uj,i)
where a comma stands for partial differentiation define the operator DIDt by
.fij = l)(i.j),
with respect to spatial coordinates.
DQ=aQ+w,aQ
We also
.
rG=Q,
at
Dt
(2.3)
(2.4)
where w=
L(p1u + p2v).
(2.5)
P
3. BALANCE
LAWS
By generalizing the energy balance postulate enclosing a region V, we postulate d dt”I[
given in [27,28] for a fixed closed surface A
pi UlUj +
p,U,+p,U,+2@,u’+p,S)]dV+~[
= (plFl’)Ui +P2Fj2)IJi +pr) I V
In this equation,
(tj’)Ui
dV +
+
p2U2Vi
tj2)Ui
+
+
5 (pIU2Ui
h(,!‘Ui,j
+
+
hjf’Ui,j
p*V%i)
+
4)
1
?li dA
dA.
(3.1)
I A
U, is internal energy per unit mass of constituent
s,, Ff”’ is the external
1425
Theory of mixtures of nonsimple fluids
body force per unit mass applied to s,, tj”) is the partial stress vector associated with s,, r is the heat supply function, 4 is the flux of heat across A per unit area and unit time, hj:’ is the partial hypertraction associated with s,, and ni is the outward unit normal to A. Without loss of generality we assume that the dipolar body forces are absent (cf. [29]). The method used is essentially the same as that given by Green and Naghdi [5,27]. The equation (3.1) may be written in the form ”
d”‘U --+p2dt =
d”‘U, +p~~~“~~+p2~~2’V~+~~(U~+~~2)+~~(U~+~~)]d~ dt
IV
(plFl’)ui + pzFj’)ui + pr) dV +
IA
(ti’)Ui + ti(‘bi + hj/)Ui,j + h~~‘Ui.j+ 9) dA,
(3.2)
where ml
=
2
+
lll.2
(Plui),i,
=
2
+
(3.3)
(pzUi),i*
We suppose that the body has arrived at a given state at a time t through some prescribed motion. We consider a second motion which differs from the given motion only by a constant superposed rigid body translational velocity, the body occupying the same position at time t. ,, , r and q are unaltered by such superposed rigid We assume that pa, U F@’ - a@‘, t!“) I@) body velocity. If we iie ;he equatidn’(3.2) with Ui and Vi replaced by Ui + ai and Ui+ ai, respectively, where ai is an arbitrary constant, then we obtain Ui
[pl(F!” - a,“‘) + p2(Fi2) - a!“) -
PTZlUi -
+I
WZzUi]
dV (ml+mz)dV=O.
(t!1’+t~2))dA}-~~iai/
(3.4)
V
A
From (3.4) we get (ml + m2) dV = 0, v [pl(Fi’) - al’)) + pz(Fi” - ~1”) - mlUi - LPZzUi] dV +
IA
(ti” + t!“) dA = 0,
(3.5)
for an arbitrary domain V. It follows that
(tji + sji),j + pJ+”
Ml +mz=o, $1) + t!2) = (t.. + s..)n.
(3.6)
i p2FI;) = p:bjl’
(3.8)
(3.7)
1 p:hj2’ + rnlUi + m2ui,
where tji and Sji are the partial Cauchy stress tensors associated with s1 and s2, respectively. Taking into account (3.6)-(3.8), the equation (3.2) reduces to
J-F V
Pr-Pl
d”‘U,
d”‘U, -dt
”
--m*lJl
-m2U2
dt
+ i (PlFf”
-
p2Fi2’)(Ui
-~(p~lZ{1’-p2U~2’)(Ui
-
J-F
Vi) + d (tji + Sji)(Ui
+ A i (ti” -
On applying (3.9) to an arbitrary tetrahedron, T(Ui
-
Vi) + (hJ/’
-
/L,PZ,)Ui,j
t$“)(Ui
-
-
Ui)
+ Ui),j
1
Vi) + hj/‘Ui,j
5[tj”
+ hjf'Ui,j + q] d/4 =
0. (3.9)
we get + (hjf’
-
v,jin,)Ui,j
+ 4 -
where qi is the heat flux vector, p,ji and q,ji are partial hyperstress constituents sl and s2, respectively, and T =
dV
- ti(‘) - (tki - S&)nk]e
qinj
=
0,
(3.10)
tensors associated with the
(3.11)
D. IE$AN
1426
With the help of (3.8) and (3.10), equation (3.9) reduces to (3.12) where PU
= PIG
Q, = (PI Vj”U, + p2V$*‘U*),;,
+ P2Gt
VW I = u.I - w.If
V!” I = u. - w.I>
(3.13)
and
?r;= ; (tji - Sji)*j + ;
pl(Fj” - a,“‘) -
$p*(Fj*’ -
a(*‘).
