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Vol. 2s. NO. 6. pp. 723-728.

1990

0020-7462/W 31.00 + .oO Pergnmon Press pk

MIXED CONVECTION OF AN INCOMPRESSIBLE VISCOUS FLUID IN A POROUS MEDIUM PAST A HOT VERTICAL PLATE H. S. TAKHAR,*V. M. SOUNDALGEKAR’ and A. S. GUPIA* *Department of Engineering, University of Manchester, Manchester, Ml3 9PL, U.K. ‘31A/12, Brindavan Society, Thane (4OOoOl),India. *Department of Mathematics, Indian Institute of Technology, Kharagpur, (721302), India (Received 3 January

1989; in revised form 25 August 1989)

Abstract-This paper analyses steady two-dimensional mixed convection of an imcompressible viscous fluid in a porous medium past a hot vertical plate. Assuming Darcy-Brinkman model for the Row in a porous medium, the boundary layer equations are integrated numerically to obtain the non-similar solution for the velocity and temperature distribution for several values of the permeability and viscous dissipation parameters. It is shown that for a fixed value of Prandtl number Pr and dissipation parameter E, the skin-friction at the plate decreases with increase in the permeability parameter K I. However for the same value or Pr and E, the heat transfer rate at the plate increases with increasing K,. The dimensionless velocity and temperature functions in the flow are plotted for several values of E and K, with Pr = 0.73. It is also shown that for fixed values of K, and Pr, the skin-friction increases with increase in the dissipation parameter E.

1. INTRODUCTION

In recent years a great deal of interest has been shown in the area of convective heat transfer in flows in fluid-saturated porous media. This is due to numerous and wide-ranging applications of convective flow through such media. In the context of geophysics, the problem is of interest in the geothermal use of energy when the onset of thermal convection takes place in the earth’s mantle which can be regarded as a porous medium. Using Darcy’s law for flow in a porous medium, Lapwood [l] investigated the breakdown of stability of a layer of fluid subject to a vertical adverse temperature gradient due to heating from below. It may be noted that in a porous medium (say a porous rock), the path of an individual particle cannot be traced analytically. But a gross effect as the fluid percolates through the pores or the rock may be represented by a macroscopic law (Darcy’s law) applied to masses of fluid large compared with the dimensions of the porous structures of the medium. This law is based on the hypothesis that fluid flowing with macroscopic velocity V through a porous medium experiences a resistance gV/K, per unit mass, where g is the acceleration due to gravity, and K is a constant (known as the permeability of the medium) which increases with the ease of percolation. Convective flow through porous media also arises in industrial problems such as petroleum reservoir modelling, thermal insulation techniques, chemical and nuclear engineering, solar power collectors, regenerative heat exchangers containing porous materials, and burying drums containing heat-generating chemicals in the earth. A comprehensive account of convection in a porous medium could be found in the monographs by Bejan [2] and Cheng [3]. On the other hand Nield [4] recently provided extensive state-of-the-art reviews on convective heat transfer in flows through a porous medium. Theoretical study of free convection from a horizontal cylinder embedded in a porous medium was made by Schrok, Fernandez and Kesavan [5]. Numerical calculations and experimental investigations of the above problem were carried out by Fernandez and Schrok [a]. Similarity solutions of the governing equations that describe free convection about a cylinder in a porous medium were obtained by Merkin [7] while Nilson [8] carried out boundary layer analysis on two-dimensional and axisymmetric surfaces in fluid-saturated porous medium. Ingham, Merkin and Pop [9] studied the collision of free convection boundary layers on a horizontal cylinder embedded in a porous medium.

Contributed by K. R. Rajagopal. 723

724

H. s.

TAKHAR

et af.

In the field of combined free and forced convection in a porous medium, not much work has been done. Cheng [lo] studied the mixed convection about a horizontal cylinder and a sphere in a fluid-saturated porous medium. Minkowycz, Cheng and Chang [l l] investigated mixed convection about a non-isothe~al cylinder and sphere in a porous medium. Cheng [ 121 examined free and forced convection flow about inclined surfaces in porous media. Huang, Yih, Chou and Chen Cl33 also studied mixed convection flow over a horizontal cylinder or a sphere embedded in a saturated porous medium. Mixed convective heat transfer from a heated horizontal plate in a porous medium near an impermeabIe surface was investigated by Oosthuizen [14]. In all these investigations, however, Darcy’s law was used so that the effects of non-linear convective terms as well as the viscous term vV2V were ignored. The neglect of non-linear convective terms may be justified in slow motion in seepage through a porous medium. If the fluid occupies almost all the parts of the porous medium, the viscous term vV2V should be taken into account as in the Darcy-Brinkman model for flow through a porous medium. This provides the motivation for the present study where we analyse mixed convection (i.e. combined free and forced convection) of an incompressible viscous fluid in a porous medium past a hot vertical plate held at a constant temperature higher than that of the ambient stream.

