Taylor stability of a thermoviscoelastic fluid in couette flow

TAYLOR STABILITY OF A THERMOVISCOELASTIC FLUID IN COUETTE FLOW Department

of Mathematics,

Oregon State University,

Corvallis, OR 97331, U.S.A.

Abstract-Taylor stability in Couette flow of a thermoviscoeiastic fluid between two rotating coaxial circular cylinders is investigated employing the canstitutive theory of Eringen and Koh. The stability problem is reduced to a characteristic value problem which is solved to obtain stability criteria in terms of critical values of Taylor numbers. The results are compared with those of the classical viscous Ruids and viscoinelastic fluids and the stabiii~ing effects of the thermoviscoelasIjc nature of the fluid are discussed.

1. INTRODUCTION THE

of Couette flow of a viscous fluid between two coaxial rotating cylinders was first investigated by Taylor]l] and the theoretical results obtained were satisfactorily confirmed by his own experiments. Chandr~sekhar~2,3] developed a rigorous mathematical basis for investigating the stability of a wide variety of hydrodynamic and hydromagnetic flows. His concepts include reducing the stabiJity problem to a characteristic value problem employing variational methods and expansion in orthogonal functions. These classicdeveiopments in the field were followed by a number of works of fundamental impartance, i.e. Lin[4], Di Prima351 and a host of other papers on various aspects of the field. Among these papers quite a few were devoted to the investigation of stability of non-Newtonian fluid t?ows as their use in many laboratory experiments as well as technological and b~ophysical studies became crucially important. To cite only a few of these recent works. we have the papers by Craebel[6] and Narasimhan{?]. In the present investigation we study the stability of Couette flow of a thermoviscoelasti~ fluid based on the constitutive equations formulated by Eringen and Koh[8]. Our procedure consists of ending first the solution for the steady state Couette flow problem for the thermoviscoelastic fluid in a concentric rotating cylindrical annuius and then superposing axisymmetric, time-dependent small disturbances on the steady flow, Next, the problem is reduced to a characteristic value problem. The requirement that nontrivial solutions exist for this problem leads to a characteristic equation between the parameters (T, a, LY, IV;),where T is the Taylor number. n represents the wave number of the superposed disturbances and (Y represents the angular velocity ratio of the rotating cylinders and I’$ represent geometric parameters, and thermoviscoelastic properties of the fluid. The characteristic value problem consists in finding those sets of values of these parameters which represent the solution of the system at marginal stability, that is, one of determining the mode of unstable motion which will appear at the onset of instability. This would give a sequence of values of T for different values of a, the lowest of which would be of interest in obtaining marginal or neutral stability because the mode described by this value would appear before others. Thus the characteristic value problem becomes one of finding the minimum of these lowest positive, real values of T for various values of a. The stability criterion is then determined in terms of these critical Tayior numbers based on the prjnciple of exchange of stabilities and the orthogonal development technique of Chandrasekhar [3]. The numerical results obtained are compared with the existing stability investigations for viscous fluids as well as some non-Newtonian fluids for which some data from earlier works are avaitable. STABILITY

2. FORMULATION

OF THE PROBLEM

We consider an incompressible laminar flow of a thermovis~oelastic ftuid between two rotating circular coaxial cylinders with radii RI and R2, lR, > Rt). The inner and outer cylinders

are assumed to rotate about their common axis with different anguIar velocities 0, and GLIS respectively. and are maintained at constant temperatures 8r and 8?, respectively, tBz> 8,). We introduce a cylindrical coordinate system (r, 4,~) where 2 is chosen along the common axis of 303

M.N.L.NARASIMHAN andN.N.GHANDOUR

304

the cylinders, r and 4 are chosen as the radial and azimuthal coordinates, respectively. The velocity components are denoted by U, v and w along the radial, azimuthal and axial directions, respectively. The basic governing equations of incompressible thermoviscoelastic fluid flows are: equation of motion pij = T$ + pf',

(2.1)

Uii = 0,

(2.2)

equation of continuity

equation of heat-conduction

pci=T’” d,,

- q(, + pQ,

(2.3)

constitutive equations [8]:

T = a,1 + a2ii +a,(i;B - Bi) + a,(AB - BA).