Let us now consider a motion of the body which differs from the given motion only by a superposed uniform rigid body angular velocity, the continua occupying the same position at time t. We assume that pa, U,, q;, x;, t;j, S;j, pr;i, qr;j and r are unaltered by such motions. in this case Ui and Vi are replaced by Ui + E;jkb+k and vi + E;jkb+k, respectively, where &ii&is the alternating symbol and bi are arbitrary constants. From (3.12) we obtain (Zii + ai;)Ej;pb, = 0, for all arbitrary constants b,. It follows that Zji + Uji
(3.16)
= Zij + aij*
In view of (3.16), the equation (3.12) becomes PIQ + Q, - pr - qi,i = Xi(Ui- Vi) + ‘rc/jjd;j$ U(ij)Jj f Zbi]rij f
&blj;Krji
+ r)rj;Yrj;f
(3.17)
where Kijk
r..
‘I
=
,!!)
11
-
,!?)
=
fJk,ijr
?ijk
1
11 ’
OJi;’ = i (Ui,j
We restrict attention to a single temperature inequality is (cf. [ll, 27,281)
-
=
Uj,;)
vk,ij,
(2) -
= U[i,j]>
Oij
-
u[i,jJ*
(3.18)
1”(>0) for the mixture. The entropy production
where q4 is the entropy per unit mass of the constituent combine to yield
s,. The terms involving qa in (3.19)
r (~4 + W dK
JA
where Y = (PI V”‘7jl C p*Vj2’r/2) Ii * t
(3.20)
If we now apply the entropy production inequality to an arbitrary tetrahedron bounded by coordinate planes at xi and a plane with unit normal nk, we obtain q - q;tZ;3 0. From this, following Green and Steel [30], we get (3.21) 4 = Qini. Thus, the local form of the entropy production
inequality is
pTf/ + T’IJ - pr - q;,; + iq;T;
2 0.
(3.22)
Let A,=
ua - Trlnt
PA = PEAI
+
~42
=
P(U
- 7%).
(3.23)
1427
Theory of mixtures of nonsimpie fluids
From (3.12) and (3.17) we find that - p(A + Pq) - Z + $qtTi
+ Iti(Ui - Vi) + r(ij,U’v+ So,jj f r~~lI’~f
PrjiKrji
+ qaYr/i
20,
(3.24)
where qi* = qi - T(p, Vi”Tj* + p*Vj2)v2).
2 = (pi V{‘)At + pzV$2)Az),i,
(3.25)
Following [27,31] we introduce the notations t/j =
S@= $j - $&ij*
7Ei= ifi + p,ip
(3.26)
where (3.27)
rp= ?(A,-A,)=p,(A,-A). The inequality (3.24) then becomes 1 -P(A + fn) + ,qrTi
f #i(Ui - Ui) + Ztij,dii + Ci;,ij,$j+ rtijlI?ij+ FijkKijk + q[jkyijk 30,
(3.28)
where (3.29)
ZQ= xj + p$cL,ij,s, ~ij = &j + ?$f$j,re It follows from (3.8) and (3.15) that tiw-ri+p,F:*)-~
’ (m*Ui + m2Ui) = plaf’),
Sji,j+ ni + p2FIZ’- ’ mlUi + mZUi)= p2Uj2)* 2( Moreover,
(3.30)
using (3.23), (3.25), (3.26) and (3.29), the energy equation (3.17) becomes
p(A + ?n + T4) - pr - q?i = $(&i - vi) + ?{ij,dij + @,ij,Aj + r[ji)r@ + /kijkKijk+ qijkyijks (3.31) We observe that the parts of tji, Sji and Ei in (3.26) which arise from 9 do not contribute equations of motion (3.30), the energy equation (3.31) and the total stress.