2. MATHEMATICAL

FORMULATION

AND

ANALYSIS

Consider the steady two-dimensional laminar flow of an incompressible viscous fluid past a hot vertical plate embedded in a porous medium. Taking x-axis along the plate and y-axis normal to it (Fig. l), the governing equations assuming the boundary layer approximations are, in usual notation, given by

(3) where Boussinesq approximation has been used to account for the buoyancy force term in (2) and the permeability K is assumed constant. It may be noted that the viscous dissipation term in (3) is generally neglected in heat transfer problems in porous medium although this may not be true in certain flow situations. Let us introduce the stream function I&X, y) such that

Y t ”

t

”

Fig. 1. A sketch of the physical problem.

Mixed convection

725

where U, is the constant free steam velocity, T, and T, are the constant temperature of the plate and the ambient temperature, respectively. We define AT, = T, - T,

t(x) = g/?AT,(x - x,)/U&

‘I = y/d(x),

(5a)

and 6(x) = [2(x - x0)v/U,]“2,

(5b)

x,, being a reference point. Equations (4) and (5) using (l), yield

d6

“C -

‘Oft+

‘O’ar

a_f a< __ax

1

af dd 9q;i;;“0

*

(6)

Using (4), (5) and (6) in (2) and (3), gives (7)

(8)

U6

E=

F

U?l =x’

s’

pr - t!s - k’

(9)

Here 1 is a representative length and Gr, E and Pr denote the Grashof number, Eckert number and Prandtl number, respectively. It should be noted that the representative length 1 in this problem can be taken as v/U, so that R = 1. A non-similar solution of (7) and (8) is now sought in the form f(L

It) =fo(‘t)

+ 5fi(tt)

ei59 rt) = eo(rl) + W,h)

+ 52f2(tl) + * * *,

(10)

+ r2Ml)

(11)

+ . * .*

Substitution of (10) and (11) in (7) and (8) gives the following set of ordinary differential equations f’d’+fJ-b:=o,

(12)

f;“+fof;‘+3fif’d-2fbf;+2eo-K,fb=O,

(13)

f;“+fof;+3fif;‘+5f2fb:-2(f;2+2fbf;)+2e1-Klf;=0,

(14)

e,d + prfoeb

e;, +

pr[foe;

- 2fbe,

+ PrE. f g2 = 0, + 3fiegj

(15)

+ 2prEf;

.f;‘=

0,

(16)

e~+p~~foe;+5f2e~+2f~e;-4f~e2-2e,f;~+p~E~2f~f;’+f;‘2~=0,

(17)

where a prime denotes derivative with respect to q. The no-slip boundary conditions and specified temperature at the plate give fo=f1=f2=o;

fb=fi=O

eo= 1,

e,=e,=o

at q = 0,

(18a)

atq=0;

(18b)

the boundary conditions at infinity are fb = 1,

f;

=f;

= +o;

8, = 8, = 8, -+ 0

asq+

oc).

(19)

It can be seen from (12)-(17) that the flow and heat transfer in the boundary layer past the hot plate are characterized by the three dimensionless parameters K,, Pr and E. Following Takhar [15], these equations are integrated numerically on a computer for several values of the above parameters. Figures 2-4 show the functions f b(q), f i(q), f i(q) vs q for the xcomponent of the velocity for several values of E and K1 with Pr = 0.73. On the other hand,

726

H. S.

TAKHAR

et al.

Fig. 2. Velocity function f;,

fi

Fig. 3. Velocity function f’,

Pt=O.73

E

K,

0.1

1

0.1 0.1

2 II 3 m

03

1 Is!

-0.6 Fig. 4. Velocity function J;.

I

727

Mixed convection

as-

0.6-

0.4.

0.2-

0-f 0

2

L

6

61

Fig. 5. Temperature function 8,.

a2

0

I

-0.2

z

I -aLJ

0.31 i!z

Fig. 6. Temperature function 0,.

a?