(2.4)

H = /3,B + &(Bi + LB) + /3,(BA+ AB),

(2.5)

where p = mass density, c = specific heat of the fluid, f’ = body force per unit mass, v’ = velocity vector, T = (T’j) = stress tensor, 0 = absolute temperature of the fluid, 4’ = heat flux vector, Q = heat energy supply per unit mass and B = (Bii) = (e&r), H = (Hii) = (eiikqk),

(2.6)

are the temperature gradient bivector and heat flux bivector, respectively; the kinematic tensors 1 and i are respectively.

Jiij

=

2dij =

(Ui;j

+

Uj;;).

~ij

=

Ui:j

+

Uj:;

+

2U,:jV ~,

(2.7)

Ui being the acceleration vector; is the well known Levi-Civita alternating tensor; ai and pi are the material and thermal response coefficients. In the above equations the subscript semi-colon followed by an index denotes the usual covariant differentiation with respect to space variables, the superposed dot(.) denotes the well known material derivative and the summation convention is applied over repeated indices. The boundary conditions of the problem are: eijk

u=O=w

at r=R,,

r=Rz.

(2.8)

v(R,)= f&R,, v(RJ = R2R2.

(2.9)

NR,) = 01, 8(Rz) = Oz.

(2.10)

3.STEADY STATE FLOWPROBLEM

We now consider a fully developed, steady state Couette flow of the thermoviscoelastic fluid between two long rotating coaxial circular cylinders. Due to axial symmetry, we have (a/ad) ( ) = 0; Due to infinite length of the channel in the z-direction, we have (a/&)( ) = 0. In the

Taylor stability of a lhermoviscoelaslic kid in couette flow

305

absence of body forces and heat supply, we have f = 0 and Q = 0. Now on comparison with incompressible Newtonian fluids, we can set crl = - p. the hydrostatic pressure, and ty2= ,u, the coefficient of viscosity, Y= p/p being the kinematic coefficient of viscosity. We readily find from the basic equations of the last section that the solutions for the steady state flow problem are : u = 0 = w, u,Jr) = (A/r) + Br,

(3.1)

8,(r) = & i- (C~2~) log (r’ + (A/B)] + D,

(3.2)

(#Ii%) = (p/r)[(Alr) + BrJZ- cr3[(B2/r)+ 2(AB/r3) - IS(A’lr’)] - a4CZ/r(A f Br2)2,

(3.3)

where the subscript s denotes the steady state solutions and the constants A, B, C and D determined by the boundary conditions (2.8)-(2.10) are given by A = i+R:R$,fR:

- Rj), 3 = &[R; - (a + l)R:]/(R: - R;),

C = 2B(& - et)/ tog [(A + ~R~)~(A + BR:)],

(3.4)

D = (0, - 8J log (A + BR:)/ log [(A + BR:)/(A + BR:)], (Y = (St,/&) - 1.

We note here that the solution for the velocity of the steady state problem remains the same as that of a similar problem for Newtonian ffuids, while the thermal and the pressure gradient fields are different.

4. STABILITY

ANALYSIS

OF THE COUETTE

FLOW

We now superpose axisymmetric, time-dependent disturbances with small amplitude on the steady state system obtained above in the following manner. Denoting the steady state field quantities by the subscript s as above and the perturbed fields by the subscript, p, we have in the vector form: v = v, + vp = (e’“‘f(r) cos (Az), v, + e’“‘g(r) cos (hz), e’“‘h(r) sin (hz)} 8 = 0, t 8, = &(r) t e’“‘q(r) cos (AZ), p = ps +p, = p,(r) + e’“‘n(r) cos (AZ),