4, CONSTITUTIVE
We assume that the mass of each constituent
to the
EQUATIONS
is conserved, so that
ml =m2=0,
(4.1)
and that the constituents of the mixture are Newtonian compressible fluids. We exclude the heat conduction phenomena, and assume that A,, qa, ii;:, $j+ Ei,i,/hijk, %jgkand qT are nonlinear functions of p,, p2, pl,i, p2.i and T, and linear functions in the variables pl,ij, pz,ij, & = ui - ui, dij, $1, I,, Kijk and yijk* The axiom of objectivity reqUireS that the constitutive functionals must be hemitropic. In what follows we restrict our attention to isotropic fluids. It follows from (3.28) that A =A(P~, ~21 p~,it P2,i, 7’), and
r)=-2,
aA
qp=o,
(4.2)
D. lE$AN
1428
In view of (2.4), (3.2) and (4.1) we obtain PA2 = -PPZLr
PPI = -PPldrr - p2pl,i&, PP1.i = -P(P~,kuk,i
-I pt,drr
-P(P2,kUk,i
+ P2,i.L
Pb2.i
=
+ PlP2,iViF
PI Kiss) - P2Pl,ijy,
+
(4.4)
+ P*Yiss) + PI PZ,ijV;:-
From (4.3) and (4.4) we conclude that A =&PI,
PZ, ~1, K2, T),
KI = Pl,iPt.i,
(4.5)
K2 = P2,iP2,it
and Qj, = -pq + z&,,
G(jjj = -4ij + ffQ>,
&=Qi+n:, Pij =
9ij =
&jk =
PPI g+
(
-sijk
2PK12
PP2 g
>
+ 2PK2 2
21
+
b$k,
4j + 2P e
=
PI e
lijk
=
P2
2
@l,iajk
+
Pl,jaik),
(P2,iajk
+
P2,jsik),
=
-rijk
-t
q$k,
Pl,iPl,j, 1
8, + 2P g
Vi=P,Pl,i~-PIP,,i~-P1~Kl,i+P,~K2,i,
%jk
%jk
2
p2,ip2,jy
1
2
(4.6)
2
where the functions 7fij),a&,, p$k, q$;, I@ and Tuilare given by z&j = h,&rSjj + 2pldij + A,f,sij + 2/++$j, afij, = h,.IZ$, + 2pzf;j.+ h&,&j + 2Pddij, Z[ji]
@ = d& + bl K,,j+ b2Kirr+ dl ‘Yrri + dzyirr,
= Z[ ji] = CT,
1 F$k
=
i
@I(Kmiajk
+
Krr$ik)
KjrrSik)
$
i
$
?jrraik)
f
k,SijVk + kz&ikF + k,6jkq,
+
yjrraik)
+
pl(?rriajk
+
+
+
6YI(Krrtajk
Kjrraik)
LZiY3K,rk6ij
’
+
-6 i
f
2Kkrrhj
+
2Ykdij
@3yrrkhj
2v3Yrrkaij
+
+
263KrrkSij
2ff~Kijk
’
+
+
Yrrj&k)
zv4Yijk
+
@2(Kdjk
f
+
v5(Ykji
+
+
f
Kkij)
+
ykij)
PZ(yidjk
&(ykji
+
Krrjaik)
&LKijk
(Tg(Kkji
+
2e4yijk
+
2Kkrraii
+
f
Ytcij)
f;(Kirrajk
fCS(Kkji
+
Kkij)
(4.7)
4h,Sijvkk+2~ik~++h3~jkV;..
Here 8, is the Kronecker delta, and the constitutive coefficients are functions of ~1, PZ, K3
=
Pt,iP2,i
and
r
KI,
~2,
1429
Theory of mixtures of nonsimple fluids
The entropy inequality (4.3) yields the following restriction 1rF6 + Z$j,dij + ob,fij +
CI’ijTv
+ /L&Kijk
+ qJky/jk 2
0.
(4.8)
In view of (3.27) we find that Q = b + 456
+
6fir
63~1.m +
+
where the coefficients are prescribed functions of pl, p2, (3.14), (3.26), (3.29) (4.6) (4.7) and (4.9) that
K~,
K2,
~~
QS,
+
A,drrSij + 2pldij + h,f,6,
+ 2p3&j +
Q6,
+
A2f,6,
h,d,r6,
+ 2pddij -
rij = -pij + -
(4.9)
54~2,ss,
Uij
=
-qij
G
=
Qi + & +
+
2p2Jj
+
and T. It follows from
CT,, CT,,
(4.10)
Q,i + bl Krri + b2Kirr + dl ‘Yrri + d,Yirr.
