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Thermal instability in a micropolar fluid layer subject to a magnetic field
II& 1. fhgwg ki Vol. 18, pp. 741-7%7 Prinlcd inGnal @ Pcrgamm Press Ltd., 1980.
Britain
THERMAL INSTABILITY IN A MICROPOLAR FLUID LAYER SUBJECT TO A MAGNETIC FIELD K. V. RAMA RAO Department of Mathematics, Indian Institute of Technology, Kharagpur 721302,Indiat Abstnxt-In the present paper we have conside~d thermal instability in a heat conducting scrolls fluid layer under the influence of a transverse magnetic field. Assuming the bounding surfaces to be rigid the eigenvalue problem is solved using finite-difference and Wilkinson’s iteration techniques. Here it is seen that the instability sets in not only for adverse temperature gradient but also for positive temperature gradient. Both the microtation and the magnetic field are seen to stabilize the fluid Iayer. However, the sta~~i~ effect of ~c~otation becomes less s~nificant when the strength of the rn~t~ fiefd is large. In the case of heating from below, the critical wave number is seen to be insensitive to increase in the strength of the magnetic field, while it increases significantly when the fluid is heated from above. I. INTRODUCTION
INCONTRAST with Ne~onian fluids, micropolar fluids contain molecular constituents whose size is not negligible when compared with the characteristic length of the geometry. Eringen [I] has developed the theory of these fluids taking into account the effects arising from the micromotions of these constituents. Using this theory many authors have studied problems eoncerned with flow, heat transfer and stability. An excellent review of these works has been presented by Ariman et al. [2]. Thermal instability in a layer of micropolar fluid heated from below has been initiated by Ahmadi[3]. Establishing the principle of exchange of stabilities, he has solved the problem for the case of free boundaries and has shown that micropolar fluids are more stable than Newtonian ones. Later Datta and Sastry[4] have extended the analysis to the case of heat conducting micropolar fluids and have pointed out that the instability in heat conducting micropolar fluids may set in even for heating from above. Subsequently pointing out the possible applications of this analysis in areas like the convective process in the interior of the earth, the rising of volcanic liquid with bubbles, etc. Walzer[S] has considered the problem for rigid boundaries, However, he has concluded his analysis without any numerical results. Recently Rama Rao[6] has considered the stability of a heat conducting micropolar fluid layer with rigid boundaries. His analysis reveals that the instability may set in even for heating from above and that the convection cells in the case of positive temperature gradient are more elongated than in the case of adverse temperature gradient. In the present investigation, we study the influence of a transverse magnetic field on the thermal instability of a micropolar fluid layer. The problem is solved using a combination of finite-difference and Wilkinson’s iteration techniques. 2. FORMULATION
OF THE PROBLEM
Consider an infinite horizontal layer of an incompressible electrically conducting micropolar fluid conlined between two non-conducting rigid boundaries. Let d be the depth of the layer, and Ha be the strength of the magnetic field, applied at right angles to the fluid layer. Now, taking the origin on the lower plane, we introduce the Cartesian coordinate system (x,, x2, x3), in which x3 is measured vertically upwards. Under Boussinesq approximation, the equations governing the disturbances can be written as avi &O ahi --dp’+(p+k)~Vi+kEilr~+~~AT’ei, -(2.1) aXi PO at 47r zI
ah.
-$ = HO!$j + nV2h,, 3
(2.2)
+Present address: Simulation Division, Aeronautical Development Establishment, Indira Nagar, Bangalore 560038, India.
K.V.RAMARAO
142
d$ =(a + PI $$+YV'Vi +2k(@ -
Vi),
(2.3)
I
(-$-kVI)T’=O[v3-$$-Z)J avi _.
