UNSTEADY
FORCED
AN INFINITE
AND FREE-CONVECTION
VERTICAL
PLATE
WITH
FLOW PAST
CONSTANT
HEAT
FLUX BASANT KUMAR JHA Department of Mathematics, Banaras Hindu University, Varanasi, India
(Received 21 January, 1991) Abstract. An analytical study is performedto examinethe unsteady forced and free-convectionflow of a viscous, incompressiblefluid when there is a consdtant heat flux between fluid and plate. The expressions for the vleocityfield, the penetration distance and the skin-friction have been obtained by the Laplace transform technique. The influence of the various parameters entering into the problem is extensively discussed with the help of tables,
1. Introduction Free- and force-convection flow past a vertical flat plate have been investigated extensively. The majority of the recent studies have dealt with circular tube geometry, but increasing attention is being focused on the flat plate case. This configuration is relevant to solar energy collection, as in the conventional flat plate collector and the Trombe Wall, and in the cooling of modern electronic systems. In the latter application, electronic components are mounted on circuit cards, an array of which is positioned vertically in a cabinet forming vertical flat plates through which coolants are passed (Aung et al., 1973; Aung, 1973). The coolant may be propelled by fre convection, forced convection, or mixed convection, depending on the application. Recently, Jagahirdar and Lahurikar (1989) studied the unsteady forced and free-convection flow past an infinite vertical isothermal plate. Hence, it is now proposed to study the unsteady forced and freeconvection flow past an infinite vertical plate when the heat is supplied to the plate at constant rate. 2. Mathematical Analysis An unsteady forced and free-convection flow of an incompressible viscous fluid past an infinite vertical plate is considered. Initially, the plate and fluid are at same temperature. At time t' > 0, the heat is also supplied to the plate at constant rate. The x 1 axis is taken along the plate and in the upward directin and y'-axis normal to it. Since the plate is infinite inextent, all physical quantities are functions of y ' and t' only. The derivation of the equations governing the transient forced and free-convection flow past an infinite vertical plate is given by Jahagirdar and Lahurikar (1989). Following this treatment, the flow of an incompressible, viscous fluid under Boussinesq's approximaAstrophysics and Space Science 185: 295-298, 1991. 9 1991 Kluwer Academic Publishers, Printed in Belgium.
296
B.K. JHA
tion is governed by the following system of non-dimensional equations &
a2u
--=0+--, at Oy2 Pr
80 at
(1)
a20
(2)
ay 2
The non-dimensional quantities introduced in the above equations are defined as y = y ' Uo GrX/Z/v,
Pr -
u = u'/Uo,
t = t' U2o G r / v ,
(3)
#@ k
0 = kUo(T' - T~)/qv,
Gr = vqgfl/kU o .
The initial and boundary conditions are for
for
t < 0:
u(y, t) = O ,
O(y, t) = O ,
u(O, t) = O,
ao(0, t) ay
u(oo, t) = 10
0(o% t) = 0 .
1,
(4)
t> 0
The solution of Equation (2) under the boundary condition (4) has been obtained by Soundalgekar and Patil (1980) and Georgantopoulos et al. (1979). If we apply the usual Laplace transform technique, the solution of Equation (1) subjects to the boundary condition (4) is given by (403/2 [~exp(-r/2) u = erf(t])+ (pr ~ i~ ~ / ~ ] [ . - ; 7 ~
;~
(~ t/erfc (tl)
~)} +
(1 + t12 Pr) - ~/x/~ erfc(t/x/~ )
-
+
(5) where t 1 = y/2 x~tt.
By use of the expression (5) the penetration distance is given by Xp=L-
p-~(y,p)
=t+(pr_
1)x/~k
x/_7~
• {-6t12 Pr - 4t/4 pr2 _ ~} + erfc(t/x/~ ) •
297
UNSTEADY FORCED AND FREE-CONVECTION FLOW
6q25
154 ~ / 4 - 8 )
+
%
+ ~2r/t exp( - q2) _ t erfc(q) (1 + 2r] 2) 1 .
(6)
Also by use of expression (5) the skin-friction at the plate is given by 0u
1 =
t +
+
3. Discussion and Results F o r the purpose of discussing the results, some numerical calculations are carried out for velocity field (u), penetration distance (Xp), and skin-friction (~). In order to be TABLE I Variations of velocity field and penetration distance for different values of Pr and t Pr
t
y
u
Xp
0.71
0.2
0,0 0.2 0.4 0.6 0.8 1.0
0.0000 0.2639 0.4913 0.6725 0.8048 0.8928
0.0000 0.0709 0.1145 0.1395 0.1541 0.1642
0.4
0.0 0.2 0.4 0.6 0.8 1.0
0.0000 0.2135 0.3961 0.4393 0.6743 0.7729
0.0000 0.0778 0.1306 0.1633 0.1827 0.1959
0.2
0.0 0.2 0.4 0.6 0.8 1.0
0.0000 0.2497 0.4739 0.6577 0.7943 0.8862
0.0000 0.0829 0.1354 0.1667 0.1840 0.1930
0.4
0.0 0.2 0.4 0.6 0.8 1.0
0.0000 0.1811 0.3493 0.5004 0.6306 0.7374
0.i)000 0.1253 0.2168 0.2818 0.3265 0.3563
7.0
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B.K. JHA
realistic, the values of Prandtl number (Pr) are chosen to be 0.71 and 7.0 which corresponds to air and water, respectively. The variation o f velocity field and penetration distance are given in Table I. F r o m this table it is clear that with the increase o f Prandtl number and time parameter (t) velocity decreases and penetration distance increases. The variations o f skin-friction are given in Table II. F r o m this table we conclude that skin-friction decrases with increase of Prandtl number and time parameter. TABLE II Variations of skin-friction (~) for different values of Pr and t Pr
0.71
t 0.2
0.4
1.3901 1.2820
1.1495 0.9333
References Aung, W.: 1973, Bell System TechnicalJ. 52(6), 907. Aung, W., Kessler, T. I., and Beintin, K. I.: 1973, ICEE Trans. Parts, Hybrids, Packing 9(2), 75. Georgantopoulos, G, A., Nanousis, D. N., and Goudas, C. L.: 1979,Astrohys. Space Sci. 66, 13. Jahagirdar, M. D. and Lahurikar, R. M.: 1989, Indian J. Pure Appl. Math. 20(7), 711. Soundalgekar, V. M. and Patil, M. R.: 1980,Astrophys. Space Sci. 70, 179.
