Inr. J. Engng Sci. Vol. 27, No. 11, pp. 14211428, 1989 Printed in Great Britain. All rights reserved
OlEO7225/89 $3.00 + 0.00 Copyright @ 1989 Pergamon Press plc
FINITE ELEMENT SOLUTION OF FLOW AND HEAT TRANSFER OF A MICROPOLAR FLUID OVER A STRETCHING SHEET R. S. AGARWAL, Department
RAMA
of Mathematics,
BHARGAVA
and A. V. S. BALAJI
University of Roorkee, Roorkee247667 (U.P.) India
AbstractA steady, incompressible micropolar fluid flow and heat transfer over a stretching sheet has been analysed. The resulting system of nonlinear ordinary coupled differential equations is solved by finite element method using variational Ritz model. Numerical results obtained for velocity, microrotation and temperature distributions are shown graphically.
1. INTRODUCTION In recent years, the dynamics of micropolar fluids, originated from the theory of Eringen [l, 21 has been a popular area of research. The equations governing the flow of a micropolar fluid involve a microrotation vector and a gyration parameter in addition to the classical velocity vector fleld. This theory may be applied to explain the flow of colloidal solutions, liquid crystals, fluids with additives, suspension solutions, animal blood, etc. The boundary layer concept in such fluid has been first studied by Peddieson and McNitt [3] and Wilson [4]. While extending the theory of micropolar fluids, Eringen [5] developed the theory of thermomicro fluids taking into account the effect of microelements of fluids on both the kinematics and conduction of heat. The flow characteristics of the boundary layer flow of a micropolar fluid past a semiinfinite plate in different situations have been studied by several authors [681. In the present study, we have analysed the problem of micropolar fluid flow and heat transfer over a stretching sheet. Flow over a stretching sheet may have applications to polymer technology where one deals with stretching plastic sheets. To be more specific, it may be pointed out that many metallurgical processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid and that in the process of drawing, the strips are sometimes stretched. Mention may be made of drawing, annealing and tinning of copper wires. In all these cases, the properties of the final product depend to a great extent on the rate of cooling. By drawing such strips in a micropolar fluid, the rate of cooling may be controlled and the final products of desired characteristics might be achieved.
2. MATHEMATICAL
FORMULATION
Consider the flow of an incompressible micropolar fluid past a porous wall coinciding with the plane y = 0, the flow being confined to y > 0. Two equal and opposite forces are introduced along the xaxis (see Fig. 1) so that the wall is stretched keeping the origin fixed. Also we assume that the wall temperature remains steady at T, and the free stream fluid temperature remains constant at T,. The effects of viscous dissipation will be assumed to be negligible. The basic equations for the steady flow are, in the usual notation
du+i?!=o
(2.1)
a~ ay
(2.2) pi(.~+v~)=$(y~)k(2v+~)
aTav .g+,z =k,dZT+(Y ( > (ax ay
p
ay
a?
1421
(2.3)
aTav ay ax
>
(2.4)
1422
R. S. AGARWAL
et al.
Fig. 1. Sketch of the physical problem.
where p is the density, p the coefficient of gyroviscosity, j the microinertia, y the micropolar material constant, u and u are the velocities along and perpendicular to the plate, Y the microrotation, cp the heat at constant pressure, T the temperature of the fluid, k, and (Yare the heat conduction coefficients. Following [9,8] the boundary conditions on velocity, temperature and microrotation are taken as: u = I/;
v = v”;
u+o;
Y =
T=T,
0; T+T,
v0;
as
at
y=O
y+w
(2.5)
where V, is the suction velocity. In accordance with [9], introducing
(ef)l’>(q) y= (cp>lf2g(v)
u = Cxf’(q);
v
q= T Ir;;
(> T

e(v) =Tw
=

cx
P
T,
(2.6)
Tm
where prime denotes differentiation with respect to 11and U = Cx represents the velocity of the wall with C > 0. Clearly u and v in (2.6) satisfy (2.1). Now substitution of (2.6) in (2.2)(2.4) give (1 + R)f”’ +p Ag” + A,(&’
+ Rg’ = 0
f”
f’g)
 R(2g +f”)
(2.7) = 0
(2.8)
0” + Pr(f6’  Lu*ge’) = 0
where R = k/p the micropolar
(2.9)
parameter, A=EC
and
A 1 =pic
CL2
are the micropolar variables, Pr = k,/pc,
P
the Prandtl number and ff*=_.CYC PCP
The boundary conditions (2.5) reduce to
f(O)=
v, (pc,p)l,2
f’(w) = 0;
=
Way);
dJ) = 0;
f ‘(0) e(m) =
= 1;
0.