(3.5)

where the amplitudes of the corresponding disturbances, f, g and k are small compared to vs; 4 is small compared to & while T(T) is small compared to ps. A( > 0) denotes the wave number in the z-direction, and w, in general, would be a complex constant. Re(iw) = amplification factor, and ~~(~~/~) = phase velocity. According to whether Re(iw) is positive, zero, or negative we have an ampli~ed, neutral, or damped disturbance, respectively. Substituting eqns (3.1f-(3.5) into (2-l)-(2.7) and neglecting quadratic terms of disturbances, we obtain the equations governing the amplitude functions of the disturbances. Since our main interest lies in determining the mode of unstable motion which will appear at the onset of instability we only need to consider the case of marginal stability, that is, the case of neutral disturbances characterized by w = 0, as shown by Pellew and Southwell[9]. Furthermore, we note here that solving the resulting set of field equations is an extremely difficult matter. Hence following Chandrasekhar~31, we consider the case of small spacing between the cylinders. The cyIindrical geometry of the fluid layer can be approximated sufficiently we11by a plane layer when RI - RI e f(R, + R,) = Ro.

(4.2)

306

M.N.L.NARASIMHANandN.N.GHANDOUR

This plane

yields the following equations for the amplitudes of the

layer approximation

disturbances: (D’- u’)‘f +

$[(N,+N,&)D’ + u’(N, + N,()D

t [(N, f N&D f

+

Nxt Nbt a'(M,t M,[)]g

dN,+ N,,Olq = 0.

a'Tft (D2- a’)g + [u'(N,, + p4

t

(4.3)

ap4[)D2 t (N,, t N&D

+ u’N,d + N,c + (2u’N,4+ y3N,Xlf = 0,

(4.4)

[a2{(N2z f N:,) + (Nz, f N,,)[}D + (M, -t u'N,,) t (MSt u’N&]f + [(MJ f MvW + CM,+ u?Nd + (M,+ u’N,z,)tlg + [Nz,D’t

u’Ms-

N,,[]q

= 0.

(4.5)

where T is the Tuylor number defined by

T = - 4ABR;/v2

(4.6)

with A and B given by (3.30). 5, N, and Mi are non-dimensional quantities defined by YI = 2(& - &)/log (20 + 2). y? = 2[2 - (I/F)],

5=(r-R,)/d,

OS[SI.

D=d

d5’

u=AR?.

y1 = R,/2R,,

y4 = R,/d,

d=l?-R,,

R0 = (R, t R:)/?, a = (CL/R,) - 1, p, = RJR,: N, = NJ2 + 3p,y4/41. N: = - N3y?, N3 = qv/4pR,

d4. N4 = 3pN,y,d/R,.

N( = - (N,/y4)[2 + (2/y,) + (3~2 dlRo)l, NC,= 4N&, + 3dy31.

N, = - (p2/4)[(a d/R,) t (a t 16)/S], pz = ~y,(& - eJl4p.n; d” R;, /.LJ= 4~3 d*fl:/pv’.

p4= -~~YI(Yd/pv'R,,,

Nx = - FJ48d’+ R0 d(1 - cy’)t a’R0R,]/32R,,R, d, Ns = - pzay3/2, N,o = pJ(d/R,) - a31/2. N,, = ~41 + (YP;YJY~), NE = &-h[(l/y4) - ~21.

N,? = - d(l/~4)+ rzllr4,N,4= ~P;PCLJP~YJ. NIT = - 3~3~2 d/2R,y,, N,6 = (a4/12v2d2)[(Z,+ (a,fIfR$4pRiR,

d3)Z2t (a4yf/4pR,R:

d?)ZJ.

I, = R:[3aR, d(p., - 3) - 18a’~R; - a’R, d - 3d*(l t aR;/2R0 d)‘] Z, = 9a’ - [(16R:, d’ log R,)/R:](I + aR;/2R,, d) - 4a((~ - 2R,, d/R;), I-(= R:(a -2R0d/R;)t4dR010g((uy~/8)

N,, = (d3/4u’)J, t (a$I;R;/4pR;Rf J, = 2fl;[(uy, d(p, - 3) - 3a’~,R,(

d’y,)J, t (a,y$SpaR:R&, 1 - log R,) - a d(1 + aR;/2R,,

d)‘],

J? = 9~’ d t (4aR;R, d’/R;)( I + aR;/2R0 d)’ - 24R,(a’ - 2R,, d/R;), J1 = - 4R;a’ d(u - 2R0 d/R;,’ t 2aR,R:(a

- 2R, d/R;)

+ 8aR;d[2 - log (~yyJ8)+ RfR,Ju - 2R,, d/R;)‘. N,R= ((~&ISpv)[yJl f cu’)(3- p,) + 3a’~,y41. N,s = (~JMpv)[3a’y4(7p,

-4) - cuy,(p, - I)- 3a’~,/21.