at’--
(2.5)
9
ahi=0
(2.6)
’
aXi
(2.4)
where pois the density, g the acceleration due to gravity, A the coefficientof volume expansion, 4 the temperature gradient, C’the time, Cv the specific heat at constant volume, k, the thermal diffusivity, pr the magnetic permeability, 7 the magnetic diffusivity, p the viscosity, j the microinertia, k, a, /3, ‘y,S are material constants of micropolar fluid, and ai, tz:,hi (i = 1,2,3), p’ and T’ are the perturbations in velocity, micro-rotation, magnetic field, pressure and temperature respectively. In the above equations v2 =
a2
a2 a2 z+ax’+ 1
avi
zp=ani
2
and 1 Wi= 2 $jk 2 9ei = (090, l), J
and Q is the alternating tensor. Now, proceeding in a similar manner as given in Chandrasekhar[7], and introducing the nondimensional variables, given by (X,
y,
2)
t =
pt’/~d2,
=
h/d,
x2/d w
=
X3/49
ujdlk,, v = v3d2/kc, h = h#o,
r = T’l&d, $1 = wd2/k,,
$2
=
td31kc,
$3
=
@IHo
and R = R3d3/k,
in which
we obtain the equations governing the disturbances as (2.7)
$VW-%&
n,$=B;
Fh = (1 f
(
(62+z
>
R)V4w$ RV2!-l+R,V:T,
(2.8)
+AV%+R(&-2v),
(2.9)
Thermal instability in a micropolar fluid layer subject to a magnetic field
nl g
743
(2.10)
= AV*fI - 2RR - RV’w,
(2.11)
(2.12) a+3 1 a41 -_=--+1v”43, at
pI a2
(2.13)
p2
(2.14) where
pI(
= p/p&), p2( = p/pov) are the thermal and magnetic Prandtl numbers, Q = (~J-Zfd2/47r~r)) is the Chandrasekhar number, R, = (p&$d4/&) is the Rayleigh number, and R( = k/p), A( = r/~&), B( = a + B/h&), nl ( = j/d’) and 8( = S/poCvd’) are the micropolar parameters. Now, under the usual normal mode analysis, we assume the perturbations to be of the form
in which 2, x, X, W, Y, G, 0 and H are all functions of z, u is the non-dimensional frequency parameter, and k, and ky are the non-dimensional wave numbers in x and y directions respectively. Using the relation (2.15), the equations governing the disturbances can be written as [(l+R)(D%r2)-u]X+(%+R)DZ+Ro2Y=0, (2.16)
[A(ti-a2)+Bti-2R-n,a]Y+BDZ+RX=O, [A@ - a’) - Ba2 - 2R - n,cr]Z - Ba’DY - RDX = 0, (Dh2-p,cr)~+eDX=O PI (02 - a2)[(1 + R)(p - u2) - a] W + s
9
D(p - u2)H + R(p - u2)G - R,u20 = 0,
[A@-a2)-2R-n,c~]G-R(b-u2)W=0, (02-u2-p,a)O+
W-&G=O,
(p-a’-p,a)H+BDW=O PI
’
(2.17)
where d D=zandu2=k:+k:. As the boundaries are rigid, we have W=DW=@=G=O
x=
Y=Z=O.
I
ati=
1 T2’
(2.18)
144
K.V.RAMA RAO
where
Now we assume the boundaries adjoining the fluid medium to be non-conducting. Then from the fact that no currents can flow across a non-conducting boundary, we have x = 0. Further, at the interface the magnetic field inside the fluid medium must be continuous with the external field. Thus, following Gibson[S], we have DH + aH = 0 at f = 7 l/2. Hence, in the case of non-conducting boundaries, the appropriate conditions for the components of the current and the magnetic field are given by x=0
and DH-taH=O
at f= 7112.
(2.19)
It can be readily seen that the equations (2.16), together with the boundary conditions X = Y = 2 = x = 0, yield the trivial solutions X = Y = 2 = x = 0, everywhere inside the fluid. Now, confining our attention to the case of steady marginal state (a =0), and then eliminating H, the eigenvalue problem governing the disturbances at the onset of instability can be written as {(1f R)(D’ - a2)2- Qo’} W + R(p - a*)G - R,a20 = 0, (2.20) A(@--a2)G-R(@-a*)W-2RG=O, (D2-a*)@+
(2.21)
W-&=0,
W=DW=G=O=O
(2.22)
at 1=+1/2.
(2.23)
The above system of equations suggests that the solutions decompose into even or odd. Here, we confine our attention only to even solutions, as it is well known that these will result in the lowest eigenvalue. In what follows, to examine the salient features of the problem, firstly, we find an approximate solution by using Gaierkin’s method. Then, in order to improve the accuracy of the solution, we employ finite-difference method in conjunction with Wilkinson’s iteration technique. 3.SOLUTION OF THE PROBLEM
Satisfying the boundary conditions (2.23), we select even solutions of the form W = a,( 1 - 4f2)2, G = bl(l - 4f2), and
Now, on using Gaierkin method [9], we get the secular relation given by R ~(a2+10)[28(A(a2+10)+2R~(1iR)(a*+24n2+504)+12Q)-3R2(3a2+28)*] (I 9a2[3(A(a2 + 10)+ 2R}- 6R(3a2 + 28)]
.
(3 Ij .
From the above relation, it can be readily seen that the numerator is greater than zero, while the denominator ‘, 0 depending upon
I
I’* = a* (say).
