Oscillations of a plate in a binary mixture of incompressible Newtonian fluids

Int. J. Engng Sci., 197 1, Vol. 9, pp. 1177-l 192.

Pergamon Press.

Printed in Great Britain

OSCILLATIONS OF A PLATE IN A BINARY MIXTURE OF INCOMPRESSIBLE NEWTONIAN FLUIDS R. E. CRAINE School of Mathematics and Physics, University of East Anglia, Norwich, Englandt Abstract - Constitutive equations are derived for a mixture of two incompressible Newtonian fluids and the flow induced in such a mixture by the steady oscillation of an infinite plate is then investigated. Explicit results are obtained in the limiting cases in which a suitably non-dimensionalized frequency is very small or very large, the nature of the solution in the intermediate frequency range being discussed numericahy. 1. INTRODUCTION IN RECENT years much work has been published on the continuum theory of mixtures and the subject now appears to have emerged from a period of controversy into a position in which the principal authors are substantially in agreement about the basic results and procedures. Up-to-date accounts of the continuum theory of mixtures can be found in the book by Truesdell[ l] and in papers by Green and Naghdi[21 and Craine, Green and Naghdi[3], and a comprehensive list of references has been provided by Gurtin and de La Penha[4]. Mixtures of fluids which are viscous in the sense that their constitutive equations contain gradients of the velocity fields of the constituents have been considered in the context of the continuum theory by Green and Naghdi[2], Craine, Green and Naghdi[3], Adkins[5], Mills[6], Miiller[7], Doria[8], Bowen[9] and Bowen and Garcia[ 101. Eringen and Ingram[ 11,121 have investigated mixtures of chemically reacting fluids. In addition there is a considerable literature in which general two-phase flows are viewed from a different viewpoint, much of this work being concerned with suspensions of solid particles in fluids. Saffman[ 131 has given equations governing the motion of a dusty gas, and Michael and Miller[l4] have discussed some applications of his theory, among them the flow produced by an oscillating plate. However, the approach of Michael and Miller assumes that the partial stress of the gas is symmetric and does not depend on the velocity gradient of the other constituent, the dust. The basic equations adopted in [ 141 consequently differ from those derived in this paper. In the work described here a binary mixture of incompressible, Newtonian fluids is considered. In section 2 the governing field equations of the continuum theory of mixtures are briefly stated, and in section 3 they are used to develop an appropriate constitutive theory. The resulting equations represent perhaps the simplest model of a physically realistic binary mixture the bulk behaviour of which is akin to that of a liquid. The damped oscillatory motion of the mixture generated by a sinusoidally vibrating plate is studied in section 4. A simple form for the solution is assumed and it is shown that this form is possible only if the composition of the mixture remains constant. The problem is well-posed and has a unique solution, but explicit results are obtainable only in the low- and high-frequency limits. These limiting forms of the solution are found in section 5 and their interpretation discussed. The behaviour at intermediate frequencies (appropriately non-dimensionalized) is inferred from numerical results in section 6. tPresent address: Department of Mathematics, University of Southampton, Southampton, England. 1177

1178

R. E. CRAINE

The solution obtained in this paper indicates that useful if limited conclusions follow from a simplified form of the general constitutive theory, but that a full interpretation of the final results involves certain ‘non-standard’ material constants for which experimental results are not available. Some comments are made in section 6 on the sensitivity of the solution to these constants. 2. BASIC

EQUATIONS

A mixture of two continua, in motion relative to each other, is considered. At an arbitrary time t it is assumed that at each place x in the region of space occupied by the mixture two particles are situated, one belonging to each constituent. This is a basic assumption of the continuum theory of mixtures and its validity requires that the mixture appears homogeneous when viewed on the scale of the applied disturbance. The velocities of the l- and 2-constituents are denoted by P and 9) respectively, and we define (2.1) where xi are the coordinates of x relative to fixed rectangular Cartesian axes and the usual summation convention applies to italic (but not to Greek) indices. Here and henceforth the index (Ytakes the values 1,2. The governing kinematic equations are well known (see, for example, Craine, Green and Naghdi[3]). Here, for ease of exposition, attention is restricted to the case in which the constituents do not interact chemically and have a common temperature T. The relevant equations?, in point form, are (2.2)