Thus, the constitutive equations are given by (4.2), (4.5)-(4.7), (4.9) and (4.10). We note that in the present theory the free energy can depend also on the gradient densities. Taking into account (4.2) and (4.6), the energy equation (3.31) reduces to pT7j = Z$dij + U$fij + ZI$;j + ?$V, +
CL~~Kij~
+
TlJkyijk
+
pr.
(4.11)
In view of (3.14) and (4.1), the equations of motion (3.30) may be written in the form rji,j - /.Llji,rj- Ici + ~1FI” = plUj’), Uji,j- qlji,rj + Xi + p2Fi2) = p2aj2).
(4.12)
From (3.3) and (4.1) we get $
+ (Pl”i),i
2
= Or
+ (pzUi),j
= 0.
(4.13)
To the above equations we must add boundary conditions and initial conditions. Let B be a regular region of three-dimensional Euclidean space bounded by the smooth surface LIB. It follows from (3.1) (3.7), (3.10) and (3.21) that the total rate of work over the surface tIB is P = I,,(f!‘)Ui
+ ti(2)Ui +
hjj)Ui,j
+
h~f'Ui,j)dA
=
l,B(ljiU;
+SjiUi
+
/aLjrsU*,,
+
qjrsVs,r)nj
dA.
(4.14)
Following Toupin [18] we can write P =
I aB
(Sjl)Ui + Sj2)Ui+ R$“DUi + Rj2)Dq) &I,
(4.15)
where Of =fini and $I’
= (rji - Prji.r)nj
- nrDsllsri
+ /&si@rs - Pmmnrns)~
Sl” = (flji - l)rji,r)nj - Osrlsri + 7)&m R!”, = CLi-s,.n I n s, Rt2’ = 77 .n I IS, r n S’
- PmdVs)~
(4.16)
form of the surface aB and Dj/.tsjriis the surface gradient
In (4.16), Pij is the second fundamental of the hyperstress, i.e.
DjPjri = Pjri,j - njD/-+. It follows from (4.15) that P = i I,
[(Si” + S$“)(ui +
Vi)
+
(Si”
+
Many types of boundary
-
Si(2’)(Ui
-
Vi)
(RI” + Ri2’)D(ui + Vi) + (RI” - R$2’)D(ui - Vi)] dA.
conditions have been suggested (cf. [ll, 121). The first boundaryby the following mechanical boundary conditions
value problem is characterized Ui
=
fii,
(4.17)
Vi
=
Oi,
Dui = A,
Dui=gi
on
aBXZ,
1430
D. IE$AN
where 1zi, 6(, A, gi are prescribed functions and I = [0, a). In the second boundary-value problem the mechanical boundary conditions are $1’ Rj” + Ri2’ = &, & = Vi, D&=& on MXI, I + $2’ = p.It where pi, cl/, pi and di are given. The thermal boundary conditions have the classical form (see f321). We consider the following initial conditions P,(xt 0) = p?(x), Ui(X,
0) z
P2@,
Up(X),
0)
r)(X,
=
Ui(X,
PW,
0)
0) =
U:(X),
x E B,
= TO(X),
where pf, pf, up, up and 71’are prescribed functions. Let us assume that the constitutive coefficients Ai, cli (i = 1, 2,3,4), c, CY,,&, v~,f; {s = 1,2,3,4,5) are constants. Then, in view of (2.3), (3.18), (4.7) and (4.10), the equations of motion (4.12) can be written in the form (7, - PZ~A)AU,+ (22 - m2 A)U,ri f (~3- m,A) Au) + (24 - m4A)u,, - d(ui - Ui)+ Ai + PIFir’ = praf’)* (xl - rlA) Aui + Cxz- rzA)u,ri + 013- r3A)Aui f (~4 - rdA)u,,ri + d(ui - Vi)+ I’i + p2F12)= PZU~~‘,(4.18) where A is the Laplacian, Ai = skji,kj
