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A thermal instability problem in a rotating micropolar fluid
Inr. 1. Engng Sci. Vol. 30, No. 9, pp. 1117-1126, 1992 Printed in Great Britain. All rights reserved
A THERMAL
OCQO-7225/92 $5.00 + 0.00 Copyright @ 1992 Pergamon Press Ltd
INSTABILITY PROBLEM IN A ROTATING MICROPOLAR FLUID Y. QIN and P. N. KALONI
Department
of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada N9B 3P4
Abstract-An analysis of the thermal stability problem of an incompressible, rotating, micropolar fluid heated from below is given. Consideration is also given to the possibility of oscillatory convection. It is found that, depending upon the values of various micropolar parameters and the low values of the Taylor number, the rotation has a stabilizing effect. The effect of micropolar parameters on the condition whether stationary convection or oscillatory convection will prevail is briefly pointed out.
1. INTRODUCTION
In 1964, Eringen [l] developed a continuum theory of micro-fluids which takes into account the local motions and deformations of the substructure of the fluid. In its present form, this theory is very rich. In our opinion, however, because of its generality, its potential has not thus far been fully exploited. Several special forms of this theory [2,3] have also been given by Eringen. In a recent paper Eringen [4], has also developed a continuum theory of dense rigid suspensions. When the substructure particles are assumed to be rigid, the special form of the above theory is called the theory of micropolar fluids. The equations for such a theory were also given by Eringen [5]. This theory, because of its simplicity and analytical tractability has been the subject of numerous investigations. Thermal effects in microfluids have also been discussed by Eringen [6]. These have been again specialized to generate a theory of isotropic thermomicropolar fluids. Various successful applications of these theories to liquid crystals, fluid suspensions, polymeric fluids, turbulence and blood flow have been reported in the literature. In the past 15 years there have been several investigations dealing with the thermal instability of a micropolar fluid between two horizontal planes heated from below (or above). This so-called BCnard problem has been studied by Dutta and Sastry [7], Ahmadi [8], Lebon and Parez-Garcia [9] and by Payne and Straughan [lo]. The last mentioned authors also discuss the oscillatory and non-linear convection in an isotropic thermomicropolar fluid. Depending on whether or not a thermal coupling was considered in the energy equation, differences in the results were obtained. Payne and Straughan [lo] have documented an interesting discussion concerning this point. In this paper we present an analytical solution of the thermal stability problem of an incompressible, rotating micropolar fluid heated from below. Some consideration is also given to oscillatory convection. We believe that such studies have relevance to instability problems in geophysics and astrophysics. We remark that some aspects of the stability of a hot rotating micropolar fluid layer have been studied by Sastry and Rao [ll] and Bhattacharyya and Abbas [12]. In [ 111, the convection problem between two rigid boundaries was studied but without taking into account the rotation effect in the angular momentum equation. Bhattacharyya and Abbas [12], on the other hand, used the correct form of the equations and studied the rotating layer problem when the boundaries are free. These authors, however, in our opinion, discussed the characteristic equation from a different point of view. As noted below, we are not able to reproduce their results from our calculations even when we assign values to different parameters as chosen by these authors. We have also compared our results with Chandrasekhar [13], who studied the corresponding problem in a viscous fluid. We point out the effects of consideration of different micropolar parameters. In this study we follow the notations and 1117
1118
Y. QIN and P. N. KALONI
techniques, as much as possible, employed by Payne and Straughan [lo], who have studied the corresponding non-rotating BCnard problem.
2.
BASIC
EQUATIONS
The conservation equations of mass, linear momentum, angular momentum incompressible isotropic therm0 micro-polar fluid in a rotating frame are: Uj,i= 0,
and energy of an (1)
(2) rnki,k
Eiklfkl
+
fklbkl
+
dVi dVi -$ + uj z +
P/i = Pi
+
EijkQjVk
I
mklVk.1
+
qk,k
+
ph
=
pi.
(4)
Here bkl = uk,[ - t+p,, ui is the velocity vector, vi the spin vector, h the body force, Ii the body couple, p the density, i the tiCrOinertia COnStaId, tkl the stress tensor, mk[ the couple stress tensor, e the internal energy, qi the heat flux vector and h the heat supply. The rotation of the frame is denoted by the constant angular velocity vector 52. The constitutive equations for an incompressible micropolar fluid are [6]: tkl
=
-nbkl
mkl
=
+
a,,
p(uk,l
8k1+
qk
=
+
&.k)
bk.l
Ke,k
+
+
+
K(ul,k
vh,k
+
-
&klrvr),
@kd,mt
(5)
(6)
&EklmUI.m,
(7)
where x is the pressure, j& i?, &, 6, jj are viscosity coefficients and K is the heat conduction coefficient. The two coupling terms in equations (6) and (7) have the coefficients 6, and &. All the above constitutive coefficients, which could possibly be functions of temperature 8, are restricted as [6]: 052D-+K, osy+p,
0 5 K,
05
K,
OI3&+p+r,
(6, - &o-1)2 12K0-1(7
- 6).
(8)
The field equations for u, v and 0 are thus obtained by inserting equations (5)-(7) in equations (l)-(4). W e note that each of equations (6) and (7) has one mechanical-thermal coupling term which is of solenoidal nature. Thus these coupling terms do not contribute to div m in equation (3) and div q in equation (4). The term involving 6, in (6), however, survives in the energy equation (4), through the term m k[ u,& and has been the subject of special attention [lo]. In the Boussinesq approximation, which we adopt here, we note that all material coefficients can be assumed constant. For small temperature differences it can be shown that the density p can also be treated as a constant except in the external force term (cf. Chandrasekhar [13]). In fact, there it is governed by P = ~dl
+ 48,
- e)i,
(9)
where p. is the density at some properly chosen mean temperature e. and c~is the coefficient of volume expansion of the fluid. On substituting (5)-(7) in (4), we get @ + Kv28 = pc,, $f + ~l&,&,,,~,k, where
(10)
Thermal instability problem in a rotating micropolar fluid
1119
In (lo), Q, represents the dissipation function involving quadratic terms and the last term on the right-hand side is the coupling term. Following the lead of previous authors ([lo] and [13]), we assume Q, is negligible and following [lo], we also assume that 6, = -b, b > 0. We note that neglect of Q is justified because it contains nonlinear terms which are negligible in the linear theory while the later assumption implies that stationary convection is allowed only by heating from below. Moreover this assumption is not inconsistent with (8). The full system of equations to be studied thus is (l)-(3), (5), (6), (9) and (10) with @ neglected. We assume that the fluid is contained in the layer z E (0, d) with gravity in the negative z-direction and is rotating with constant angular velocity S2 about the z-axis. The adverse temperature gradient is maintained by keeping the planes z = o, d, at constant temperatures 0, and e2 with 8, > 13~. When no motion is present, we have 0= -Bz + e,,
V = 0,
ii=o,
(11)
where /I = (0, - 8,)/d is the adverse temperature gradient. To study the stability of the system (ll), we now consider the perturbation equations on the basis of Boussinesq’s approximation. Accordingly we write,
e= a+ 81, and non-dimensionalize
u=ii+u,
v=t+v ,
the various quantities as: u=L
x = x*d,
u=u*u,
dPo’
Pr=L, KPo
u
v=-v*, d
t* = tplpod2,
G = (& + /!j)/pd’.