(2.3)

(2.4) where a comma denotes differentiation with respect to xk keeping t fixed. The quantities pa, acQ)and Ey”’are in turn the density, partial stress and body force per unit mass of the c&h constituent. The diffusive force vector, o, may be interpreted as the drag exerted on one constituent due to the motion of the other. Conservation of angular momentum in the mixture implies that

where suffixes enclosed in square brackets refer to components of the antisymmetric part of a second-order tensor and circular brackets have a similar meaning in relation TThese equations can be derived from integral balance equations referred to either material regions or regions fixed in space.

Oscillations of a plate in a binary mixture

1179

to the symmetric part. When the total stress is defined to be the sum of the partial stresses, as in Green and Naghdi[2], equation (2.5) states that the total stress is symmetric. The energy equation for the mixture may be written as

where r, the heat supply per unit volume, and q, the heat flux vector, refer to the mixture as a whole, and U1, U, are the internal energies per unit mass of the l- and 2-constituents respectively. The density p of the mixture is given by p=

(2.7)

p1+p2*

Admissible thermo-mechanical processes in the mixture (i.e. sets of functions which, for some choice of F*’ and r, satisfy the field equations (2.2)-(2.4) and (2.6) and an appropriate set of constitutive equations) must be compatible with an entropy production inequality. For binary mixtures the following point form of the ClausiusDuhem inequality has been given by Truesdell [ 11, Green and Naghdi [2] and Bowen and Wiese[lS]:

(2.8) where S, and S, are the entropies tively. 3.

per unit mass of the l- and 2- constituents

CONSTITUTIVE

respec-

THEORY

The formulation of constitutive equations for a mixture of Newtonian fluids is now considered, attention being restricted to the case in which both constituents are incompressible. Such a mixture was considered by Mills [6], but only the total entropy and free energy of the mixture appeared in his theory, and an earlier version of the Clausius-Duhem inequality, different from (2.8), was used. Since the inequality (2.8) is now accepted as the correct one the theory in [6] requires modification. Let the densities of the l- and 2-constituents of the mixture, when separated, be plo and p20 respectively. In view of the incompressibility assumption, plo and p20 are constants. Introducing a composition factor y, defined as the proportion by volume of the l-constituent, and assuming that the mixture does not contain voids, it follows that the densities of the two fluids are given by PI =

-YPro,

P2 =

(1 -Y)P20.

Hence fi+fi= p10

1. P20

(3.1)

It seems preferable to work with the total density p in place of y, and it is easily shown,

1180

R. E. CRAINE

using (2.7) and (3. I), that (3.2) Substituting relationt

(3.2) into equations

(2.2) and eliminating @/at between them gives the

(fy +2” 1

=

’

0,

(3.3)

,I;

or, using (3.2), (Pzo--P)4,‘+

b-plo)@-~k~k

=

(3.4)

0,

where

are in turn the total density gradient, the relative velocity and the stretching components associated with the two velocity fields. Equation (3.4) is a constraint on admissible processes in the mixture and in developing the consequences of the Clausius-Duhem inequality it must be incorporated into the left-hand side of (2.8) in association with a Lagrange multiplier A. For convenience partial Helmholtz free energies A, are introduced, defined by the relations A, = U, - TS,.

(3.5)

Using equations (2.6) and (3.5) the inequality (2.8), as modified by the constraint (3.4), may be written D”‘A

Dt2’A

-Pt3+P23f--PS1

Dt

+ wai

-

A{

( pzo -P)dE+

(P-P~o)d;?-tk%)

3

0,

(3.6)

where bk = T.k.