-
Pji,j-
Qi,
ri = rkji,kj
-
4ji.j
Qit
+
(4.19)
and
z3 =
y3
+
;c-
dl + kl,
Zq=A3+p3-!c-dd2+k2+k3, 2
1
X,=pa+;c+bl-hl,
xz=h4+C14
-2cCbZ-hZ-h3,
X3=p++dl+hir
X4=&+p2+;c+dp+h2+h3, m2 = 2(al f a2 + as),
ml = 2(a3 -+-a4),
m4 = 2@1+ P2 + Ps), rl = 2(6x + fbh r4 = 2(v, + v2 + vs). r3 = 2(Y3 -t “4)~
5. LINEARIZED
m3 = 2(& + &a), 5
=
WI
+
32
+
Sd
(4.20)
THEORY
In this section we consider the linearized version of the theory developed in Section 4. We assume that there exists an equilibrium state of the mixture in which the constituent densities pl, p2 and the temperature T have the uniform values py, p! and To, respectively. In what follows we assume that the motion takes place under isothermal conditions. We introduce the notations th=Pc-Pt
Jtz=pz-P%
(5.1)
and assume that 3/i, I@~, II and v are small, i.e. JIr = &I&,e2 = E&, u = EU’and v = EV’,where E is a constant small enough for squares and higher powers to be neglected, and ++I;,JI;, u‘, vf are independent of E. To the second order, A is taken in the form A =A,+ YI& + KI
=
@l.iJll,i,
~292+$71J;:+
K2
=
?#‘2,i’!‘2,i,
fl2&+hh3/2+
~4~1
+GK2),
(5.2)
1431
Theory of mixtures of nonsimple fluids
where Ao, yI, y2, us (s = 1,2, . . . , 5) are prescribed constants. It follows from (4.6), (4.19) and (5.2) that Pij = (a10 + all+l +
9ij = (a2o + a214+1
a12+2cz)sij9 1
Qi = (c2+2Cjk
=
i
ri
=
B
Cl +l).i,
&5($2.i~jk
W2.i
sijk
+
-
(b21
i
=
pya,(Jll,iajk
Ai
I(I2,jhk)t
J/l
+
=
a
+
$l,jsik),
A+l,i
-
(bll
JIl
+
bl2+22),i,
(5.3)
b22J12),i,
where a11 = Yl(PO
a10 = P0Ph9 a12 = YIP:
+ P”Ph
Y2(P0 + Pi)
a22
=
hl
= a11 -cl, a
=
+ Pa
020
=
P0Ph
=
a21
Cl = Ph
+ P0P%2> bl2
+ Pod%
a12
+
b2l
c21
=
a21
=
Y2P!: + P0P%39
c2
=
+
Cl7
PO=P(:+P;,
PYY2, b22
=
a22
-
~2,
P = Ph.
Ph
By (4.9) and (5.1), Q = b + 51&r+
52.h
+
~3’7h.s~
+
(5.4)
64@2,ss,
where the coefficients are prescribed constants. Here and in what follows we denote the constitutive coefficients and their values in the equilibrium state by the same letters. From (3.14), (3.26), (3.29), (4.6) and (4.7) we get rij =
-(UIO+~II$‘I
+a12$2
Q)aij +A~drraij +2~1dij+h3f,6ij +2/~3f;, +Crijt
-
aij=-(a20+UZI+I+a~$2 + Q)Sij + h,f,aij +~/Jz& +b&r6ij +2/~4dij -Crijt ni= (czICIZ - CI'!'I + Q),i + d% + blKrri + b2Kirr + d,Yrri + &3/irr, hjk
=
-sijk
+
qijk
p;k,
=
-rijk
+
(5.5)
q$k,
where the constitutive coefficients are constants. The entropy is determined energy and initial conditions. It follows from (4.13) that a*, at +
Ph,i
a*2
at
= 07
+ P:Ui,i =
by the equation of
0.
(5.6)
The equations of motion reduce to r1i.j-
ffjisj
-
~kji,kj
?)kji,kj
-
Ei
+
+
Zi
+
p~FI” = p’: ~, p!Fi2’ = pi:.