This value of u is plausible if A < 8R < (30A + 6R)/28. This result indicates that, in the case of heat conducting micropolar fluids (sf O), there is an asymptote which separates the two zones
Thermal instability in a micropolar fluid layer subject to a magnetic field
745
of ins~bility co~esponding to R, > 0 and % < 0. It is also worth noticing that the value of a at which the asymptote exists, and the physical phenomena, viz. occurrence of instability for Ra >O and R,
U2h2)
+
(3.2)
2Rh2}Gi + AG;+l- RWi-,
+ R(2+a2h2)wi-RM$+~ ~0,
(3.3)
@i-1-(2+ a2h2)Oi+ @;+I+ h2(w - 8Gi) = 0,
(3.4)
where i takes the values 1 to N, and h( = l/N) is the step length, As we have confined our analysis to even solutions for W, G and 8, we consider the problem in the range 0 to 112.Thus, the eqns (3.2)-(3.4) can be rewritten as A,@+B,&-R
au2h46=0 3
A&‘+ B& =O,
(3.5) (3.4) (3.7)
in which Al, BI, AZ, B2 and B are the coefficient matrices of order n x n, and @, d and 0 are column vectors each having n components, where n( = (N-1)/2) or N/2, depending upon N is odd or even. Now, in view of the boundary conditions (2.23), we get
iv,=P-1,G,= G-,,0, = ix,,
(3.8)
and
where m = (N + 1)/2 or N/2 depending upon N is odd or even. From the relations (3.6) and (3.7), we have (3.9) where B3 = (B2 + c%A&‘A2B and A3 = $B, - B. Now, on using the above relations, we eliminate w and d from (3.5), and then obtain the algebraic eigenvalue problem given by (C-Al)Q=O, (3.10) where C=-$(AIAI+
B1B3) and A = Rah4.
146
K. V. RAMA RAO
To solve the algebraic eigenvalue problem, we have adopted Wilkinson’s iteration technique[lO]. Prescribing the values of R, A, 8 and Q, we have calculated the lowest eigenvalue of C for various values of a (taken around the approximate value determined by Galerkin method). Thus, obtaining R. for various values of a, we have determined the critical value of Ra (minimum value of R,J and the corresponding value of a. Using this method the classical BCnard problem with magnetic field is solved. The values of R, thus obtained for Q =0 and Q = 10 with h = l/21 are 1704 and 1947, respectively. These values are quite comparable with the values of Chandrasekhar given by 1708 and 1949. 4. DISSCUSSION
OF THE
RESULTS
In carrying out the numerical computations, we have taken various values for Q, R, A and L?. For R, >O, neutral stability curves are drawn in Figs. 1-3, and the critical wave numbers are exhibited in Figs. 4 and 5. On the other hand, for R. ~0, the critical Rayleigh number and the corresponding wave number are presented in Tables l-3. Throughout the computation we have taken h = l/25. When the fluid is heated from below (R. > O), the onset of instability is delayed as R or 8 increases, while it is hastened as A increases. The mechanism underlying this physical phenomena can be understood by having a probe into the nature of micropolar fluids. The microrotation induced to the fluid particles increases with R. As a part of the kinetic energy of the system will be consumed in developing gyrational velocities of the fluid particles, the disturbances are reduced. Thus, delaying the onset of instability. When 8 increases, the heat induced into the fluid is more; thus reducing the heat transfer from the bottom to the top. This inhibition of heat transfer is responsible for the delay in the onset of instability. On the other hand, when A increases the couple stress of the fluid increases; this increase in the couple stress causes the micro-rotation to decrease; rendering the system prone to instability. Nevertheless, the above phenomenon is true whether the magnetic field is present or not. 7
f
t
E
2 = z
.!
!
-. -----
R-0
-
R-2
R:L
I
I
1
2
’ 3,
3
ao-
2
3 Log10Q -
4 CbJ
Fig. l(a). Neutral stability curves showing effect of R (A = 0.001, 8= 0.1). (b) Effect of R on critical R,
IA =O.OOl, i=O.l,.
Thermal instability in a micropolar fluid layer subject to a magnetic field
I
I
1.0
2.0
I
3.0
I
LO
a-
Fig. 2. Marginal stability curves showing effect of &A = 0.001, R = 2).
t
6-
d 8 -I
=10
1.0
2.0
3.0
t.0
Fig. 3. Marginal stability curves, showing effect of A.(R= 2,s'=0.1). I.I.E.S. 18/S--o
K. V. RAMA RAO
748 3.5
s =0.05
r
rob G
_I__________-----2.5
E:o
Fig. 4. Effect
of f on the critical
wave number
1
a, (R = 2, A = 0.001).
Az0.1
I 1
1.51
I 2
I 3
I L
Lo!310Q Fig. 5. Effect
Table
I. Effect
of A on the a, fR = 2, s=O.l).
of R on critical
values
of R, and u (A = 0.001,
S=O.l) R
Q
2
IO’ Iti 10 lo’ lo4 IO 10’ lti IO IO’ lo4
10
4
6
8
‘Figures
R.