The basic constitutive

postulate made in this work is that

tFor a mixture of Y incompressible fluids, (3.1) and (3.3) generalize to

(3.7)

1181

Oscillations of a plate in a binary mixture

are functions of p, T, Tr, bkr v$ v;:. Objectivity

arguments require that the last four variables should be replaced by akr

@‘,

rik

where w{$ = t&i,

Ilk = w’i,?/ - wj;‘,

(3.8)

are respectively the components of spin associated with the velocity fields and the components of relative spin. A total free energy A and a total entropy S are defined by PA = WI1 +p2&

(3.9)

PS = PlSl +p,sz.

Using standard procedures for exploiting the Clausius-Duhem inequality a few general results can now be found. The residual inequality remains complicated, however, and in order to make further progress some linearity assumptions are usually introduced. For simplicity we make these additional assumptions at the outset, taking the constitutive equations to be linear in their vector and tensor variables. From representation theorems under the full orthogonal group (see Spencer [ 161) the appropriate forms of these equations are PI.4 1 = A 10+ c,dg’ + c&,

(3.10)

p‘#l* = A 20+ c&) + c*&),

(3.11)

plS, = Slo+ c,d;;‘+ c,d$

(3.12)

pz S2 = S,, + c,d;;) + c,d;;),

(3.13)

mt = w+PSi+rbi,

(3.14)

qr = - T(bi

(3.15)

+ k,S, + k&i),

CL{’= (-PI + x,d$j’ + A&‘) &k + 2/&d$’ + 2/&z’ + &jr&,

(3.16)

Cr# = (-pz + Xdd$’+ AZ@) 6ik + 2/&d(ii’+ 2&@ + AGIl/,+,

(3.17)

where Alo, AzO, cl,. . . , c8, CGp, Y, Slol Szo, kl, kB, kt, pl, all functions of p and T. It follows from equation (2.5) that

p2, A~,. . . ,

A~, pl,

p4 are

A5+Ag = 0.

(3.18)

&)’ = @’ - i@‘&k.

(3.19)

Let

Then with use of equations can be put into the form

(2.1), (2.2), (3.2) and (3.8) to (3.19), the inequality

(3.6)

R. E. CRAINE

1182

--Pn+

+

aa&

+

( p

(p-pm)

(P

+

A )a&

+

2~-h)+c,a,b,Jf2~,~~)‘rf,p’+2(~+~~)~~)~~~)~

AJd’i~

+

(

S20 +

~2 %+y+kl aT

)

aibi+k2bi[i+k3bibi

2 0.

The quantities D”‘T/Dt, D’*‘d$/Dt, dE:j, which appear only once in this inequality, can be specified arbitrarily and independently of the remaining terms. Their coefficients are therefore zero, giving the results Cl

=

c2 =

c3 =

CJ =

0,

A ~=&~p,~),

s--2! -

A

(3.2 1)

dT'

Since the residual inequality holds for all admissible processes ses are given by (3.16) and (3.17) it follows that c,=c,=c,=c3=0,

Pl=-_(Pm

a 3 0,

A5 2

AI+&

3 0,

h2+3k

k, 2 0, 0, 5 0,

P2 =

(P-P,,)(P~~--h).

(p2 (S2+$$)+y+k,}2

jLI a 0,

/L-22 0,

and the partial stres-

k2=0,

/3=-A,

-P)(PI$+$

(3.20)

=A(p.T),

(cL3+jd2

{A~+A~++(Ps+P~))~

6 4ak3,

(3.22)

s 4/w4.-z~

(3.23)

c ~(A,+~FI)(Az+&).

(3.24)

The reduced forms of the constitutive equations for a binary mixture of incompressible, heat conducting Newtonian fluids are thus (3.20), (3.21) and S, =

SAP,

T).

pi=

aai-At(

qf= - T(M

(3.25) +ybi,

(3.26)

+ k3bt),

(3.27)

Oscillations of a plate in a binary mixture

where the coefficients cr, A,, . . . , As, y, kl, ka, pl, . , . , p4 are functions of p and T, and satisfy the inequalities (3.22) to (3.24)t. On substituting these results into the energy equation (2.6), and making use of equations (2.2), (3.2) and (3.4) we obtain the residual energy equation

