Thermal interaction between free convection and forced convection along a vertical conducting wall

A theoretical study is presented in this paper to investigate the conjugate heat transfer across a vertical ®nite wall separating two forced and free convection ̄ows at different temperatures. It is assumed that the heat conduction in the wall is only in the transversal direction. We also assume that countercurrent boundary layers are formed on the both sides of the wall. The governing equations of this problem and their corresponding boundary conditions are all cast into a dimensionless form by using a non-similarity transformation. These resulting equations, which are singular at the points nc ˆ 0 and 1, are solved numerically using a very ef®cient singular perturbation method. The effects of the resistance parameters and of the Prandtl numbers on heat transfer characteristics are investigated and presented in a table and ten ®gures. List of symbols A constant b thickness of the plate C constant f reduced stream function g acceleration due to gravity hx local heat transfer coef®cient k thermal conductivity L length of the plate Nux local Nusselt number Nu average Nusselt number Pr Prandtl number qx local heat ̄ux Q overall heat ̄ux Re Reynolds number Rt forced convection thermal resistance parameter R t free to forced convection parameter t ambient temperature Dt characteristic ambient temperature T ̄uid temperature u; v velocity components U1 free stream velocity x; y Cartesian coordinates Greek letters a thermal diffusivity b coef®cient of thermal expansion h dimensionless temperature k dummy variable m kinematic viscosity n; g reduced coordinates w stream function