0,
- 0.52798516f4)’ - 0.93095547(4) - 0.22401688(5) - 0.77974102(4) -0.12269605(5) - 0.27290262(5) - 0.10338568(5) - 0.15145633(5) - 0.32027512(5) -0.12867156(5) -0.17950246f5) -0.36511942(5)
6.9 9.8 15.2 6.9 9.6 14.1 7.0 8.9 13.5 7.0 8.6 13.0
in the parentheses
denote
powers
of IO.
J 5
Thermal instability in a micropolar fluid layer subject to a magnetic field
149
Table 2. Effect of A on critical R,, and a (R = 2, i= 0.1) A
Q
a
R.3
I@ lo4 IO lo2 Iti 104
6.9 7.5 9.8 15.2 6.6 7.0 8.3 II.4 7.7 1.9
- 0.52798516(4)’ - 0.58859922(4) - 0.93095547(4) - 0.22401688(5) - 0.17375320(5) - 0.19252087(5) - 0.32576676(5) - 0.1077243l(6) -0.63188949(5) - 0.67361625(5)
II.4 8.8
- 0.10044338(6) 0.30548806(6)
0.001
0.050
0.100
t
S
1:3
+Figures in the parentheses denote powers of IO.
Table 3. Effect of 8 on critical Ra and a (R = 2, A = 0.001)
0.10
0.05
IO lo2 I@ lti 1: lo-’ lo’
6.9 7.5 9.8 15.2 9.0 9.4 II.0 15.8
- 0.527985lq4)’ -
0.58859922(4) 0.93095547(4) 0.22401688(5) 0.16744367(5) 0.17608574(4) 0.23561699(5) 0.49402094(5)
+Figures in the parentheses denote powers of IO.
When the strength of the magnetic field increases, the system becomes highly stable-a result which is seen in the Newtonian case. On the other hand, when the microrotation and the magnetic field are simultaneously present (see Fig. l), the stabilizing effect of R is reduced; being counteracted by the magnetic field. In literature [7], similar phenomenon has been noticed, when the system is subject to both rotation and magnetic field. The above phenomenon can be physically explained as follows: When the magnetic field permeating the medium is considerably large, it induces voscosity into the fluid, and the magnetic lines are distorted by the convection. Then these magnetic lines hinder the growth of disturbances, leading to the delay in the onset of instability. However, the viscosity produced by the magnetic field lessens the rotation of the fluid particles; thus controlling the stabilizing effect of R. From Figs. 4 and 5, we notice that the critical wave number a, increases with A and decreases as d increases. From this, we infer that the width of the cell at the onset of instability decreases by the couple stress, while it increases due to the heat induced by microrotation. However, in the case of heat conducting micropolar fluid (heated from below) a, is less sensitive to magnetic field, while it shows considerable increase in Newtonian fluid. This indicates that, in the case of Newtonian fluid, the convection cells are elongated with the magnetic field, while they do not exhibit such a behaviour in heat conducting micropolar fluid. When the fluid is heated from above (Ra < 0), the instability is delayed as R or A increases, while opposite effect is seen with 8. It is also seen that the system becomes more stable, when it is subjected to a magnetic field. Further, we notice that a, increases with Q. This indicates that, in heat conducting micropolar fluids heated from above, the cells at the onset of stability are significantly elongated. From the above findings we conclude that the stabilizing effect of microrotation is controlled by the presence of the magnetic field. Ack~owlcdgrenf-Theauthorwishes to thank Prof. Suhubi for his suggestions which led to improvement in the presentation and Dr. V. U. K. Sastry for his helpful discussions and encouragement. Thanks are also due to C.S.I.R. for providing the financial support.