4. FLOW

INDUCED

BY STEADY

OSCILLATIONS

OF A PLATE

The literature on the treatment of fluid mixtures from the continuum viewpoint contains solutions to a number of particular problems, among them the propagation of acoustic waves [5, 171 and certain steady helical motions including Poiseuille and Couette flows[6]. These latter flows can also be studied on the basis of the theory developed in sections 2 and 3, but the end-results are closely similar to those given by Mills[6]. It seems preferable, therefore, to devote the remainder of this paper to a fresh application of the basic theory for binary mixtures of incompressible Newtonian fluids and we examine the motion generated in a semi-infinite body of such a mixture by steady sinusoidal rectilinear oscillations of an infinite plane boundary. For convenience set x1 = X, x2 = y, x3 = z, and let the y-axis be directed normally to the plate, with y = 0 on the plate and y > 0 in the mixture, the x-axis being in the direction of motion of the plate. It will be assumed that body forces are absent and that the motion of the mixed fluid takes place under isothermal conditions. The second of these requirements is strictly satisfied only if the heat created by viscous dissipation and diffusive resistance is extracted through the agency of an appropriate negative heat supp1y.S The physical artificiality of this situation is of little account so long as the rate of dissipation of energy is ‘small’ and on this argument rests the standard practice of regarding the energy equation as irrelevant in problems of ‘slow’ viscous flow. Since the temperature T is no longer a variable, the coefficients p1 etc. appearing in the constitutive equations depend only on the total density p. A solution is sought tThe expressions (3.26)-(3.29) are similar, but not identical, to those given by Mills[6] when the latter are adjusted to allow for invariance under the full orthogonal group. SSubstitution of the solution (4.1) into the residual energy equation (3.24) gives

R. E. CRAINE

1184

in which v(l)= (u(y,t),O,O),

v=)=

(u(y,t),O,O),

(4.1)

p=p(y,t),

and, taking the velocity of the plate to have components (LI cos rzOt,0, 0), the application to the velocity fields (4.1),.* of the no-slip condition at the plate provides the boundary conditions u(0, t) = u cos not, V(0, t) = u cos nor. (4.2) Substitution

of (4. I ) into equations (2.2) gives

whence ap/at

= 0 and (4. 1)3 simplifies to

(4.3)

P=P(Y>*

Using (4.1) and (4.3), the constitutive stresses reduce to o= (1) -

Ull

(1) -

ff22 -

equations

for the diffusive force and partial

(cx(U--),-Adp/dy,O),

(4.4)

42 =

(4.5)

b20-P+l~+A),

(4.6)

u&l)= (p 1 +‘A ’

(2) -

(+I1

(2) u22 -

)%+(ps.s-+A

5 ay

u33(‘)=

-

5

)*

(4.7)

ay’

(4.8)

(p--p,o)(p.$$-A),

(4.9)

(4.10) (U) -

Cl3

(a) u23

-

u3l

(a) = &$’

(4.11)

= 0

On substituting from equations (4.4) to (4.11) into (2.3) and (2.4) there results the following system of partial differential equations with dependent variables u, V,p and A: (~no-~)~+~{(~l+jh~)~+

(/&As)$}-a(~-0)

=p$,

(4.12)

1185

Oscillations of a plate in a binary mixture

-*As)E+ b-Plo)g+-${ (CL4

(/&+&As)$]+a(u-v)

=/I?$,

(4.13)

~[(P20-P)(P~~+A)}+A~= 0,

(4.14)

$(- (P-Plo)(P*~-A)}-A~=O,

(4.15)

ah - 0. at-

(4.16)

It follows immediately from (4.16) that A is independent of z. In addition (4.12) and (4.13) imply that L&/ax is a function of y and r only, whereas (4.14) and (4.15) are satisfied only if ahjay is a function of y. The most general form of A is consequently

Since the total stress has been defined to be the sum of the partial stresses, the total pressure gradient along the plate is

We suppose, however, that the pressures far upstream and downstream (i.e. as x --* -+ a)) are kept equal throughout the motion; thus fo(t) = 0 and the pressure gradient in the x-direction is zero. As a result equations (4.12)-(4.15) reduce to (4.17)

(4.18)

$ IPI(P*o--P)$f I+(p*o-p);i;.dfi = 0, a - -&J-P&$ ay L With the use of (3.1) and (3.2) elimination after some manipulation,

Since,

in general

I

+ (p-pKJd$=

0.