The non-dimensionalized
perturbation
equations,
omitting all stars, become
(12)
v-u=o,
(13)
(14) pr G + U . v8
where k = (0,0,l)
>
= RW + v20 - bR&,,v,,,,,,
(15)
and w = u - k and
R=(poF4) = temperature T=(q) Equations
+ bP&,,v,,me,k,
=Taylor
Rayleigh number. number.
(12)-( 15) are to be solved with the boundary conditions
8 = 0,
v=o, on
z = 0, 1,
(16)
and either u = 0 on z = 0, 1 or stress free boundary condition there. As is customary, we also assume periodicity in the x and y-direction. We remark that the principle of exchange of stabilities is not valid in the present case because of the presence of rotation and also because of the inclusion of the coupling term. ES SOS-C
Y. QIN and P. N. KALONI
1120
STATIONARY
3.
CONVECTION
Within the limit of linear theory we assume u(x, t) = e‘%(x), with similar expressions for v and 8. On taking curl and curl curl of linearized version of (13), taking curl and curl curl of the linearized version of (14), respectively and retaining the third components (both in u and v) of the resulting equations we get Lw = -k(V
x V x v) . k - T”*Dw,
(17)
- kAc + T’12Dm,
LAW = -RA*8
(18)
MC = kAw + f T”*c$,
(19)
M(V X V X V) - k = kVo + 2 jT112Dc,
(20)
NB = -Rw
- Rb&
(21)
Ov,+GDE+ko=O.
(22)
L, M, N and 0 are defined as
In the above equations operators
L = (1 + k)A - CT, N = (A - P,a),
M=(rA-2k-ja),
0 = (M + CD*),
(23)
and w = (V X u) . k,
D=-$,
c=(VXv).k,
A*=$+$,
e=(:+$),
G=(a+O),
(24)
2
and A is the three-dimensional Laplacian operator. Equations (17)-(22) are six equations for six unknowns, w, w, v3, g, C and 8. 0 n eliminating different variables, the equation determining w becomes [{&122OD - k2AD(M +kMD(M
+ GA)}{(KNA
+ GA){--:MND2
- R*bA*)(LM
- (LNA - R2A*)(LM
{ -R*A*D(k& where ill = T’” and A2= (1/2)jT”*. R2A* M(LM [ - T(;
+ k*A)(M
+ k*A) + illA2kND2} + K*A)}
- )3,bM) + A,kLNAD
+ A,(LO + K*A*) - il,kNNAD}]w
= 0,
(25)
On solving this equation for R* we get
+ GA) + kb(M + AG)A(LM
+ k*A)
(LO + k2A*) + f kb(M + AG)D*]]w
= N A(M + AG)(LM [
+ k’A)* + T (M + GA)M2D2
1
-$
TOD2 -$(LO
+ k*A*)LA
11 (26) w.
We now restrict our attention to the problem of stress free surfaces and zero micro-rotation the surfaces. Thus following the usual mode form we write w = W(z)@@, Y), where A*$ = -a2@ and a is the wave number. (27) imply Do = 0 at z = 0, 1, we select
Since the boundary
W(z) = sin rmz.
conditions
on
(27) and choice of
(28)
Thermal instability problem in a rotating micropolar fluid
On employing (27) and (28) in equations
(23)-(26)
1121
we obtain
R2a2 {(A+Nc+a)(~A+2k+ja)-k2A}{~A+2k+ja+GA}{(~A+2k+ja)+kbA) [ - T ; [(A + Ak + a)(l-‘A + 2k + ju + Gn*n*) - k*u*]
+ ; kb(TA + 2k + ju + GA)n*n*
=
(A + p,u)[A(I’A
+ 2k +ju){(A
II + Ak + u)(rA + 2k +ju) - k*A}* .2
(I’A + 2k +ju + GA)(TA + 2k +ju)*n*n* - $ T(l-‘A + 2k +ju + Gn*n*)n*n*
-f
[(A + Ak + u)(rA + 2k +ju + G~*Jc*) - k*u*][(A + Ak + u)A]]].
(29)
For the stationary marginal state we set u = 0 in (29) and then obtain [I-(1 + k)A + 2k + k2]A
R&li3
a A[(T+kb)A+2k]-#B+kbC} +z
{ (f’h + GA + 2k)(l-‘A + 2k)*n*n*}
u2A[(r+kb)A+2k]-T;[$3+kbC} --- T j’ { T(f’A + 2k + Gn2~2)n2n2 + [A(1 + k)(TA + 2k + Gn*n*) - k*u*](l u* 4 A[(T+kb)A+2k]-T;{$+kbC}
+ k)A*}
(30) where A = {[(r + G)A + 2k][A(l+
k)T + 2k + k*]},
B = ((1 + k)A(TA + 2k + Gn*n*) - k2a2}, c =
{[(r + G)A
+ 2k]h*),
A = n2n2 + a*.
(31)
For the lowest value of R*, it appears natural to assume n = 1. At least for lower values of T, this seems to be reasonable. When T = 0, equation (30) reduces to the expression obtained by Payne and Straughan [lo]. However, when T # 0 but j = 0, (30) corresponds to the situation considered by Sastry and Rao [ll]. In this case, the result reads R2=1\3[I’(l+k)A+2k+k2] u*
[(l- + kb)A + 2k]
(I-A + 2k)‘n’ +r u2 [(I’(1 + k)A + 2k + k*][(l? + kb)A + 2k] * Since the second term at the right-hand side of (32) is always positive, the effect this case, is likely to have a stabilizing effect (cf. Sastry and Rao [ll]). However, case, i.e. when T # 0, j # 0 several interesting possibilities, depending upon the various parameters, seem to occur and hence mathematical corroboration has
(32) of rotation, in in the general magnitudes of some interest.
Y. QIN and P. N. KALONI
1122
From (30) we note that all j terms are related with the Taylor number T and thus whereas there is no effect of j in the stationary convection of a non-rotational problem, the presence of j (micro-inertia) affects the value of R2 in a rotational problem (cf. [12]). Bhattacharyya and Abbas [ 121, consider equations equivalent to our set (17)-(22). While we found concise expressions (26), (29) and (30), these authors found it difficult to solve their characteristic equation analytically. In the final section we discuss numerical results pertaining to equation (30) and find results which appear different from these authors. In fact, for reasonable values of the different parameters and for low values of T, we find that the presence of the last expression in (30) is not very significant in characterizing the effect of inhibiting convection.