g(0) = 0;
6(O) = 1 (2.10)
Flow and heat transferof a micropolar fluid
1423
3. METHOD OF SOLUTION The nonlinear coupled differential equations (2.7)(2.9) subject to the boundary conditions (2.10) have been solved using finite element method. Equation (2.7) can be decomposed into a pair of lower order equations f’h=O
(3.1)
(1 + R)h” +fh’  h2 + Z?g’= 0
(3.21
Ag” + A,(fg’  hg)  R(2g + h’) = 0
(3.3)
wk Pr(fe'a*gey =o.
(3.4)
The boundary conditions (2.10) are changed as f =a;
h=l;
h*0;
g=o; g*0;
ei 6+0
as
at
q=O
~300,
(3.5)
For the computational purposes [7], 01,has been fixed as 8. The domain is divided into a set (say, N) of line elements, each element having two end nodes. Variational formulation The variational form associated with equations (3.1)(3.3) over a typical element (a twonode line element) (
[email protected],v~+~)is given by I).?+1
f 'Ir
w,(f’  h) dq = 0
%+t w2[(1 + R)h” +$h’ h2+Rg’]dr]=0
J9e
(3.6) (3.7)
%+1 G‘W’ I
b
+
Adfg’  hg)  R(2g + h’)] dq = 0
w.,[ 8"+ Pr(
[email protected]’  a*gW)] dq = 0
(3.8) (3.9)
where w,, wz, w3and w4are arbitrary test functions and can be viewed as the variations off, h, g and 8, respectively. Finite element formulation The finite element model of the equations may be obtained by substituting finite~lement approximations of the form h = 5 hjvj j=l
f =,$lf;*j; WI=
W2 = W3 = l)i
(3.10)
into the variational fo~ulations (3.6)(3.8). Thus the finite element model of these equations is given by (3.11)
R. S. AGARWAL
1424
et al.
where [
[email protected]] and {F”} ((u, p = 1,2, 3) are defined as
(3.12)
where h, f and g are velocity and microrotation components that are assumed to be known. Clearly the element coefficient matrix, and hence the global coefficient matrix (by assembly of element equations to obtain the equations of the whole problem [lo]) depends on the velocity and microrotation field which is not known a priori. Therefore an iterative solution procedure is required. At the beginning of the first iteration, the velocity and microrotation field is set to zero and the global equations are solved for the nodal velocities and microrotation. This procedure is repeated till the required accuracy is achieved. Similarly the finiteelement model of the equation using the approximation of the form (3.13)
6 = 5 ejqJ, I=1 and W, = pi in the variational form (3.9), we obtain i
[email protected])
[email protected] _ GI” J=l
=
0
(3.14)
where
GI” = [
&($1;:”
6 and g, are the known velocity and angular velocity components from the previous calculations. After assembly of all element equations (3.14), etc., the system of equations have been solved by Gaussianelimination method.
Flow and heat transfer of a micropolar fluid
1425
1 00
0
so
0 60
0 70
0 60
f’
0 50
0 40
0 30
0 20
0 10
I 0
0 70
1 40
2 10
2.60
350
420
490
560
1
Fig. 2. Velocity distribution for various values of R (A = 1.0).