(4.19)

(4.20)

of df,/dy between (4.19) and (4.20) gives,

p Z plo, p Z pzo and d*(pA)/dp*

# 0 we must have dpldy = 0,

1186

R. E. CRAINE

whence p = po, a constant. From (4.19) and (4.20) it follows that f= fro, also a constant. The foregoing argument shows that a solution exists in the simple form (4.1) only when the composition of the mixture remains constant throughout the motion. Since p has been proved to be constant and isothermal conditions have been assumed, all the coefficients in equations (4.17) and (4.18) are constant. It is convenient at this point to introduce dimensionless variables and material constants f, 7, & (i = 1,*. .74), X, and j& defined by f=

at/PO, 7 =

((Y/cL)1'2Y, pf = pfip,

h, = hglp, pa= pulpI),

p being a typical viscosity coefficient in the problem. Replacing (4.18) by the sum of (4.17) and (4.18) the surviving pair of governing equations can be written (4.21)

(4.22) We look for a solution of the form u = URe[efniF(jj)],

v = URe[efnTG(g)],

(4.23)

where n = nopo/~ and Re denotes the real part. Equations (4.2 1) and (4.22) then yield

Up+3LP2-

1-i~In}F+{(~-$ig)D2+1}G

=O,

(4.24)

=O,

(4.25)

G = {- (p1+j&)D2+ip,n}V,

(4.26)

{(jZ~+jZ4)D2-ijTln}F+{(&+~)D2-i~2n}G where D = dldy. Setting F = {(ih++)D2-ip2n}V,

equation (4.25) is satisfied identically and (4.24) becomes

(D4+P,D2+/32)v= o,

(4.27)

where ~0~,=-((p,+Fz+~+~~Lq)-in(P,F2+A~,+~~~), POP2 =

-

&j52n2+ in,

Po=Pl~-~~4++~5(p,+~+~++4).

Seeking a solution of (4.27) of the form V = B exp A7 leads to a bi-quadratic equation for A with roots

1187

Oscillations of a plate in a binary mixture

AT= s, + it, =3P-P1+(8:-48Ph

(4.28)

“; = S2+it2 = ${-&-(fl;-4&)1’2},

s, and t, being real constants. The complex numbers S, + it, have unique square roots with negative real part, say - I, + im,, where (4.29)

1, = 2-lj2{( S: + tz)1’2+ s,} 112, m, = - tJ21,.

To obtain physically reasonable expressions for u and ZI,it is necessary that V remains finite as 7 + M,The appropriate general solution for V is hence (4.30) where B and C are disposable complex constants, and it follows from (4.26) that F(Y) =B[(~+~)A~-i~2n]exp{(-lI,+im,)~}+C[(&+CTi,)h~-i~n]

,

xexP{(-I,+im2)ji), ’

(4.31)

G (Y) = B[- ( p1 + i&) A:+ i&n] exp { (-II + im,)Y} + C [- ( j& + &) A;+ i&n] xexp{ (-lz+im,)y}. Applying the boundary conditions (4.2) to equations (4.23) gives F(0) = G(0) =

(4.32)

1.

Thus, setting B= BI+iB2,

C= C,+iC2,

equations (4.3 1) and (4.32) provide four equations for the four unknown real numbers B,, C,. Define A= (~1--s#+

(tl-r2)2,

(4.33)

#J=

Then subject to the conditions? A # 0,

ih+p4

#

0,

ih(/32+&)

+

P2(&+~4i4),

these equations have the unique solution AB, =

92t1

(4.34)

tA = 0 only if A: = hi, and in this case the required general solution for V is not (4.3) but

where B’ and C’ are complex constants. If & + & - 0 and/or A(& + fiS) = p&z1 + B,). we return to equation (4.25) which is considerably simplified. In either case the appropriate solution can easily be obtained but no details are given here.