4. OSCILLATORY
CONVECTION
To study oscillatory convection, we follow Chandrasekhar [13] and write o = ioi, oi E R, in (29) and then separate the real and imaginary parts. The complete set of expressions becomes quite unwieldy and we, therefore, consider situation when T is low and j is small. Equating real and imaginary parts, in this case, gives R2a2 = A(EA
- P,a:)(FF’
+ j2u:) _ j u:(EP,
(F’2 +j2uf)
+ j”uf)(EA
+~‘c$[(F’~
+j’uf)(E’+
(FF’ +j’uf)(EP,
+j2a:)[(Ff2
+ Tn2 (Fr2 + j2uf)[(F’2
+ Tn2(F’2
-ju:P,)]
-j&)
+ k4A2]
- A) - k2A(jA + FP,)] a:) - 2k2A(EF
- EjA) _ k2A2(P,F’
(F’2 +j2u;) (FF’ + j2u:)[(F2
j&u:)
+
(F’2 +j’u:)
- k2A(FA
uf) - 2k2A(EF
+j’u:)(E’+
+ A)A _ kbA’(jP,u:
(F’2 + j’u:)
-
+ Pp:)
[(Fr2 + j2a:)(EP, - Tn2kbAu’j(F’2
k2A2(F’A
(F” +j2&)
(FF’ +j’u:)[(F” + Tn2(Ft2
+ A)kbA
-+a:)
+ k4A2] ’
(33)
- jA)
(F’2 + j’u;) + j2u:)(EPr
+j2uf)(E2
- A) - k2A(jA + Fp,)]
+ 6)
- 2k2A(EF
- ju:) + k4A2]
kbAj{ (F2 + j’u:)(EA
+ P,u;) - k2A(FA - ju:P,)}
+j2uf)[(F2
+ a:) - 2k2A(EF
+j’u;)(E’
+a:)
+ k4A2] =”
(34)
where E = (1 + k)A,
F = (I-A + 2k),
F’ = (IA + 2k + kbA).
We note that when T = 0, the above equations reduce to equations (21) and (22), respectively, of [lo]. Equation (33) and (34) are two equations for two unknowns R2 and 6. In principle, we should solve (34) for u: (which turns out to be cubic in a:) and substitute this value in (33) and then minimize R2 with respect to u2. This clearly can be accomplished numerically. However, since the parameters, k and j are likely to be small we simplify the above equations by neglecting 0(k2) terms and considering the case when b = 0. As a result on setting FF’ = F12 and simplifying (33) and (34), we find R2=[(1+k)A2-Pruf];;I+7
A
T3t2 (1 + k)A2 + P,u: (1 + k)‘A2 + a:
(35)
and u2
=
’
1- P,(l + k) T-x2-
l+P,(l+k)
A2
(1
+ k)*A2.
(36)
1123
Thermal instability problem in a rotating micropolar fluid
At this point it is convenient
to define (cf. [13])
R:=$, $.=$
a2 X-_-
Jr
2’
With the use of (37), equations (35) and (36), respectively p
= l
(1+ kN
+
4”
+
P,r12T,
+
(l+~)2(l+x)2+~z
(1
x
+x)
[(I
?j=$. become
+
k)@+
42
_
X
1-P,(l+k) Tl n2=1+P,(1+k)(1+X)
(37)
p
r
q2]
(38)
f
- (1 + k)2( 1 + X)“.
(39)
Equation (39) can be written as (40) and using this equation in (38), we get (41) For over-stability to be possible, it follows from (39) that P, must be less than l/(1 f k), i.e. P, < l/(1 + k). Since k > 0, this value is smaller than that of the viscous flow case. Even in the situation when this condition is met we obtain real frequencies n2 only if T
1-k et1 + k)
1_p,(l+k)(1+~)2t1+~)3.
>
1
For a given T,, overstable
solution is possible only for x
When x =x*,
l-P,(l+k) Tl l-I_ P,( 1+ k) (1-t k)Z *
(43)
setting q2 = 0 in (38) leads to
+l
x*
* [&)+(1+x”).(l+k)]
It is clear that RT given by (44) corresponds to the approximate value as obtained in stationary convection for a wave number corresponding to x*. This x* is obtained by solving (43) for the given P, and cl. When n >x*, overstability is not possible, and the onset of instability as stationary convection remains the only possibility. When x < x *, overstability is possible, and following [13], we assume that the manner of discriminating between two is decided by a solution corres~nding to the lower Rayleigh number. In the (x, R,) plane (see Fig. l), we have first the curve, labelled “convection”, (45) which defines the locus of states which are marginal with respect to stationary convection. The overstable solutions branch off from this locus at the point x*, and for x
[1+ =x1 + k)l(l +x) # (1+k)
=i (1+ k)[l+
P,(1+
X
Ql[ cl+ XI3+ f1+ p;;+ r
k)]2
ij
*
(46)
Y. QIN and P. N. KALONI
1124 3.0
2.5
R,
2.0
1.5
1.0 _
u
1
2
3
4
5
X
Fig. 1. The disposition of the curves for marginal stability in the (x, R,) plane for a Taylor number, T = 104, for the parameter k = 0.01. The curve labelled “convection” applies for all values of P, and defines the locus R’,“. The remaining curves are the overstable loci RI”) for values of P, by which they are labelled. The minimum for the curves C(P, = 0.5054) and for the convection curve occurs for the same value of R,.
The first of (46) shows that R?*x$$,, then for all x
CT, 2r3 + 3x2 = ’ + [l + P,(l t k)12
(47)
with X, so determined, as the root of this cubic equation, we substitute this value in (35) and determine the required critical Rayleigh number R$, for overstability.
5.