1

3.2

R.O.73

E K, 0.1

1' I

0.1 2 II 0.1 3 m 0.3 1 H

n

Fig. 7. Temperature function &.

Figs 5-7 display e,(q), 0,(q) and 0,(q) vs q for the temperature distribution for different values of E and Ki with Pr = 0.73. Table 1 gives the values of f;‘(O), f;‘(O), Pi(O) and e;(O) for several values of E and the permeability parameter K, with Pr = 0.73. The numerical values of f;(0) and &JO) are already available in the literature. It should be noted that numerical integration of (12) subject to (18a) and (19) gives f;(O) = 0.506. Using relations (10) and (11) and Table 1, both the skin-friction and heat

H. S. TAKHAR et al.

728 Table

I. Values ofjj’(0)

and f?:‘(O) (i = 1, 2)

E

K,

f Y(O)

f ;‘CO)

0.1 0.1 0.1 0.3

I.0 2.0 3.0 1.0

1.11 0.516 0.02 I 1.15

- 0.675 0.225 1.15 - 0.751

0; (0)

e;(o)

- 0.129 0.049 0.185 -0.110

0.220 - 0.068 - 0.221 0.238

Table 2. Values of f”(O) and 0’(O) - B,(O)

K,

f”(O)

1 2 3

0.61025 0.56015 0.5196

Q’(0)-

ago,

- 0.0107 0.0042 0.0162

transfer at any position on the plate can be computed. In particular for the position 5 = 0.1 with Pr = 0.73 and E = 0.1, the dimensionless skin-friction f”(0) and plate heat transfer rate 6’(O) are shown in Table 2 for several values of K,.

3. DISCUSSION It can’be seen from Table 2 that for fixed values of Pr and E, the skin-friction at the plate decreases with increase in the permeability parameter K,. However the heat transfer rate at the plate increases with increase in K,. Again for Pr = 0.73 and E = 0.3 we find on using Table 2 that at 5 = 0.1, f”(0) = 0.61349 for K, = 1. Comparing with the corresponding value 0.61025 of skin-friction for E = 0.1 and K, = 1 with Pr = 0.73 from Table 2, we find that for given values of K, and Pr, the skin-friction increases with the Eckert number E. Since E is a measure of viscous dissipation, we may interpret the result physically in the following manner. Enhanced viscous dissipation results in increased heating of the boundary layer fluid with a concomitant rise in the local velocities near the plate. Hence the shear stress at the plate rises with increase in E, the other parameters being held constant.

REFERENCES 1. E. R. Lapwood, Proc. Camb. Phil. Sot. 44, 508 (1948). 2. A. Bejan, Conuection Hear Transfir. Wiley, New York (1984). 3. P. Cheng, Natural Convection: Fundamentals and Applications (Edited by S. Kakac et a/.). Hemisphere, Washington, D.C. (1985). 4. D. A. Nield, Proc. CSfRO/DSIR Seminar on Convective jbw in Porous Media, D.S.I.R., Wellington, New Zealand (1985). 5. V. E. Schrok, R. T. Fernandez and K. Kesavan, Proc. Inc. Hear Transfer Co& Paris-Versailles, Vol. VII CT-3.6 (1970). 6. R. T. Fernandez and V. E. Schrok, Proc. Int. Heat Transfer Conf, Munich, Vol. II, p. 335 (1982). 7. J. H. Merkin, Int. J. Heat Mass Transfer 22, 1461 (1979). 8. R. H. Nilson, ASME J. Heat Transfer 103, 803 (1981). 9. D. B. Ingham, J. H. Merkin and I. Pop, Q. J. Mech. appl. Math. 36, 313 (1983). 10. P. Cheng, Int. J. Heat Mass Transfer 25. 1245 (1982). P. Cheng and C. H. Chang, Numer. Heat Transfer 8, 349 (1985). 11. W. J. Minkowycz, 12. P. Cheng, Int. J. Heat and Mass Transfer 20, 807 (1977). 13. M. T. Huang, K. A. Yih, Y. L. Chou and C. K. Chen, ASME J. Heat Transfer 108, 469 (1986). ASME J. Heat Transfer 110, 390 (1988). 14. P. H. Oosthuizen, 15. H. S. Takhar, J. Fluid Mech. 34, 81 (lY88).