750
K. V. RAMA RAO
REFERENCES [I] A. C. ERINGEN, .l. Math. Mech. 16. 1(1!%6). [2] T. ARIMAN, M. A. TURK and N. D. SYLVESTER. Int. J. Engng .Sci. 12, 273 (1973). [3] G. AHMADI, ht. J. EngngSC;.14, 81 (1976). [4] A. B. DATTA and V. U. K. SASTRY. Int. J. Engng Sci. 14, 631 (1976). [5] U. WALZER, Ger. Beit.Geophysik 85, 137 W76). [6] K. V. RAMA RAO. Acta Mechanica 32,79 (1979). [7] S. CHANDRASEKHAR, Hydrodynamic and Hydromagnetic Stability. Oxford University Press. Oxford (l%l). [8] R. D. GIBSON, Proc. Camb. Phil. Sot. 62, 287 (1966). [9] U. H. KURZWEG, Princeton University Technical Report, No. 1959,20, 11-29 (l%l). [lo] J. H. WILKINSON, Proc. Camb. Phil. Sot. SO. 536 (1954). (Received 22 March
1979)
Britain
THERMAL INSTABILITY IN A MICROPOLAR FLUID LAYER SUBJECT TO A MAGNETIC FIELD K. V. RAMA RAO Department of Mathematics, Indian Institute of Technology, Kharagpur 721302,Indiat Abstnxt-In the present paper we have conside~d thermal instability in a heat conducting scrolls fluid layer under the influence of a transverse magnetic field. Assuming the bounding surfaces to be rigid the eigenvalue problem is solved using finite-difference and Wilkinson’s iteration techniques. Here it is seen that the instability sets in not only for adverse temperature gradient but also for positive temperature gradient. Both the microtation and the magnetic field are seen to stabilize the fluid Iayer. However, the sta~~i~ effect of ~c~otation becomes less s~nificant when the strength of the rn~t~ fiefd is large. In the case of heating from below, the critical wave number is seen to be insensitive to increase in the strength of the magnetic field, while it increases significantly when the fluid is heated from above. I. INTRODUCTION
INCONTRAST with Ne~onian fluids, micropolar fluids contain molecular constituents whose size is not negligible when compared with the characteristic length of the geometry. Eringen [I] has developed the theory of these fluids taking into account the effects arising from the micromotions of these constituents. Using this theory many authors have studied problems eoncerned with flow, heat transfer and stability. An excellent review of these works has been presented by Ariman et al. [2]. Thermal instability in a layer of micropolar fluid heated from below has been initiated by Ahmadi[3]. Establishing the principle of exchange of stabilities, he has solved the problem for the case of free boundaries and has shown that micropolar fluids are more stable than Newtonian ones. Later Datta and Sastry[4] have extended the analysis to the case of heat conducting micropolar fluids and have pointed out that the instability in heat conducting micropolar fluids may set in even for heating from above. Subsequently pointing out the possible applications of this analysis in areas like the convective process in the interior of the earth, the rising of volcanic liquid with bubbles, etc. Walzer[S] has considered the problem for rigid boundaries, However, he has concluded his analysis without any numerical results. Recently Rama Rao[6] has considered the stability of a heat conducting micropolar fluid layer with rigid boundaries. His analysis reveals that the instability may set in even for heating from above and that the convection cells in the case of positive temperature gradient are more elongated than in the case of adverse temperature gradient. In the present investigation, we study the influence of a transverse magnetic field on the thermal instability of a micropolar fluid layer. The problem is solved using a combination of finite-difference and Wilkinson’s iteration techniques. 2. FORMULATION
OF THE PROBLEM
Consider an infinite horizontal layer of an incompressible electrically conducting micropolar fluid conlined between two non-conducting rigid boundaries. Let d be the depth of the layer, and Ha be the strength of the magnetic field, applied at right angles to the fluid layer. Now, taking the origin on the lower plane, we introduce the Cartesian coordinate system (x,, x2, x3), in which x3 is measured vertically upwards. Under Boussinesq approximation, the equations governing the disturbances can be written as avi &O ahi --dp’+(p+k)~Vi+kEilr~+~~AT’ei, -(2.1) aXi PO at 47r zI
ah.
-$ = HO!$j + nV2h,, 3
(2.2)
+Present address: Simulation Division, Aeronautical Development Establishment, Indira Nagar, Bangalore 560038, India.
K.V.RAMARAO
142
d$ =(a + PI $$+YV'Vi +2k(@ -
Vi),
(2.3)
I
(-$-kVI)T’=O[v3-$$-Z)J avi _.
