DARCY-BRINKMAN FREE CONVECTION ABOUT A WEOGE AND A CONE SUBJECTED TO A MIXED THERMAL BOUNDARY CONDITION

The Darcy-Brinkman free convection near a wedge and a cone in a porous medium with high porosity has been considered. The surfaces are subjected to a mixed thermal boundary condition characterized by a parameter m; m=0, 1, ∞ correspond to the cases of prescribed temperature, prescribed heat flux and prescribed heat transfer coefficient respectively. It is shown that the solutions for different m are dependent and a transformation group has been found, through which one can get solution for any m provided solution for a particular value of m is known. The effects of Darcy number on skin friction and rate of heat transfer are analyzed.

1. NTRODICTI ON.The problem of free convection adjacent to a heated vertical surface has received a great deal of attention.These studies assume that the surface is subjected to a prescribed temperature or a prescribed heat flux.In the existing literature these two cases have been studied independently.The present paper aims to present a unified treatment of these cases.It also i::l,des the case of prescribed heat transfer coefficient hitherto not consi.eredby earlier researchers.
Further the free convection on heated surfaces subjected to mixed thermal boundary condition has not received sufficient attention.In this paper we shall consider Darcy-Brinkman free convection [1,2] on n -edge and a cone in a porous medium ,ith high porosity.The free convection on a vertical plate subjected to a prescribed temperature and prescribed heat flux are obtained as special cases.

ANALYSIS.
The configuration of free convection adjacent to a wedge and a cone is shown in Fig. 1.The surfaces are subjected to a mixed thermal boundary conditions.The boundary layer equations governing the Darcy-Brlnkman free convection are (rnu} x + (rnv)y O, n 0 for wedge 1 for cone (2.1) u u x + v Uy o Uyy (o/K) u + gfl(T-T(R))coso (2.2)   u T x + v Ty (o/P r) Tyy where u,v are the velocity components along x and y directions respec- tively.T is th.e temperature and T(R) is the ambient temperature.The symbols g, , o and Pr denote gravitational acceleration, coefficient of thermal expansion, kinematic viscosity of the ambient .fluidand Prandtl number respectively, a O, a I 0, a 2 0 are prescribed constants.

Property
The equations (2.7) (2.9) are invariant under the transformation, * A Da* A 2 Da f*(* ,Da*) f(,Da)IA, e*(*,Da*) (,Da)/A 4 (3.1)where A is any positive real number.Property 2 If f{e,Da), e{,Da) is the solution of the boundary value problem {2.7) (2.10) for any particular value of m, say m 0, then the solution for any m is given by the equations {3.1) provided A is the positive root of the equation,  3. Values of A for transition Table 3 gives the values of the parameter A required for transition from one case to the other.The transition is illustrated by the following example for cone case n 1) with Da -I 0.1.
1.For PT we have, 0'(0) = 0.80193, f"(0) 0. The results of free convection on a vertical plate subjected to prescribed temperature or prescribed heat flux can be obtained from the present study as special cases of m 0 or respectively when n O, a 0 and Da -I O.

O
A 5 (l-m) A e(0,Da) + m e'(O,Da) 0 (3.2) Property 3 If the solution of the boundary value problem (2.7) {2.10) is same for any two distinct values of m, then the solution is same for all values of m, The mixed boundary conditions {2.10) includes the following as special cases 7'92 G. RAHANA]AH AND V. KUNARAN 1. Prescribed Temperature (PT)': a 0 > O, a Heat Transfer Coefficient (PHTC) a 0 < O, a 1 > O, a 2 O. Hence L -al/a0, m (R) and equation (2.10) becomes o(o) / o'

Table 2 .
. It is observed that the porous medium transports larger Values of e(o

Table 4 ,
critical values of Pr for different values of Darcy number, for which the solution is independent of m (property 3) are given.An interesting aspect of this c is that it bifurcates the class of solutions for different Pr as value of Pr' say Pr follows:c the values of 0(0), -'(0) and -O'(O)/O(0) decrease with m whereas

Table 4 .
Critical values of Pr for different Darcy numbers