R. E. CRAINE

1188

cl =-B1,

C, =-B,-$dn.

(4.36)

In order to display this solution explicitly formulae for s,. t,, f, and m, in terms of the dimensionless constants pl, etc. are required, but the general forms of these results are very cumbersome. It is comparatively easy, however, to obtain approximate results valid in the low- and high-frequency regimes defined by the strong inequalities 12e 1 and n % 1. We proceed now to examine these limiting cases.

5. APPROXIMATE

SOLUTIONS

FOR

Low- and high-frequency approximations the expressions (4.3 1). In the low-frequency are identical, being given by

LOW

AND

HIGH

to F and G can be derived directly from limit the first approximations to F and G

I( Pl+E72&+P4

F(P) = G(J) =exp -

FREQUENCIES

)l’z(l+i)J)[l+O(n)l.

(5.1)

The velocities of the two fluids therefore coincide and the mixture behaves as a single fluid with average properties. Some difficulty is experienced in obtaining the high-frequency limit. Consider first the special case of a mixture for which /is = Fir;,= h5 = 0. In this case the partial stresses are symmetric and each one depends only on the velocity gradient of that constituent, no cross-terms being included. The expressions (4.3 1) for F and G are then simplified, reducing to

(5.2)

The velocities of the individual constituents are now seen to be different, and in this limit each fluid moves independently of the other. When all the coefficients appearing in the partial stresses are non-zero, the character of the high-frequency limit is not so clear-cut. The velocities of the two fluids again differ, but viscous interaction between the constituents now occurs and the results (5.2) no longer hold. The numerical solutions described in the next section reinforce these comments. The approximate formulae (5.1) and (5.2) can also be obtained by a simple, but non-rigorous argument proceeding directly from the governing equations (4.21) and (4.22). From (4.23), au/at and au/at are O(n). Thus, from (4.22), a2&J2 and a2v/ay2 are also O(n). The low- and high-frequency limits are now considered in turn. (i) n Q 1. In this case (4.2 1) implies u - v = 0 (n), and so u = v to a first approximation. Equation (4.22) becomes

Oscillations of a plate in a binary mixture

1189

and the required solution is (5.1). (ii) n % 1. All the terms in (4.21) except u - v are 0 (n) and therefore much larger than u - u. In the leading approximation u - v can hence be neglected and we get

These simultaneous equations can easily be decoupled; we then have two independent linear combinations of u and v satisfying independent diffusion equations. The process is trivial when p3 = ,!&= X, = 0; then _ a2u

_ au

_ a2v

_ av

l-h-+=

I%-$’

1*2,92=

PZdf’

and the appropriate solutions of these equations are (5.2). The results outlined in this section are in qualitative agreement with those given for a mixture of v fluids by Adkins[5]. The constitutive theory used by Adkins assumes that each partial stress depends non-linearly on the velocity gradient of that constituent, but is independent of the velocity gradients of the other fluids. Consideration of the propagation of small amplitude waves shows that for low frequencies the mixture behaves as a single fluid with average properties, whereas in the high-frequency range the solutions correspond to wave propagation in each fluid in the absence of the others. 6. NUMERICAL