DISCUSSION
We shall now provide a brief discussion of the roles played by various micropolar parameters and the rotation on the flow and instability characteristics. Tables l-4 show the trend of the variation of critical Rayleigh numbers and the critical wave numbers as parameters are varied. For various physical reasons we have assumed that stationary convection is possible only by heating from below. Besides assuming b > 0, for low values of T, we further require that A[(T + kb)A + 2k] > T(j/2)[(j/2)B + kbC], where A, B, C are defined in (31). For reasonable values of the parameters, for example, similar to those chosen in [lo], we have found that the
Thermal instability problem in a rotating micropolar fluid Table 1. Stationary convection
results when r = G = k = j = 0.1
T = 1.0
T=O
1125
T = 10.0
T = 5.0
b
4
R:
4
RZ
4
RE
a2
Rf
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
4.923 4.947 4.967 4.985 5.000 5.014 5.026 5.036 5.046 5.055 5.063
719.348 661.106 611.581 568.954 531.878 499.336 470.545 444.891 421.889 401.147 382.349
5.013 5.042 5.066 5.087 5.106 5.122 5.137 5.150 5.161 5.172 5.182
728.734 670.092 620.175 577.172 539.740 506.865 477.761 451.817 428.543 407.548 388.514
5.410 5.460 5.503 5.541 5.574 5.603 5.629 5.652 5.673 5.692 5.709
768.427 708.117 656.555 611.973 573.045 538.761 508.341 481.166 456.744 434.678 414.643
5.994 6.078 6.150 6.213 6.268 6.316 6.359 6.397 6.432 6.463 6.491
823.599 760.999 707.159 660.380 619.364 583.112 550.849 521.950 495.924 472.357 450.920
E l:o
Table 2. Stationary convection results when F = G = k = j = 0.01
T = 1.0
T=O
T = 10.0
T = 5.0
b
aI
RZ
6
RZ
4
RZ
4
R:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4.934 4.957 4.978 4.995 5.024 5.037 5.047 5.057
663.695 609.954 564.258 524.927 490.718 460.692 434.127 410.458 389.235 370.098 352.754
5.027 5.056 5.082 5.104 5.123 5.140 5.155 5.168 5.180 5.191 5.201
672.617 618.563 572.541 532.885 498.363 468.037 441.188 417.250 395.774 376.400 358.834
5.436 5.492 5.540 5.581 5.618 5.650 5.679 5.705 5.728 5.749 5.768
710.273 654.940 607.572 566.569 530.738 499.155 471.113 446.048 423.509 403.135 384.628
6.035 6.132 6.215 6.288 6.351 6.407 6.457 6.501 6.541 6.577 6.610
762.434 705.401 656.212 613.376 575.739 542.422 512.726 486.091 462.074 440.307 420.490
::EE 5.088
Table 3. Stationary convection results when r = 2.0, k = j = G = 1.0
T = 1.0
T = 5.0
T = 10.0
T = 50.0
T = 100.0
b
2
R:
d
R%
4
R:
a:
R:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4.926 4.935 4.944 4.952 4.959 4.966 4.972 4.978 4.983 4.988 4.993
1299.29 1241.27 1188.22 1139.51 1094.64 1053.16 1014.72 978.980 945.672 914.556 885.422
5.041 5.061 5.079 5.095 5.111 5.125 5.138 5.150 5.162 5.173 5.183
1319.99 1261.52 1208.01 1158.84 1113.53 1071.62 1032.76 996.610 931.413 901.910
5.193 5.227 5.258 5.286 5.313 5.338 5.361 5.382 5.403 5.422 5.439
1346.69 1287.67 1233.61 1183.91 1138.07 1095.64 1056.26 1019.62 985.437 953.472 923.510
6.764 6.964 7.151 7.324 7.486 7.636 7.775
1601.22 1540.12 1483.74 1431.48 1382.91 1337.58 1295.15 1255.39 1217.99 1182.78 1149.56
;:FZ 8.141 8.248
4
RZ
9.758 10.237 10.664 11.046 11.389 11.698 11.979 12.234 12.466 12.680 12.876
Table 4. Stationary convection results when r = k = G = 0.01
T = 1.0 j 0.001 0.0 0.001 0.0 0.001 0.0
b 0 0 0.1 0.1
4 2
4.956 4.956 4.981 4.979 5.003 5.000
T = 10 Rf
665.741 665.672 611.917 611.763 566.138 565.924
ac 2
5.155 5.148 5.190 5.172 5.221 5.194
T=ld Rf
683.785 683.090 629.249 627.693 582.757 580.600
2 =c
6.784 6.698 6.931 6.727 7.061 6.753
Rf
839.023 831.338 779.823 763.230 728.322 705.427
2066.20 2007.93 1951.91 1898.01 1846.16 1796.31 1748.53 1702.65 1658.62 1616.50 1576.10
Y. QIN and P. N. KALONI
1126
above condition is satisfied when T has lower values. In order to make meaningful comparison to detect the effect of rotation we have given values to different parameters, similar to those in Tables l-4 of Payne and Straughan [lo]. We point out that for T = 0 our results in Table 1 agree with those of Payne and Straughan [lo]. From Table 1 we draw the conclusion that stationary convection with rotation also predicts reduction in Rz values for increasing coupling number b. The effect of rotation, however, appears to slow down the reduction in Rz values. Moreover, as T increases, the value of R: increases, which clearly indicates that in a micropolar fluid also, the rotation has a stabilizing effect. From Table 2 we also note that for smaller values of the micropolar parameters the critical values of Rayleigh number become lower. In Table 3 we have considered very high values of the parameters. In this case we again found that the effect of increasing b is to decrease R& and that Rz increases as T increases, indicating again that rotation has stabilizing effect. Table 4 shows the variation of j. A comparison with Table 2 values shows that as j decreases so does Rz, but the effect of variation of j alone is not very significant for small values ofj. From the above we conclude that for the cases we considered, the effect of rotation in a stationary convection, is stabilizing and that higher the values of the micropolar parameters, the more the stabilizing effect. For oscillatory convection we note that for the simplifications we introduced, there is not much difference between our graph 1 and the graph for a Newtonian fluid (cf. [13], p. 117) for low values of the parameter k. Hence the classification which is applicable there also applies in the present case. Acknowledgements-The work reported in this paper has been supported by Grant #A7728 of the N.S.E.R.C. of Canada. The authors gratefully acknowledge the support thus received. They also thank a referee for helpful comments.
REFERENCES C. ERINGEN, Int. /. Engng Sci. 2, 189 (1964). C. ERINGEN, Int. J. Engng Sci. 7, 115 (1969). C. ERINGEN, Znt. /. Engng Sci. 18, 5 (1980). C. ERINGEN, Rheol. Acta 30,23 (1991). C. ERINGEN, J. Math. Mach. 16, 1 (1966). C. ERINGEN, J. Math. Anayt. Appl. 38, 480 (1972). B. DA’I’TA and V. I. K. SASTRY, Int. J. Engng Sci. 14, 631 (1976). AHMADI, Inf. J. Engng Sci. 14, 81 (1976). LEBON and C. PEREZ-GARCIA, Int. J. Engng Sci. 19, 1321 (1981). E. PAYNE and B. STRAUGHAN, Int. J. Engng Sci. 27, 827 (1989). U. K. SASTRY and V. R. RAO, Int. J. Engng Sci. 5,449 (1983). [12] S. P. BHATTACHARYYA and M. ABBAS, Int. J. Engng Sci. 23, 371 (1985). [13] S. CHANDRASEKHAR, Hydrodynamics and Hydromugnetic Stability. Dover, New York (1981). [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo] (1 l]
A. A. A. A. A. A. A. G. G. L. V.
(Revision received and accepted 13 January
1992)
A THERMAL
OCQO-7225/92 $5.00 + 0.00 Copyright @ 1992 Pergamon Press Ltd
INSTABILITY PROBLEM IN A ROTATING MICROPOLAR FLUID Y. QIN and P. N. KALONI
Department
of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada N9B 3P4
Abstract-An analysis of the thermal stability problem of an incompressible, rotating, micropolar fluid heated from below is given. Consideration is also given to the possibility of oscillatory convection. It is found that, depending upon the values of various micropolar parameters and the low values of the Taylor number, the rotation has a stabilizing effect. The effect of micropolar parameters on the condition whether stationary convection or oscillatory convection will prevail is briefly pointed out.