Fig. 3. Velocity distribution for various values of 3, (R = 1.0).
630
1426
R. S. AGARWAL
0
600
500
4.00
3.00
200
1 00
et al.
?
Fig. 4. Transverse velocity distribution for various values of R (A = 1.0). 4.
RESULTS
AND
DISCUSSION
The numerical results for the velocity field are obtained correct to six decimal places for R and A ranging from 0 to 10 and 0 to 5 respectively. The heat transfer results are obtained for the Prandtl numbers 1.0 and 5.0 with varying R and A. In all the cases the calculations are carried out for the region 0 I n I 8 dividing it into 40 elements. It is observed that the results 0
18
0.06
R.
50
R:
30
R=
1 0
0 04
0°2LL 0
0 70
1 40
Fig. 5. Microrotation
2 10
260
350
4.20
4.90
560
distribution for various values of R (A = 1.0).
630
1427
Flow and heat transfer of a micropolar fluid
1 00
0 60
g
060
0 40
0
20
Fig. 6. Microrotation distribution for various values of A (R = 1.0).
obtained are sufficiently high degree of accuracy even for small number of elements, say 10,15, 20, etc. Also it has been noticed that the effect of R on velocity field is more pronounced when compared with the effect of variation in A and Al. So that A and Al are fixed as 1.0 and CY*= 1.0 throughout the study. Figures 2 and 3 show the velocity profiles. In Fig. 2, it is observed that the velocity as compared to the viscous case R = 0, increases on increasing micropolar effects in the fluid for a tixed L = 1.0. The velocity profiles as a whole are lowered much with increase in suction (see Fig. 3). Further, it may be seen from Fig. 4 that for a fixed il, the transverse velocity increases
0.60
0
0.60
Fig. 7. Temperature
1 20
distribution for
160
2L IO
variousvaluesof Pr and A (R
3.00
= 1.0).
1428
R. S. AGARWAL
PPr 
0
:
pr ::
et al.
1.0 5.0
0.60
2.40
3.00
Fig. 8. Temperature distribution for various values of Pr and R (A = 1.0).
with increase in R. The microrotation distribution is shown in Figs 5 and 6. From Fig. 5, it is clear that the microrotation increases with increase in R. The microrotation near the plate increase with increase in A but the situation is completely reversed as one moves away from the plate (see Fig. 6). Figures 7 and 8 depict the temperature profiles for various values of I and R with the Prandtl numbers 1 and 5. It is clear from Fig. 7 that an increase in suction leads to a decrease in the temperature of the fluid throughout in the boundary layer irrespective of Pr. As would be expected, the temperature inside the boundary layer decreases with the increase in micropolar effects which results to cooling of the fluid. But the effect is reversed and not much prominent near the plate. Further, it is clear that for high Prandtl number, the micropolar effects on the fluid are not much significant. The effect is pronounced for large values of R only. Also for application of this problem to liquid metals [9], one must take Pr <<1.
REFERENCES 1) A. C. ERINGEN, Int. J. Engng Sci. 2,205 (1964). 21 A. C. ERINGEN, J. Math. Me&. 16, 1 (1966). 31 J. PEDDIESON and R. P. McNI’lT, Recent Adv. Engng Sci. 5,405 (1970). 4] A. J. WILLSON, hoc. Camb. Phil. Sot. 67,469 (1970). 51 A. C. ERINGEN, 1. Math. Anal. Appt. 38,480 (1972). 61 G. AHMADI, ht. 1. Engng Sci. 14,639 (1976). 71 R. S. R. GORLA, Int. J. Engng Sci. 21,25 (1983). 81 H. S. TAKHAR and V. M. SOUNDALGEKAR, ht. J. Engng Sci. 23,201 (1985). 91 A. CHAKRABARTI and A. S. GUPTA, Q. Appl. Math. 37,73 (1979). 0] J. N. REDDY, An Introduction to the Finite Element Merhod. McGrawHill, New York (1985). (Received 13 March 1989)