N:,, = - (~$I,a’y, d/SpvR,,. Nz, = 5a&, y> d(p, - 1)/SpvR,,

Taylor stability of a thermoviscoelastic fluid in couette flow %, = ff2~~y~132pu2R~, Ntj = (UN&~, Nz4 = - (rgy:cr2/32p~, vR,R:

307

d,

(4.7)

Equations (4.3)-(4.5) will be used next to establish the characteristic value problem the solutions of which will yield the stability criterion for the problem on hand. The boundary conditions for the problem, (2.8)-(2.10), become in view of (3.5): f =O=Df,

g=O at {=O and t= I,

q=Br at f=O; q=&at 5. THE CHARACTERISTIC

Orthogonal decetopment

.$=I.

(4.8)

VALUE PROBLEM

method

Our starting point in the stability analysis of the problem is the set of eqns (4.3)-(4.5) and (4.8). Owing to space fimitations in this paper, we shall confine our attention only to an outline of the main steps in deriving the characteristic equation. We shall first represent the unknown function f by a doubly infinite series of the form: I

x

f = C C emnsin (m7rO sin inn& m=I n=l

(5.1)

where c,,,, are coefficients which clearly satisfies the boundary conditions. We next substitute (5.1) into (4.4). The solution g of the resulting second order linear differential equation with constant coefhcients subject to (4.8) is straightforward. Then the resulting expressions for f and g are both substituted into (4.5) whose formal solution is readily obtained subject to (4.8) in a simple manner. In order to concretize the above procedure, we shall illustrate the stability analysis by considering, for the sake of simplicity, certain types of thermoviscoelastic fluids, governed by the relation: N35= NZ6, that is, MS = 0. Equation (4.5) becomes after substituting the known series solutions for f and g:

CD’- s3)q = Q@,.

(5.2)

where s = (N36/N26)“3and Q(5) is a known function of 5 which results from (4.5) after substituting for f and g the orthogonal series solutions. ft is readily recognized that (5.2) is the non-homogeneous Airy’s equation whose general solution is

s(S)= p,,Ai(sl) + q,,,,Bi(sf) f q,,(t)

(5.3)

308

M.N.L.NARASIMHANandN.N.GHANDOUR

where Ai

and Bi(s[) are the well known Airy’s functions given by:

Ai

= c,go 3k(f), $$

- c& 3k(5), s. (5.4)

with k3k= (3a + 1)(3a +4). . . . . . . . (3~ +3k-2)

(5.5)

for arbitary (Y and k = 1,2,3.. . . . .; cl and c? are the well known Airy’s constants (Abramowitz[lO]; and q,(t) is the particular integral which is readily obtained by the method of variation of parameters; pmn and qmnare determined by the boundary conditions on q(t), given by (4.8). After this straightforward but long algebra, we substitute the solutions for f, g and q obtained above into (4.3) and then integrate the result with respect to 5 from 0 to 1 obtaining $, z1 cmn]Rm”(a,T. Niqa)] = 0, i = 1.2,. . . . ., 36

(5.6)

where F,,,, are known functions of a, T, (Yand Ni. Equations (5.6) form a set of linear homogeneous equations for the coefficients c,,. Non-trivial solutions for c,, exist if and only if the determinant of the coefficient matrix in (5.6) vanishes:

for m,n=1,2 . . . .. i=1,2 ,.... 36; that is the determinant of this matrix should vanish. This leads to the characteristic equation. Thus the solving of this characteristic equation for the minimum lowest, positive real value of T constitutes the characteristic value problem. For given values of Ni, RjR, and &- 8,, a value of u is chosen and eqn (5.7) is solved for the lowest value of T. This procedure is repeated for different values of a until the minimum of these lowest values of T is found. The solving of the infinite order characteristic equation corresponding to (5.7) is accomplished by the approximate method of setting the finite determinant made up of the first k rows and columns equal to zero and solving for T. The usefulness of this method is determined by how rapidly the lowest value of T approaches its limit as k + x. For the classical viscous fluid case, Chandrasekhar [3] has found that a very rapid convergence occurs. For viscoinelastic non-Newtonian fluids, Narasimhan[7] has shown that this holds true and in our present investigation also the same is found to be true. 6.NUMERlCAL RESULTS ANDDISCUSSION