1. INTRODUCTION
In 1964, Eringen [l] developed a continuum theory of micro-fluids which takes into account the local motions and deformations of the substructure of the fluid. In its present form, this theory is very rich. In our opinion, however, because of its generality, its potential has not thus far been fully exploited. Several special forms of this theory [2,3] have also been given by Eringen. In a recent paper Eringen [4], has also developed a continuum theory of dense rigid suspensions. When the substructure particles are assumed to be rigid, the special form of the above theory is called the theory of micropolar fluids. The equations for such a theory were also given by Eringen [5]. This theory, because of its simplicity and analytical tractability has been the subject of numerous investigations. Thermal effects in microfluids have also been discussed by Eringen [6]. These have been again specialized to generate a theory of isotropic thermomicropolar fluids. Various successful applications of these theories to liquid crystals, fluid suspensions, polymeric fluids, turbulence and blood flow have been reported in the literature. In the past 15 years there have been several investigations dealing with the thermal instability of a micropolar fluid between two horizontal planes heated from below (or above). This so-called BCnard problem has been studied by Dutta and Sastry [7], Ahmadi [8], Lebon and Parez-Garcia [9] and by Payne and Straughan [lo]. The last mentioned authors also discuss the oscillatory and non-linear convection in an isotropic thermomicropolar fluid. Depending on whether or not a thermal coupling was considered in the energy equation, differences in the results were obtained. Payne and Straughan [lo] have documented an interesting discussion concerning this point. In this paper we present an analytical solution of the thermal stability problem of an incompressible, rotating micropolar fluid heated from below. Some consideration is also given to oscillatory convection. We believe that such studies have relevance to instability problems in geophysics and astrophysics. We remark that some aspects of the stability of a hot rotating micropolar fluid layer have been studied by Sastry and Rao [ll] and Bhattacharyya and Abbas [12]. In [ 111, the convection problem between two rigid boundaries was studied but without taking into account the rotation effect in the angular momentum equation. Bhattacharyya and Abbas [12], on the other hand, used the correct form of the equations and studied the rotating layer problem when the boundaries are free. These authors, however, in our opinion, discussed the characteristic equation from a different point of view. As noted below, we are not able to reproduce their results from our calculations even when we assign values to different parameters as chosen by these authors. We have also compared our results with Chandrasekhar [13], who studied the corresponding problem in a viscous fluid. We point out the effects of consideration of different micropolar parameters. In this study we follow the notations and 1117
1118
Y. QIN and P. N. KALONI
techniques, as much as possible, employed by Payne and Straughan [lo], who have studied the corresponding non-rotating BCnard problem.
2.
BASIC
EQUATIONS
The conservation equations of mass, linear momentum, angular momentum incompressible isotropic therm0 micro-polar fluid in a rotating frame are: Uj,i= 0,
and energy of an (1)
(2) rnki,k
Eiklfkl
+
fklbkl
+
dVi dVi -$ + uj z +
P/i = Pi
+
EijkQjVk
I
mklVk.1
+
qk,k
+
ph
=
pi.
(4)
Here bkl = uk,[ - t+p,, ui is the velocity vector, vi the spin vector, h the body force, Ii the body couple, p the density, i the tiCrOinertia COnStaId, tkl the stress tensor, mk[ the couple stress tensor, e the internal energy, qi the heat flux vector and h the heat supply. The rotation of the frame is denoted by the constant angular velocity vector 52. The constitutive equations for an incompressible micropolar fluid are [6]: tkl
=
-nbkl
mkl
=
+
a,,
p(uk,l
8k1+
qk
=
+
&.k)
bk.l
Ke,k
+
+
+
K(ul,k
vh,k
+
-
&klrvr),
@kd,mt
(5)
(6)
&EklmUI.m,
(7)
where x is the pressure, j& i?, &, 6, jj are viscosity coefficients and K is the heat conduction coefficient. The two coupling terms in equations (6) and (7) have the coefficients 6, and &. All the above constitutive coefficients, which could possibly be functions of temperature 8, are restricted as [6]: 052D-+K, osy+p,
0 5 K,
05
K,
OI3&+p+r,
(6, - &o-1)2 12K0-1(7
- 6).
(8)
The field equations for u, v and 0 are thus obtained by inserting equations (5)-(7) in equations (l)-(4). W e note that each of equations (6) and (7) has one mechanical-thermal coupling term which is of solenoidal nature. Thus these coupling terms do not contribute to div m in equation (3) and div q in equation (4). The term involving 6, in (6), however, survives in the energy equation (4), through the term m k[ u,& and has been the subject of special attention [lo]. In the Boussinesq approximation, which we adopt here, we note that all material coefficients can be assumed constant. For small temperature differences it can be shown that the density p can also be treated as a constant except in the external force term (cf. Chandrasekhar [13]). In fact, there it is governed by P = ~dl
+ 48,
- e)i,
(9)
where p. is the density at some properly chosen mean temperature e. and c~is the coefficient of volume expansion of the fluid. On substituting (5)-(7) in (4), we get @ + Kv28 = pc,, $f + ~l&,&,,,~,k, where
(10)
Thermal instability problem in a rotating micropolar fluid
1119
In (lo), Q, represents the dissipation function involving quadratic terms and the last term on the right-hand side is the coupling term. Following the lead of previous authors ([lo] and [13]), we assume Q, is negligible and following [lo], we also assume that 6, = -b, b > 0. We note that neglect of Q is justified because it contains nonlinear terms which are negligible in the linear theory while the later assumption implies that stationary convection is allowed only by heating from below. Moreover this assumption is not inconsistent with (8). The full system of equations to be studied thus is (l)-(3), (5), (6), (9) and (10) with @ neglected. We assume that the fluid is contained in the layer z E (0, d) with gravity in the negative z-direction and is rotating with constant angular velocity S2 about the z-axis. The adverse temperature gradient is maintained by keeping the planes z = o, d, at constant temperatures 0, and e2 with 8, > 13~. When no motion is present, we have 0= -Bz + e,,
V = 0,
ii=o,
(11)
where /I = (0, - 8,)/d is the adverse temperature gradient. To study the stability of the system (ll), we now consider the perturbation equations on the basis of Boussinesq’s approximation. Accordingly we write,
e= a+ 81, and non-dimensionalize
u=ii+u,
v=t+v ,
the various quantities as: u=L
x = x*d,
u=u*u,
dPo’
Pr=L, KPo
u
v=-v*, d
t* = tplpod2,
G = (& + /!j)/pd’.