RESULTS

AND

DISCUSSION

The previous section outlines the salient properties of the solution, and some numerical results are now given to complete the discussion. The parameters to be specified arepJj=l,..., 4), XS and pl. Estimates for p1 are easily found for typical mixtures of liquids, and reasonable values for j& and ii2 are not difficult to obtain. The remaining coefficients are unknown, so a number of computer programmes were run with a varied selection of values for these quantities, remembering that the inequalities (3.17)-(3.19) must be satisfied. The graphs shown in Figs. l-3 represent a typical selection of the results obtained, though in all the curves shown certain parameters are fixed (pl = 1, p3 = 4, ,!i4= +&, h5 = -+). The numerical results are divided into two groups. The first one gives the variation with frequency of the velocities of the fluid at a fixed time (t = 2m7r, m = 0, 1, . . .) and at a fixed distance from the plate. In the second set a fixed frequency of oscillation of the plate is assumed and the variation of the velocities with distance from the plate is found, again at a fixed time. In Fig. 1 the velocities of the fluids are normalized, so that they both approach unity as IZ tends to zero. The calculations refer to the case 7 = (2/n)1’2, and hence, using (5. l), the normalization factor is

1190

R. E. CRAINE

0.6 t

Fig. 1. Variation of fluid velocities with frequency. = 516, ,I&= 0.9; ----F, specified as follows: ---F,

In Figs. l-3 the remaining parameters are = 519. j& = 0.9; & = 579, i& = 5.

Y

F(n/2)1’*)

Fig. 2. Variation of fluid velocities with distance; n = I.

Oscillations of a platein a binary mixture

1191

The graphs confirm the accuracy of this low-frequency approximation. It should be noted that the change in velocity with frequency occurs over four decades, whereas in work on small amplitude acoustic wave propagation in binary mixtures by Atkin [ 181 and Johnson [ 171 the range is two decades. This discrepancy is expected, however, as in [ 181and [ 171the series obtained in the low- and high-frequency limits are either odd or even in the non-dimensional frequency, but investigation in the present problem shows that all terms appear in the relevant series. As may be anticipated, Fig. 1 also shows there is little variation with frequency in the velocity of a constituent when the constituents of the mixture have similar material properties. Although the curves shown in Fig. 1 have Re(G) > Re(F), this inequality does not always hold. In some cases a

Y

0

Fig. 3. Variation of fluid velocities with distance; n = 100.

given admissible choice of parameters leads to solutions in which Re(F) > Re(G ). It is clear from Fig. 1 that at certain frequencies the velocities of the constituents differ. This is investigated further for n = 1 (Fig. 2) and n = 100 (Fig. 3), again taking t=2nm (m=O, 1, . . .). As expected the velocity difference is more marked in the high-frequency range. It follows immediately from Fig. 3 that in some circumstances one fluid is almost at rest whereas there is still appreciable motion of the other. Although no results are given here changes in ~13,,i~~and X, provided little variation to the curves in all three figures. Three points must be borne in mind when interpreting Figs. 2 and 3. Firstly, in the high-frequency region it is possible that the scale of the disturbance becomes comparable with the basic scale of the mixture. In this situation the underlying assumption that the mixture can be viewed as a superposition of single continua may be violated. In addition, although there is a greater velocity difference between the constituents at high-frequency, the form of Y(= y(~rn/2~)~‘*)implies that the bulk of the disturbance is confined to a much smaller region of the mixture. To be more precise, for two given fluids the penetration depth varies as n- I’*. The whole discussion depends critically

UESVoL9No.

12-D

1192

R. E. CRAINE

therefore on the value of a/p, for unless this combination of parameters is sufficiently small the high-frequency effect described here will not in general be noticeable. Finally we note that the footnote on p. 1183 implies that the rate of energy dissipation is O(n). Thus the high-frequency regime can only be realized under isothermal conditions if a large amount of heat is removed by distributed sinks. This is an unacceptably artificial constraint on the validity of the analysis and it is clear that a satisfactory high-frequency solution requires the inclusion of thermal effects. A cknowledgmenrs - 1 am indebted to Professor P. Chadwick for helpful discussions during the course of this work. The research was supported by a grant from the Science Research Council.