at’--
(2.5)
9
ahi=0
(2.6)
’
aXi
(2.4)
where pois the density, g the acceleration due to gravity, A the coefficientof volume expansion, 4 the temperature gradient, C’the time, Cv the specific heat at constant volume, k, the thermal diffusivity, pr the magnetic permeability, 7 the magnetic diffusivity, p the viscosity, j the microinertia, k, a, /3, ‘y,S are material constants of micropolar fluid, and ai, tz:,hi (i = 1,2,3), p’ and T’ are the perturbations in velocity, micro-rotation, magnetic field, pressure and temperature respectively. In the above equations v2 =
a2
a2 a2 z+ax’+ 1
avi
zp=ani
2
and 1 Wi= 2 $jk 2 9ei = (090, l), J
and Q is the alternating tensor. Now, proceeding in a similar manner as given in Chandrasekhar[7], and introducing the nondimensional variables, given by (X,
y,
2)
t =
pt’/~d2,
=
h/d,
x2/d w
=
X3/49
ujdlk,, v = v3d2/kc, h = h#o,
r = T’l&d, $1 = wd2/k,,
$2
=
td31kc,
$3
=
@IHo
and R = R3d3/k,
in which
we obtain the equations governing the disturbances as (2.7)
$VW-%&
n,$=B;
Fh = (1 f
(
(62+z
>
R)V4w$ RV2!-l+R,V:T,
(2.8)
+AV%+R(&-2v),
(2.9)
Thermal instability in a micropolar fluid layer subject to a magnetic field
nl g
743
(2.10)
= AV*fI - 2RR - RV’w,
(2.11)
(2.12) a+3 1 a41 -_=--+1v”43, at
pI a2
(2.13)
p2
(2.14) where
pI(
= p/p&), p2( = p/pov) are the thermal and magnetic Prandtl numbers, Q = (~J-Zfd2/47r~r)) is the Chandrasekhar number, R, = (p&$d4/&) is the Rayleigh number, and R( = k/p), A( = r/~&), B( = a + B/h&), nl ( = j/d’) and 8( = S/poCvd’) are the micropolar parameters. Now, under the usual normal mode analysis, we assume the perturbations to be of the form
in which 2, x, X, W, Y, G, 0 and H are all functions of z, u is the non-dimensional frequency parameter, and k, and ky are the non-dimensional wave numbers in x and y directions respectively. Using the relation (2.15), the equations governing the disturbances can be written as [(l+R)(D%r2)-u]X+(%+R)DZ+Ro2Y=0, (2.16)
[A(ti-a2)+Bti-2R-n,a]Y+BDZ+RX=O, [A@ - a’) - Ba2 - 2R - n,cr]Z - Ba’DY - RDX = 0, (Dh2-p,cr)~+eDX=O PI (02 - a2)[(1 + R)(p - u2) - a] W + s
9
D(p - u2)H + R(p - u2)G - R,u20 = 0,
[A@-a2)-2R-n,c~]G-R(b-u2)W=0, (02-u2-p,a)O+
W-&G=O,
(p-a’-p,a)H+BDW=O PI
’
(2.17)
where d D=zandu2=k:+k:. As the boundaries are rigid, we have W=DW=@=G=O
x=
Y=Z=O.
I
ati=
1 T2’
(2.18)
144
K.V.RAMA RAO
where
Now we assume the boundaries adjoining the fluid medium to be non-conducting. Then from the fact that no currents can flow across a non-conducting boundary, we have x = 0. Further, at the interface the magnetic field inside the fluid medium must be continuous with the external field. Thus, following Gibson[S], we have DH + aH = 0 at f = 7 l/2. Hence, in the case of non-conducting boundaries, the appropriate conditions for the components of the current and the magnetic field are given by x=0
and DH-taH=O
at f= 7112.
(2.19)
It can be readily seen that the equations (2.16), together with the boundary conditions X = Y = 2 = x = 0, yield the trivial solutions X = Y = 2 = x = 0, everywhere inside the fluid. Now, confining our attention to the case of steady marginal state (a =0), and then eliminating H, the eigenvalue problem governing the disturbances at the onset of instability can be written as {(1f R)(D’ - a2)2- Qo’} W + R(p - a*)G - R,a20 = 0, (2.20) A(@--a2)G-R(@-a*)W-2RG=O, (D2-a*)@+
(2.21)
W-&=0,
W=DW=G=O=O
(2.22)
at 1=+1/2.
(2.23)
The above system of equations suggests that the solutions decompose into even or odd. Here, we confine our attention only to even solutions, as it is well known that these will result in the lowest eigenvalue. In what follows, to examine the salient features of the problem, firstly, we find an approximate solution by using Gaierkin’s method. Then, in order to improve the accuracy of the solution, we employ finite-difference method in conjunction with Wilkinson’s iteration technique. 3.SOLUTION OF THE PROBLEM
Satisfying the boundary conditions (2.23), we select even solutions of the form W = a,( 1 - 4f2)2, G = bl(l - 4f2), and
Now, on using Gaierkin method [9], we get the secular relation given by R ~(a2+10)[28(A(a2+10)+2R~(1iR)(a*+24n2+504)+12Q)-3R2(3a2+28)*] (I 9a2[3(A(a2 + 10)+ 2R}- 6R(3a2 + 28)]
.
(3 Ij .
From the above relation, it can be readily seen that the numerator is greater than zero, while the denominator ‘, 0 depending upon
I
I’* = a* (say).