The non-dimensionalized
perturbation
equations,
omitting all stars, become
(12)
v-u=o,
(13)
(14) pr G + U . v8
where k = (0,0,l)
>
= RW + v20 - bR&,,v,,,,,,
(15)
and w = u - k and
R=(poF4) = temperature T=(q) Equations
+ bP&,,v,,me,k,
=Taylor
Rayleigh number. number.
(12)-( 15) are to be solved with the boundary conditions
8 = 0,
v=o, on
z = 0, 1,
(16)
and either u = 0 on z = 0, 1 or stress free boundary condition there. As is customary, we also assume periodicity in the x and y-direction. We remark that the principle of exchange of stabilities is not valid in the present case because of the presence of rotation and also because of the inclusion of the coupling term. ES SOS-C
Y. QIN and P. N. KALONI
1120
STATIONARY
3.
CONVECTION
Within the limit of linear theory we assume u(x, t) = e‘%(x), with similar expressions for v and 8. On taking curl and curl curl of linearized version of (13), taking curl and curl curl of the linearized version of (14), respectively and retaining the third components (both in u and v) of the resulting equations we get Lw = -k(V
x V x v) . k - T”*Dw,
(17)
- kAc + T’12Dm,
LAW = -RA*8
(18)
MC = kAw + f T”*c$,
(19)
M(V X V X V) - k = kVo + 2 jT112Dc,
(20)
NB = -Rw
- Rb&
(21)
Ov,+GDE+ko=O.
(22)
L, M, N and 0 are defined as
In the above equations operators
L = (1 + k)A - CT, N = (A - P,a),
M=(rA-2k-ja),
0 = (M + CD*),
(23)
and w = (V X u) . k,
D=-$,
c=(VXv).k,
A*=$+$,
e=(:+$),
G=(a+O),
(24)
2
and A is the three-dimensional Laplacian operator. Equations (17)-(22) are six equations for six unknowns, w, w, v3, g, C and 8. 0 n eliminating different variables, the equation determining w becomes [{&122OD - k2AD(M +kMD(M
+ GA)}{(KNA
+ GA){--:MND2
- R*bA*)(LM
- (LNA - R2A*)(LM
{ -R*A*D(k& where ill = T’” and A2= (1/2)jT”*. R2A* M(LM [ - T(;
+ k*A)(M
+ k*A) + illA2kND2} + K*A)}
- )3,bM) + A,kLNAD
+ A,(LO + K*A*) - il,kNNAD}]w
= 0,
(25)
On solving this equation for R* we get
+ GA) + kb(M + AG)A(LM
+ k*A)
(LO + k2A*) + f kb(M + AG)D*]]w
= N A(M + AG)(LM [
+ k’A)* + T (M + GA)M2D2
1
-$
TOD2 -$(LO
+ k*A*)LA
11 (26) w.
We now restrict our attention to the problem of stress free surfaces and zero micro-rotation the surfaces. Thus following the usual mode form we write w = W(z)@@, Y), where A*$ = -a2@ and a is the wave number. (27) imply Do = 0 at z = 0, 1, we select
Since the boundary
W(z) = sin rmz.
conditions
on
(27) and choice of
(28)
Thermal instability problem in a rotating micropolar fluid
On employing (27) and (28) in equations
(23)-(26)
1121
we obtain
R2a2 {(A+Nc+a)(~A+2k+ja)-k2A}{~A+2k+ja+GA}{(~A+2k+ja)+kbA) [ - T ; [(A + Ak + a)(l-‘A + 2k + ju + Gn*n*) - k*u*]
+ ; kb(TA + 2k + ju + GA)n*n*
=
(A + p,u)[A(I’A
+ 2k +ju){(A
II + Ak + u)(rA + 2k +ju) - k*A}* .2
(I’A + 2k +ju + GA)(TA + 2k +ju)*n*n* - $ T(l-‘A + 2k +ju + Gn*n*)n*n*
-f
[(A + Ak + u)(rA + 2k +ju + G~*Jc*) - k*u*][(A + Ak + u)A]]].
(29)
For the stationary marginal state we set u = 0 in (29) and then obtain [I-(1 + k)A + 2k + k2]A
R&li3
a A[(T+kb)A+2k]-#B+kbC} +z
{ (f’h + GA + 2k)(l-‘A + 2k)*n*n*}
u2A[(r+kb)A+2k]-T;[$3+kbC} --- T j’ { T(f’A + 2k + Gn2~2)n2n2 + [A(1 + k)(TA + 2k + Gn*n*) - k*u*](l u* 4 A[(T+kb)A+2k]-T;{$+kbC}
+ k)A*}
(30) where A = {[(r + G)A + 2k][A(l+
k)T + 2k + k*]},
B = ((1 + k)A(TA + 2k + Gn*n*) - k2a2}, c =
{[(r + G)A
+ 2k]h*),
A = n2n2 + a*.
(31)
For the lowest value of R*, it appears natural to assume n = 1. At least for lower values of T, this seems to be reasonable. When T = 0, equation (30) reduces to the expression obtained by Payne and Straughan [lo]. However, when T # 0 but j = 0, (30) corresponds to the situation considered by Sastry and Rao [ll]. In this case, the result reads R2=1\3[I’(l+k)A+2k+k2] u*
[(l- + kb)A + 2k]
(I-A + 2k)‘n’ +r u2 [(I’(1 + k)A + 2k + k*][(l? + kb)A + 2k] * Since the second term at the right-hand side of (32) is always positive, the effect this case, is likely to have a stabilizing effect (cf. Sastry and Rao [ll]). However, case, i.e. when T # 0, j # 0 several interesting possibilities, depending upon the various parameters, seem to occur and hence mathematical corroboration has
(32) of rotation, in in the general magnitudes of some interest.
Y. QIN and P. N. KALONI
1122
From (30) we note that all j terms are related with the Taylor number T and thus whereas there is no effect of j in the stationary convection of a non-rotational problem, the presence of j (micro-inertia) affects the value of R2 in a rotational problem (cf. [12]). Bhattacharyya and Abbas [ 121, consider equations equivalent to our set (17)-(22). While we found concise expressions (26), (29) and (30), these authors found it difficult to solve their characteristic equation analytically. In the final section we discuss numerical results pertaining to equation (30) and find results which appear different from these authors. In fact, for reasonable values of the different parameters and for low values of T, we find that the presence of the last expression in (30) is not very significant in characterizing the effect of inhibiting convection.