(5.7)
From (2.3), (3.18), (5.5) and (5.7) we obtain the following form of the equations of motion (ri -
mi A) Aui + (22 - mzA)u,,ri + (~3 - m3A)Aui +
(~4
(~2
-
r2Ab,n
-
+
(x3
-
r3A)
Avi
+
m4A)u,Vi
A+i -bll$1
-d(u,-ui)+((~ (XI - M) AUi+
-
(x4
-
-b12+2),i +p~F~“=p:~p
r4A)vr,ri
d(ui - vi) + (p A$+ - b21#1- b22t,&),i+ p;Fj2)= pzz.
(5.8)
Thus,we have obtained the system of field equations (5.6) and (5.8) for the unknown functions #i, &, UI and Vi*To this system we must add boundary and initial conditions.
D. IESAN
1432
6. FUNDAMENTAL
SOLUTIONS
In the first part of this section we establish a solution of Galerkin type of the field equations within the linearized theory. Then, we use the Galerkin representation to determine the fundamental solutions of the field equations in the case of steady vibrations. Galerkin representations and fundamental solutions in the context of a single phase polar fluid were presented in various papers (see, e.g. [33,34]). In what follows, for convenience, we consider the system Dllu + D12 grad div u + D13v + D14 grad div v + D,5 grad JI1 + D,6 grad & = -p’$(“, Dzlu + Dz2 grad div u + Dz3v + Dz4 grad div v + D,, grad J/l + Dz6 grad J12= -p;F(*),
a*, at + py div u = -M(l),
a** at + p; div v = -MC*),
(6.1)
where D,,=(r,-m,A)A-p:;-d,
D12 = r2 - m2A,
D,, = (q - m,A) A + d,
D1,+=r4-m4A,
D21 = (xl - r,A) A + d,
022 = x2 - r2A,
D23=(X3-r3A)A-pgi-d,
024
=
016
=
-h*,
D15 = CYA - bl,,
X4
025
-
=
-b21,
026
r4A,
=
0
A - b2*.
Clearly, if M(i) = M(*) =O, then the system (6.1) reduces to (5.6), (5.8). We introduce notations Aa2
=
Da2
;
P:&s,
-
Au4
=
Dcr4
+
023
-
+
013
a
-9
212
at
Z2, = A2*A + D2, d at’ T,, = D Cm,
THEOREM6.1.
Z11&(4+a)
-
the
P;&,
f
a &I = &A
(6.2)
=
Al4A
LA,D23
-
Dd21,
=
Ao2Z11
-
&4Z21>
at
>
Z~Z=&A+DI,$. Ga
z12D2(4+a),
=
-
I-I =
=
z22z11
ca2
=
-
‘%x4222
z22D2(4+a)
-
Z21D1(4+a)
z12z211 -
(a
&2&2,
=
1,2).
(6.3)
Let
u = -llD23G + IID13H + (Cl1 023
-
Czl D,,)grad div G af
+ (C12D23- C22D,3)grad div H + T,, grad at + T12grad $, v = llD21G - IIDllH + (C21Dll - C,,D,,)grad
+
(C22Dll
-
div G
C 12 D 21 ) g rad div H + T,i grad $+
I,$ = pyZll D div G - P?Z~~D div H - (II + p?T,,A)f
T2*grad 2, - pyT12 Ag,
J12= -P~Z~~D div G + p!DZ2* div H - piT2, Af - (II + p!T,,A)g,
(6.4)
where G, H, f and g satisfy the equations DIIG = p:F(‘),
D IIH = p;F’*‘,
Then II, v, JI1 and x2 satisfy the equations (6.1).
II $= M(l),
II $ = M(2).
(6.5)
Theory PROOF.
of
1433
mixtures of nonsimple fluids
It follows from (6.4) that
D, 1~ + D,2 grad div u + D,sv + D14grad div u + Q5 grad ++I+46 grad +CIZ
= -DI-IG + {(D,4D21+
(Cz,D,l -
+ #42013
&2023)n
+ (cl1023
+ 04)
+ pZ1445
cl,DZl)(D,3 -
&&,)n
+ (c22&
-
+ {T,,(D~~
+ QUA)
+ { T2(D11
+
+ (c12O23
-
+ ~~~A)
-
c,2~2,)(~13
$
+ 7i4h3
-
c21~13)(011 -
~22°13)(011
i
DlzA) ; + q2(D13 + D,J)
-
i -
~Z2~~k&wddivG + 0124
&%2D&5
+ 4A)
+ D12A)
+ diQd22WgraddivH
&(n
+ dW)
p:D,s
TA
-
- Ddn
d%J&Gwdf
+ p%ANwd
g.