This value of u is plausible if A < 8R < (30A + 6R)/28. This result indicates that, in the case of heat conducting micropolar fluids (sf O), there is an asymptote which separates the two zones
Thermal instability in a micropolar fluid layer subject to a magnetic field
745
of ins~bility co~esponding to R, > 0 and % < 0. It is also worth noticing that the value of a at which the asymptote exists, and the physical phenomena, viz. occurrence of instability for Ra >O and R,
U2h2)
+
(3.2)
2Rh2}Gi + AG;+l- RWi-,
+ R(2+a2h2)wi-RM$+~ ~0,
(3.3)
@i-1-(2+ a2h2)Oi+ @;+I+ h2(w - 8Gi) = 0,
(3.4)
where i takes the values 1 to N, and h( = l/N) is the step length, As we have confined our analysis to even solutions for W, G and 8, we consider the problem in the range 0 to 112.Thus, the eqns (3.2)-(3.4) can be rewritten as A,@+B,&-R
au2h46=0 3
A&‘+ B& =O,
(3.5) (3.4) (3.7)
in which Al, BI, AZ, B2 and B are the coefficient matrices of order n x n, and @, d and 0 are column vectors each having n components, where n( = (N-1)/2) or N/2, depending upon N is odd or even. Now, in view of the boundary conditions (2.23), we get
iv,=P-1,G,= G-,,0, = ix,,
(3.8)
and
where m = (N + 1)/2 or N/2 depending upon N is odd or even. From the relations (3.6) and (3.7), we have (3.9) where B3 = (B2 + c%A&‘A2B and A3 = $B, - B. Now, on using the above relations, we eliminate w and d from (3.5), and then obtain the algebraic eigenvalue problem given by (C-Al)Q=O, (3.10) where C=-$(AIAI+
B1B3) and A = Rah4.
146
K. V. RAMA RAO
To solve the algebraic eigenvalue problem, we have adopted Wilkinson’s iteration technique[lO]. Prescribing the values of R, A, 8 and Q, we have calculated the lowest eigenvalue of C for various values of a (taken around the approximate value determined by Galerkin method). Thus, obtaining R. for various values of a, we have determined the critical value of Ra (minimum value of R,J and the corresponding value of a. Using this method the classical BCnard problem with magnetic field is solved. The values of R, thus obtained for Q =0 and Q = 10 with h = l/21 are 1704 and 1947, respectively. These values are quite comparable with the values of Chandrasekhar given by 1708 and 1949. 4. DISSCUSSION
OF THE
RESULTS
In carrying out the numerical computations, we have taken various values for Q, R, A and L?. For R, >O, neutral stability curves are drawn in Figs. 1-3, and the critical wave numbers are exhibited in Figs. 4 and 5. On the other hand, for R. ~0, the critical Rayleigh number and the corresponding wave number are presented in Tables l-3. Throughout the computation we have taken h = l/25. When the fluid is heated from below (R. > O), the onset of instability is delayed as R or 8 increases, while it is hastened as A increases. The mechanism underlying this physical phenomena can be understood by having a probe into the nature of micropolar fluids. The microrotation induced to the fluid particles increases with R. As a part of the kinetic energy of the system will be consumed in developing gyrational velocities of the fluid particles, the disturbances are reduced. Thus, delaying the onset of instability. When 8 increases, the heat induced into the fluid is more; thus reducing the heat transfer from the bottom to the top. This inhibition of heat transfer is responsible for the delay in the onset of instability. On the other hand, when A increases the couple stress of the fluid increases; this increase in the couple stress causes the micro-rotation to decrease; rendering the system prone to instability. Nevertheless, the above phenomenon is true whether the magnetic field is present or not. 7
f
t
E
2 = z
.!
!
-. -----
R-0
-
R-2
R:L
I
I
1
2
’ 3,
3
ao-
2
3 Log10Q -
4 CbJ
Fig. l(a). Neutral stability curves showing effect of R (A = 0.001, 8= 0.1). (b) Effect of R on critical R,
IA =O.OOl, i=O.l,.
Thermal instability in a micropolar fluid layer subject to a magnetic field
I
I
1.0
2.0
I
3.0
I
LO
a-
Fig. 2. Marginal stability curves showing effect of &A = 0.001, R = 2).
t
6-
d 8 -I
=10
1.0
2.0
3.0
t.0
Fig. 3. Marginal stability curves, showing effect of A.(R= 2,s'=0.1). I.I.E.S. 18/S--o
K. V. RAMA RAO
748 3.5
s =0.05
r
rob G
_I__________-----2.5
E:o
Fig. 4. Effect
of f on the critical
wave number
1
a, (R = 2, A = 0.001).
Az0.1
I 1
1.51
I 2
I 3
I L
Lo!310Q Fig. 5. Effect
Table
I. Effect
of A on the a, fR = 2, s=O.l).
of R on critical
values
of R, and u (A = 0.001,
S=O.l) R
Q
2
IO’ Iti 10 lo’ lo4 IO 10’ lti IO IO’ lo4
10
4
6
8
‘Figures
R.