In this section we illustrate our stability analysis by choosing the following sets of data: RI = 1.0 cm., Rz = 1.03cm., ((Y~v/~@,d4) = N3 = 1.0, (16d6@/v3)= 1.0 x 10m3,[(Y~Y,(&-8,)/4&R;]

= 2.0,

= 2.0.(PzyJ4pv’) = 5.0, (~?y,/4~v d*) = 5.0, (~,YId/pv'Ro)

0.5 5 N>, 5 1.

(6.1)

The relation for NZcin (6.1) was chosen to facilitate comparison with the stability analysis of other existing works in the case of viscoinelastic fluids as well as classical viscous fluids. The stability criteria are expressed in terms of the critical Taylor numbers for given values of (Yand

Taylor stability of a thermoviscoelastic fluid in couette flow

309

a. The numerica results and comparison with those of the classical viscous fluids and viscoinelastic fiuids are given in Table I below: Table I. a

a

7-1

T*

T,

0.5 1.0 1.25

4.2 4.5 5.3

1.785x 104 1.869x 104 2.305x 104

I.817 x IO’ 2.286x I@ 2.579x 103

1.486x IO” 2.125x Id 2.521X IO-’

The values of the critical Taylor number in Table 1, T,, T2 and T3 denote the values for the cases of thermoviscoelastic fluid (the present investigation), classical viscous fluid and viscoinelastic, rate-dependent non-Newtonian fluids, respectivety, The values of r, for the classical viscous ffuids have been computed as a special case of our thermoviscoelastic fluids by setting the thermal parameters to zero in our equations. The values of 7’, for the nonNewtonian fluids under similar situation have been computed from the results of Narasimhan[7] by setting the magnetic field intensity to zero. Thus the stability criteria are expressed in terms of the critical Taylor numbers corresponding to different angular velocity ratios of the rotating cylinders as well as for di~erent wave numbers of the superposed disturbances. It is clear from Table I, that the thermoviscoelastic fluids in a Couette flow are more stable than both the classical viscous fluids and the viscoinelastic non-Newtonian fluids under similar situations. This behavior of the flow is essentially due to the viscoelastic nature of the fluid under thermal as well as rotational effects. REFERENCES [I] G. I. TAYLOR. Phil. Trans. Roy. Sot. A223,289 11923). [2] S. CHANDRASEKHAR, Am. Math. Monthly 61(7). 32 (1954). [3] S. CHANDRASEKHAR, Hydrodynamic and Hvdromagnetic Stability. Clarendon Press, Oxford (1961). [4] C. C. LIN, The Theory of Hydrodynamic Stabil;ty. Cambridge University Press, Cambridge (1954). IS] R. C. DI PRIMA, J. Fluid Mech. 6,462 (1959). [6] W. P. GRAEBEL, IUT~M, Proc. Int. Symp. 2nd Order Effacfs in Elastic~fy, Plasticiiy and Fluid dynamics (1962) (Edited by M. Reiner and D. Abir), p. 636. MacMillan, New York (1964). [7J M. N. L. NARASIMHAN, Proc. 4th Inf. Cong. Rheol. fart I (Edited by E. H. Lee and A. L. Copley), p. 345. Wiley,

Interscience, New York (1963). [8] A. C. ERINGEN and S. L. KOH. Int. J. Engng Sci. 1, 199 (1963). [9] A. PELLEW and R. V. SOUTHWELL, Proc. Roy. Sot. A176,312 (1940). [IO] M. ABRAMOWITZ and A. STEGUN, Handbook oj Mathematical Functions. Dover, New York (1965). {Received

1 April 1981)