If we take into account (6.3), then we obtain D,,u + Or2 grad div u + Di3v + D14grad div v + D15grad I+Q~ + D16grad Q2 = -DIIG. Similarly, we find that D2iu + D22grad div u +
D23~ + D24
grad div v + 025 grad @I+ 026 grad $2 = -DJ%
af
z+pydivu=--II--,
w2
ag
T+p;divv=
With the aid of (6.5) we obtain the desired result.
-II%.
0
We now consider a mixture which occupies the entire three-dimensional Euclidean space. In what follows we use the solution (6.4) to determine the fundamental solutions of the field equations for the case of steady vibrations. We suppose that F(“) = Re[F*@(x)exp(
-ior)],
M’“’ = Re[M*@(x)exp(
of vibration and i = (-l)ln.
where w is the frequency form
u = Re[u*(x; w)exp( -iot)],
-it&)],
If we seek for solutions to (6.1) of the
v = Re[v*(x; o)exp( -it&)],
+m= Re[$,*(x; w)exp( -hot)], then the field equations introduce the notations
reduce to a differential
D& = (q - mlA)A + iopy - d, AZ2 = -iwD a2 Z;2 = A&A -
-
p:D,s,
u*, o*, I,$,*.We
Dz3 = (x3 - r3A)A + iop; - d,
AZ4 = -hoDa - p:Dp6, Z,*, = A&A - iwD&,
1oD13,
TL = Z&Dw+a) -
system for the amplitudes
Zif2D2(4+aJ,
T%
Z?, = A&A - ~oD;~,
Z,* = AT2A- ioDT,,
= Z2D2c4+,,
-
D* = D:ID;3 - D,3D2,,
l-I*= z,*,z:1 - ZT2Z&,
C,*, = Az2Z5 - A:,Z;,,
C:2 = A:4Z;2 - Az2ZT2.
Z&4~~+n~,
If we assume that G = Re[G*(x; o)exp( -iwt)],
f = Re[f *(x; w)exp(-iot)],
H = Re[H*(x; w)exp( -it.&)], g = Re[g*(x; w)exp( -iot)],
(64
D. IE$AN
1434
then (6.4) leads to the following representation u = -II*D&G*
+ II*&H*
of Galerkin type for amplitudes
+ (C:,D,*, - C&&)grad
div G*
+ (CWT3 - C%&)g rad div H* - ioTT, grad f * - iwT& grad g*, v* = II*D,,G* - II*DTIH* + (C$,Dh - C:,&)grad
div G*
div H* - iwT& grad f * - ioTz2 grad g*,
+ (C&DTI - CT&)grad
$7 = pyZT,D* div G* - pyZ&D* div H* - (II* + p’lTf,A)f* - p?TT2 Ag*, $,T = -p;ZT,D*
div G* + piD*Z,*, div H* - piT& Af * - (II* + p!T&A)g*.
(6.7)
The functions G*, H*, f * and g* satisfy the equations D*II*G* = pyF*“‘,
D*l-I*H* = p!$*‘2’,
wfl*f * = iM*(‘),
on*g
= iM*t2).
(6.8)
From (6.3) and (6.6) we get D* = mIdA + m,,A3 + m12A2+ m,,A + ml,,
(6.9)
where m 13 -
ml4 = mlr3 - m3r1,
d(m,+m3+h+r3)+
m 12=
m3xl
+
rlz3
-
rlr3
-+x3,
zlX3-X123-iiw(mlp~+r3~~),
m 11= -(z, +x3 + 23 + xl)d + io(rlp;
+ x3p3,
m 1o= -pyp!w2 - iwd(p? + pi).