REFERENCES [I] C. TRUESDELL, Rational Thermodynamics. A Course of Lectures on Selected Topics. McGraw-Hill (1969). 121 A. E. GREEN and P. M. NAGHDI, Q. J. Mech. appl. Math. 22,427 (1969). [3] R. E. CRAINE, A. E. GREEN and PYM. NAGHDI, Q.J. Mech. appl. Math. 23, 171 (19’70). 141 M. E. GURTIN and G. M. de La PENHA. Arch. ration. Mech. Analvsis 36,390 (I 970). iSj J. E. ADKINS, Phil. Trans. R. Sot. A 255,607 (1963). [6] N. MILLS, Int. J. Engng Sci. 4.97 (1966). [71 I. MULLER,Arch. ration. Mech. Analysis 28, 1 (1968). [8] M. L. DORIA, Arch. ration. Mech. Analysis 32,343 (1969). [9] R. M. BOWEN, J. them. Phys. 49.1625 (1968). [ 101 R. M. BOWEN and D. J. GARCIA, Znt. J. Engng Sci. 8,63 (1970). [ 111 A. C. ERINGEN and J. D. INGRAM, Int. J. Engng Sci. 3,197 (1965). [ 121 J. D. INGRAM and A. C. ERINGEN, Int. J. Engng Sci. 5,289 (1967). 1131 P. G. SAFFMAN,J. FluidMech. 13, 120(1962). [14] D. H. MICHAEL and D. A. MILLER, Mathematika 13.97 (1966). 1151 R. M. BOWEN and J. C. WIESE, Int. J. Ennnn - - Sci. 7, 689 (1969). [16j A. J. M. SPENCER, Treatise on continuum mechanics, Section IB, Chapter 3: the theory ofinvariants (To appear). [ 171 A. F. JOHNSON, Thesis. University of East Anglia (1969). [181 R. J. ATKIN, Q.J. Me& appl. Math. 21. 171 (1968). (Received

29 July 1970)

R&~II& Des equations constitutives sont deduites pour un melange de deux liquides Newtoniens incompressibles et l’ecoulement induit dans un tel melange par l’oscillation permanente d’une plaque infinie est ensuite Ctudie. Des resultats explicites sont obtenus dans les cas limite dans lesquels une frequence convenablement non-dimensiontree est tres petite ou t&s grande. la nature de la solution dans le domaine de frequence intermediaiire &ant discutt numtriquement. Zusammenfassung- Materialgleichungen werden fur eine Mischung von zwei inkompressiblen Newton’schen Flilssigkeiten abgeleitet und die Striimung, die in einer solchen Mischung durch die Schwingung einer unendlichen Platte induziert wird, wird dann untersucht. Ausdrlickliche Resultate werden in den Grenzfallen erhalten, in denen eine nicht-dimensionalisierte Frequenz sehr klein oder sehr gross ist, wobei die Beschaffenheit der Liisung im Mittelfrequenzbereich besprochen wird. Sommario-Si derivano equazioni costitutive per una miscela di due fluidi newtoniani incomprimibili e il flusso indotto in tale miscela dalla continua oscillazione di una piastta infinita e quindi investigato. Si ottengono risultati espliciti nei casi limitanti in cui una frequenza non dimensionalizzata opportuna e piccolissima o grandissima, e la natura della soluzione nella gamma di frequenze intermedia b discussa numericamente. A6crpalc~ KOCTeti

BbIBeAeHbl

II BCCJleAOBaH

THHbI. nOJIy’IeHb1

KOHCTWTYTHBHbIe

IIOTOK,

HaBeAeHHbIti

RBHbIe pe3yJIbTaTbI

He-AHMeHCHOH~H3HpOBaHH~)

xapaxrep

peJIleHHx

B AwanasoHe

YpaBHeHlifl

B IIpAenbHbIX

YaCTOTa

-

AJI5I CMCCH AByX

HeClKHMaeMbIX

B TaKOZt CMeCH CTaIGiOHapHblMA MaJIOfi

IIpOMemyTO’iHblX

Cny’iaflX, UJIU OWHb YaCTOT.

KOI’Aa Wk3pa3MepHaS?N 60nbmofi,

HbH)TOHOBCKliX

KOne6aHliKMU IIpSi+M

KOHeSHOii

XCHAIUIaC-

(T. e. COOTBeTCTBeHHO WiCneHHO

06CyXAaCTCJI