0,
- 0.52798516f4)’ - 0.93095547(4) - 0.22401688(5) - 0.77974102(4) -0.12269605(5) - 0.27290262(5) - 0.10338568(5) - 0.15145633(5) - 0.32027512(5) -0.12867156(5) -0.17950246f5) -0.36511942(5)
6.9 9.8 15.2 6.9 9.6 14.1 7.0 8.9 13.5 7.0 8.6 13.0
in the parentheses
denote
powers
of IO.
J 5
Thermal instability in a micropolar fluid layer subject to a magnetic field
149
Table 2. Effect of A on critical R,, and a (R = 2, i= 0.1) A
Q
a
R.3
I@ lo4 IO lo2 Iti 104
6.9 7.5 9.8 15.2 6.6 7.0 8.3 II.4 7.7 1.9
- 0.52798516(4)’ - 0.58859922(4) - 0.93095547(4) - 0.22401688(5) - 0.17375320(5) - 0.19252087(5) - 0.32576676(5) - 0.1077243l(6) -0.63188949(5) - 0.67361625(5)
II.4 8.8
- 0.10044338(6) 0.30548806(6)
0.001
0.050
0.100
t
S
1:3
+Figures in the parentheses denote powers of IO.
Table 3. Effect of 8 on critical Ra and a (R = 2, A = 0.001)
0.10
0.05
IO lo2 I@ lti 1: lo-’ lo’
6.9 7.5 9.8 15.2 9.0 9.4 II.0 15.8
- 0.527985lq4)’ -
0.58859922(4) 0.93095547(4) 0.22401688(5) 0.16744367(5) 0.17608574(4) 0.23561699(5) 0.49402094(5)
+Figures in the parentheses denote powers of IO.
When the strength of the magnetic field increases, the system becomes highly stable-a result which is seen in the Newtonian case. On the other hand, when the microrotation and the magnetic field are simultaneously present (see Fig. l), the stabilizing effect of R is reduced; being counteracted by the magnetic field. In literature [7], similar phenomenon has been noticed, when the system is subject to both rotation and magnetic field. The above phenomenon can be physically explained as follows: When the magnetic field permeating the medium is considerably large, it induces voscosity into the fluid, and the magnetic lines are distorted by the convection. Then these magnetic lines hinder the growth of disturbances, leading to the delay in the onset of instability. However, the viscosity produced by the magnetic field lessens the rotation of the fluid particles; thus controlling the stabilizing effect of R. From Figs. 4 and 5, we notice that the critical wave number a, increases with A and decreases as d increases. From this, we infer that the width of the cell at the onset of instability decreases by the couple stress, while it increases due to the heat induced by microrotation. However, in the case of heat conducting micropolar fluid (heated from below) a, is less sensitive to magnetic field, while it shows considerable increase in Newtonian fluid. This indicates that, in the case of Newtonian fluid, the convection cells are elongated with the magnetic field, while they do not exhibit such a behaviour in heat conducting micropolar fluid. When the fluid is heated from above (Ra < 0), the instability is delayed as R or A increases, while opposite effect is seen with 8. It is also seen that the system becomes more stable, when it is subjected to a magnetic field. Further, we notice that a, increases with Q. This indicates that, in heat conducting micropolar fluids heated from above, the cells at the onset of stability are significantly elongated. From the above findings we conclude that the stabilizing effect of microrotation is controlled by the presence of the magnetic field. Ack~owlcdgrenf-Theauthorwishes to thank Prof. Suhubi for his suggestions which led to improvement in the presentation and Dr. V. U. K. Sastry for his helpful discussions and encouragement. Thanks are also due to C.S.I.R. for providing the financial support.
750
K. V. RAMA RAO
REFERENCES [I] A. C. ERINGEN, .l. Math. Mech. 16. 1(1!%6). [2] T. ARIMAN, M. A. TURK and N. D. SYLVESTER. Int. J. Engng .Sci. 12, 273 (1973). [3] G. AHMADI, ht. J. EngngSC;.14, 81 (1976). [4] A. B. DATTA and V. U. K. SASTRY. Int. J. Engng Sci. 14, 631 (1976). [5] U. WALZER, Ger. Beit.Geophysik 85, 137 W76). [6] K. V. RAMA RAO. Acta Mechanica 32,79 (1979). [7] S. CHANDRASEKHAR, Hydrodynamic and Hydromagnetic Stability. Oxford University Press. Oxford (l%l). [8] R. D. GIBSON, Proc. Camb. Phil. Sot. 62, 287 (1966). [9] U. H. KURZWEG, Princeton University Technical Report, No. 1959,20, 11-29 (l%l). [lo] J. H. WILKINSON, Proc. Camb. Phil. Sot. SO. 536 (1954). (Received 22 March
1979)