(6.10)
The operator II* can be expressed as II* = mz4A4+ m23A3+ mz2A2+ m2,A + mzo,
(6.11)
where m24 m 23 --
m 22
=
=
-m14w2
-m13w2
-
lZl20
-m12w2 - izllo +
m20
--w
=
2
+
ml4 +
-
m1n24
m3G2
+
nl2n54 -
+
(nl2n24 +
n24tIy2
ny4ni2,
-
-
-
n 14n;2
-
-m1102 -
=
n 24
=
’
lor4
-
+
rd4
P%
z12 = -
n24 zln24
zll
+
x3n~2 + n12(iwp!j- d) - z3ni2 - dn22 -
xlfh4,
-
Ob
~2
mld4 =
22 -
-
zlnfii4
D* = m14(A + &(A + &)(A + &)(A + E:),
and ~f+~ (i = 1,2,3,4)
are the roots of the equation -
m23Y3
+
m22Y2
-
mzl
Y +
m20
=
0,
are the roots of the equation m14y4
+
(iop? -
+ iw(pyng, + p$ny2). + nS4 + ny2 + r102~)
II* = m24(A + &:)(A + &$)(A+ &)(A + E:),
-
ml3y
3
+
m12y2
-
mlly
+ml0=
0.
iw4,
r3nY2
xl& - w&
It follows from (6.8) and (6.10) that
m2,Y4
ZZ~OW,
ny2 = pyb,, - iwz2, n22 = iwr2,
x3n12
where ~3 (S = 1,2,3,4)
nLn22,
0 _
+
zlo= -d(ny,
n14n22),
n?, = p;b12 - iw4,
r3n12,
23n22
-
rn,,
n12 = iwm2 - pyff,
mlo,
nX2= pyb2, - iwx2 , m3n22
izl3w
ny2ni4
q4 = iwm,,
z13 =
-
d)nt4
Theory of mixtures of nonsimple fluids
1435
In what follows we denote by sS (S = 1,2, . . . ,8) the roots with positive real parts and assume that e, # &2# - - - # tz8# E]. If F:(I) = QS, (j fixed), Fi*(2) = 0, M*(a) = 0, then we take GF = Y6,, Ht = 0, f* = g* = 0. It follows from (6.7) that Y satisfies the equation D*rI*Y = j@. If the functions q (j = 1,2, . . . ,8) satisfy the equations (no sum, j = 1,2, . . . ,8) then the function Y can be expressed as
where z;’
=
(S = 1,2, . . . , 8).
(E; - E:), fi j=l(jZs)
Let us assume that 6 = S(x - y) where S is the Dirac delta and y is a fixed point. Then we get r ti
exp(kr),
(S = 1, 2, . . . , 8)
where r = Jx - y[. In this case we have Gf = ES,, Hi* = 0, f* = 0, g* = 0, where 8 r P(: X ZjexP(i&jr).
E=-
j=l
From (6.7) we obtain + (CTlD,*, - C,*,D,,)E,jk,
u:(j) = -II*D,*,Eajk
u:(‘) = lYI*Ql ES,, + (C~~D~~ - Cf1Dz1)E,jk, ~,2*“’= -p$Z,*,D*E,j
Icr:“’ = p~Z;r,D*E,j,
If we assume that F)(l)= I
0, FFc2) = 6(x - y)S,,
M*(p)=
(6.12)
0, then we find the solutions
uzc3+‘) = lI*D13ESjk + (CTzD,j - C,*,D,,)E,jk, vzf3+j) = -II*DflEGjk
+ (C&Of1 - Cy2D21)E,jk,
#fc3+j) = -p;Zf2D*Evj,
I,#~+” = p;Z,:D*E,.
(6.13)
= 0.Then we have G* = 0, H* = 0, Let us assume now that F*(p)=O, M*(l)=6(x-y),M *(2) g* = 0 and f* = e, where
e;l = fi
(E;- E:),
j=l(jZs)
(S = 1, 2, 3,4).
It follows from (6.7) that &7) = -itoTf,e,k, $ft7’ = -(n* Finally,
if F*(O) = 0, M*(l)
IJE(~)= -ioT$e,k,
+ pyT:,A)e,
= 0, m*t2) = 6(x -
+zt7) = -p!T&Ae.
y) then we obtain the solution
*(‘I = -iuT&e,k, Uk
vf8) = -rwT&e.k, *
JI?“‘=
+j@’ = -(II*
-p?T&Ae,
(6.14)
+ p!Tz2A)e.
(6.15)
1436
D. IE$AN
The functions z&‘, I#‘), I,@“‘, I,@“’ (s = 1, 2, . . . ,8) given by (6.12)-(6.15) fundamental solutions of the system of field equations for steady vibrations. Acknowledgement--I
represent
the
express my gratitude to the referees for their helpful suggestions.
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