4. OSCILLATORY
CONVECTION
To study oscillatory convection, we follow Chandrasekhar [13] and write o = ioi, oi E R, in (29) and then separate the real and imaginary parts. The complete set of expressions becomes quite unwieldy and we, therefore, consider situation when T is low and j is small. Equating real and imaginary parts, in this case, gives R2a2 = A(EA
- P,a:)(FF’
+ j2u:) _ j u:(EP,
(F’2 +j2uf)
+ j”uf)(EA
+~‘c$[(F’~
+j’uf)(E’+
(FF’ +j’uf)(EP,
+j2a:)[(Ff2
+ Tn2 (Fr2 + j2uf)[(F’2
+ Tn2(F’2
-ju:P,)]
-j&)
+ k4A2]
- A) - k2A(jA + FP,)] a:) - 2k2A(EF
- EjA) _ k2A2(P,F’
(F’2 +j2u;) (FF’ + j2u:)[(F2
j&u:)
+
(F’2 +j’u:)
- k2A(FA
uf) - 2k2A(EF
+j’u:)(E’+
+ A)A _ kbA’(jP,u:
(F’2 + j’u:)
-
+ Pp:)
[(Fr2 + j2a:)(EP, - Tn2kbAu’j(F’2
k2A2(F’A
(F” +j2&)
(FF’ +j’u:)[(F” + Tn2(Ft2
+ A)kbA
-+a:)
+ k4A2] ’
(33)
- jA)
(F’2 + j’u;) + j2u:)(EPr
+j2uf)(E2
- A) - k2A(jA + Fp,)]
+ 6)
- 2k2A(EF
- ju:) + k4A2]
kbAj{ (F2 + j’u:)(EA
+ P,u;) - k2A(FA - ju:P,)}
+j2uf)[(F2
+ a:) - 2k2A(EF
+j’u;)(E’
+a:)
+ k4A2] =”
(34)
where E = (1 + k)A,
F = (I-A + 2k),
F’ = (IA + 2k + kbA).
We note that when T = 0, the above equations reduce to equations (21) and (22), respectively, of [lo]. Equation (33) and (34) are two equations for two unknowns R2 and 6. In principle, we should solve (34) for u: (which turns out to be cubic in a:) and substitute this value in (33) and then minimize R2 with respect to u2. This clearly can be accomplished numerically. However, since the parameters, k and j are likely to be small we simplify the above equations by neglecting 0(k2) terms and considering the case when b = 0. As a result on setting FF’ = F12 and simplifying (33) and (34), we find R2=[(1+k)A2-Pruf];;I+7
A
T3t2 (1 + k)A2 + P,u: (1 + k)‘A2 + a:
(35)
and u2
=
’
1- P,(l + k) T-x2-
l+P,(l+k)
A2
(1
+ k)*A2.
(36)
1123
Thermal instability problem in a rotating micropolar fluid
At this point it is convenient
to define (cf. [13])
R:=$, $.=$
a2 X-_-
Jr
2’
With the use of (37), equations (35) and (36), respectively p
= l
(1+ kN
+
4”
+
P,r12T,
+
(l+~)2(l+x)2+~z
(1
x
+x)
[(I
?j=$. become
+
k)@+
42
_
X
1-P,(l+k) Tl n2=1+P,(1+k)(1+X)
(37)
p
r
q2]
(38)
f
- (1 + k)2( 1 + X)“.
(39)
Equation (39) can be written as (40) and using this equation in (38), we get (41) For over-stability to be possible, it follows from (39) that P, must be less than l/(1 f k), i.e. P, < l/(1 + k). Since k > 0, this value is smaller than that of the viscous flow case. Even in the situation when this condition is met we obtain real frequencies n2 only if T
1-k et1 + k)
1_p,(l+k)(1+~)2t1+~)3.
>
1
For a given T,, overstable
solution is possible only for x
When x =x*,
l-P,(l+k) Tl l-I_ P,( 1+ k) (1-t k)Z *
(43)
setting q2 = 0 in (38) leads to
+l
x*
* [&)+(1+x”).(l+k)]
It is clear that RT given by (44) corresponds to the approximate value as obtained in stationary convection for a wave number corresponding to x*. This x* is obtained by solving (43) for the given P, and cl. When n >x*, overstability is not possible, and the onset of instability as stationary convection remains the only possibility. When x < x *, overstability is possible, and following [13], we assume that the manner of discriminating between two is decided by a solution corres~nding to the lower Rayleigh number. In the (x, R,) plane (see Fig. l), we have first the curve, labelled “convection”, (45) which defines the locus of states which are marginal with respect to stationary convection. The overstable solutions branch off from this locus at the point x*, and for x
[1+ =x1 + k)l(l +x) # (1+k)
=i (1+ k)[l+
P,(1+
X
Ql[ cl+ XI3+ f1+ p;;+ r
k)]2
ij
*
(46)
Y. QIN and P. N. KALONI
1124 3.0
2.5
R,
2.0
1.5
1.0 _
u
1
2
3
4
5
X
Fig. 1. The disposition of the curves for marginal stability in the (x, R,) plane for a Taylor number, T = 104, for the parameter k = 0.01. The curve labelled “convection” applies for all values of P, and defines the locus R’,“. The remaining curves are the overstable loci RI”) for values of P, by which they are labelled. The minimum for the curves C(P, = 0.5054) and for the convection curve occurs for the same value of R,.
The first of (46) shows that R?*
CT, 2r3 + 3x2 = ’ + [l + P,(l t k)12
(47)
with X, so determined, as the root of this cubic equation, we substitute this value in (35) and determine the required critical Rayleigh number R$, for overstability.
5.
DISCUSSION
We shall now provide a brief discussion of the roles played by various micropolar parameters and the rotation on the flow and instability characteristics. Tables l-4 show the trend of the variation of critical Rayleigh numbers and the critical wave numbers as parameters are varied. For various physical reasons we have assumed that stationary convection is possible only by heating from below. Besides assuming b > 0, for low values of T, we further require that A[(T + kb)A + 2k] > T(j/2)[(j/2)B + kbC], where A, B, C are defined in (31). For reasonable values of the parameters, for example, similar to those chosen in [lo], we have found that the
Thermal instability problem in a rotating micropolar fluid Table 1. Stationary convection
results when r = G = k = j = 0.1
T = 1.0
T=O
1125
T = 10.0
T = 5.0
b
4
R:
4
RZ
4
RE
a2
Rf
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
4.923 4.947 4.967 4.985 5.000 5.014 5.026 5.036 5.046 5.055 5.063
719.348 661.106 611.581 568.954 531.878 499.336 470.545 444.891 421.889 401.147 382.349
5.013 5.042 5.066 5.087 5.106 5.122 5.137 5.150 5.161 5.172 5.182
728.734 670.092 620.175 577.172 539.740 506.865 477.761 451.817 428.543 407.548 388.514
5.410 5.460 5.503 5.541 5.574 5.603 5.629 5.652 5.673 5.692 5.709
768.427 708.117 656.555 611.973 573.045 538.761 508.341 481.166 456.744 434.678 414.643
5.994 6.078 6.150 6.213 6.268 6.316 6.359 6.397 6.432 6.463 6.491
823.599 760.999 707.159 660.380 619.364 583.112 550.849 521.950 495.924 472.357 450.920
E l:o
Table 2. Stationary convection results when F = G = k = j = 0.01
T = 1.0
T=O
T = 10.0
T = 5.0
b
aI
RZ
6
RZ
4
RZ
4
R:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4.934 4.957 4.978 4.995 5.024 5.037 5.047 5.057
663.695 609.954 564.258 524.927 490.718 460.692 434.127 410.458 389.235 370.098 352.754
5.027 5.056 5.082 5.104 5.123 5.140 5.155 5.168 5.180 5.191 5.201
672.617 618.563 572.541 532.885 498.363 468.037 441.188 417.250 395.774 376.400 358.834
5.436 5.492 5.540 5.581 5.618 5.650 5.679 5.705 5.728 5.749 5.768
710.273 654.940 607.572 566.569 530.738 499.155 471.113 446.048 423.509 403.135 384.628
6.035 6.132 6.215 6.288 6.351 6.407 6.457 6.501 6.541 6.577 6.610
762.434 705.401 656.212 613.376 575.739 542.422 512.726 486.091 462.074 440.307 420.490
::EE 5.088
Table 3. Stationary convection results when r = 2.0, k = j = G = 1.0
T = 1.0
T = 5.0
T = 10.0
T = 50.0
T = 100.0
b
2
R:
d
R%
4
R:
a:
R:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4.926 4.935 4.944 4.952 4.959 4.966 4.972 4.978 4.983 4.988 4.993
1299.29 1241.27 1188.22 1139.51 1094.64 1053.16 1014.72 978.980 945.672 914.556 885.422
5.041 5.061 5.079 5.095 5.111 5.125 5.138 5.150 5.162 5.173 5.183
1319.99 1261.52 1208.01 1158.84 1113.53 1071.62 1032.76 996.610 931.413 901.910
5.193 5.227 5.258 5.286 5.313 5.338 5.361 5.382 5.403 5.422 5.439
1346.69 1287.67 1233.61 1183.91 1138.07 1095.64 1056.26 1019.62 985.437 953.472 923.510
6.764 6.964 7.151 7.324 7.486 7.636 7.775
1601.22 1540.12 1483.74 1431.48 1382.91 1337.58 1295.15 1255.39 1217.99 1182.78 1149.56
;:FZ 8.141 8.248
4
RZ
9.758 10.237 10.664 11.046 11.389 11.698 11.979 12.234 12.466 12.680 12.876
Table 4. Stationary convection results when r = k = G = 0.01
T = 1.0 j 0.001 0.0 0.001 0.0 0.001 0.0
b 0 0 0.1 0.1
4 2
4.956 4.956 4.981 4.979 5.003 5.000
T = 10 Rf
665.741 665.672 611.917 611.763 566.138 565.924
ac 2
5.155 5.148 5.190 5.172 5.221 5.194
T=ld Rf
683.785 683.090 629.249 627.693 582.757 580.600
2 =c
6.784 6.698 6.931 6.727 7.061 6.753
Rf
839.023 831.338 779.823 763.230 728.322 705.427
2066.20 2007.93 1951.91 1898.01 1846.16 1796.31 1748.53 1702.65 1658.62 1616.50 1576.10
Y. QIN and P. N. KALONI
1126
above condition is satisfied when T has lower values. In order to make meaningful comparison to detect the effect of rotation we have given values to different parameters, similar to those in Tables l-4 of Payne and Straughan [lo]. We point out that for T = 0 our results in Table 1 agree with those of Payne and Straughan [lo]. From Table 1 we draw the conclusion that stationary convection with rotation also predicts reduction in Rz values for increasing coupling number b. The effect of rotation, however, appears to slow down the reduction in Rz values. Moreover, as T increases, the value of R: increases, which clearly indicates that in a micropolar fluid also, the rotation has a stabilizing effect. From Table 2 we also note that for smaller values of the micropolar parameters the critical values of Rayleigh number become lower. In Table 3 we have considered very high values of the parameters. In this case we again found that the effect of increasing b is to decrease R& and that Rz increases as T increases, indicating again that rotation has stabilizing effect. Table 4 shows the variation of j. A comparison with Table 2 values shows that as j decreases so does Rz, but the effect of variation of j alone is not very significant for small values ofj. From the above we conclude that for the cases we considered, the effect of rotation in a stationary convection, is stabilizing and that higher the values of the micropolar parameters, the more the stabilizing effect. For oscillatory convection we note that for the simplifications we introduced, there is not much difference between our graph 1 and the graph for a Newtonian fluid (cf. [13], p. 117) for low values of the parameter k. Hence the classification which is applicable there also applies in the present case. Acknowledgements-The work reported in this paper has been supported by Grant #A7728 of the N.S.E.R.C. of Canada. The authors gratefully acknowledge the support thus received. They also thank a referee for helpful comments.
REFERENCES C. ERINGEN, Int. /. Engng Sci. 2, 189 (1964). C. ERINGEN, Int. J. Engng Sci. 7, 115 (1969). C. ERINGEN, Znt. /. Engng Sci. 18, 5 (1980). C. ERINGEN, Rheol. Acta 30,23 (1991). C. ERINGEN, J. Math. Mach. 16, 1 (1966). C. ERINGEN, J. Math. Anayt. Appl. 38, 480 (1972). B. DA’I’TA and V. I. K. SASTRY, Int. J. Engng Sci. 14, 631 (1976). AHMADI, Inf. J. Engng Sci. 14, 81 (1976). LEBON and C. PEREZ-GARCIA, Int. J. Engng Sci. 19, 1321 (1981). E. PAYNE and B. STRAUGHAN, Int. J. Engng Sci. 27, 827 (1989). U. K. SASTRY and V. R. RAO, Int. J. Engng Sci. 5,449 (1983). [12] S. P. BHATTACHARYYA and M. ABBAS, Int. J. Engng Sci. 23, 371 (1985). [13] S. CHANDRASEKHAR, Hydrodynamics and Hydromugnetic Stability. Dover, New York (1981). [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo] (1 l]
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(Revision received and accepted 13 January
1992)