Heat transfer in laminar viscous flow in a channel with one porous wall

Abstract

Similarity solutions to the two-dimensional steady Navier–Stokes and energy equations for viscous flow in a semi-infinite horizontal channel with upper porous and lower impermeable walls are presented. The finite end of the channel is closed. Both fluid suction and injection through the porous boundary are considered. Two cases of thermal conditions at the walls are studied. In one case the temperature of both channel boundaries decreases linearly along the channel. In the second one, the upper wall has the same decreasing temperature, while the lower boundary is thermally insulated. By means of similarity transformations the dynamic and heat transfer equations are reduced to three ordinary differential equations for the velocity, pressure and temperature functions. The velocity equation is solved numerically for different values of the suction/injection Reynolds number. Analytical solution of the dynamic problem in a series of small Reynolds number is also found. The thermal problem is solved numerically for arbitrary Peclet number and two values of the Prandtl number. Analytical solution for small Peclet and Reynolds numbers is also obtained. The dependence of the Nusselt number on the Peclet number at both walls is studied. The stream lines and isotherms are presented graphically for some values of the Reynolds number at the Prandtl number equal one. It is shown the strong influence of the flow behavior on the heat transfer inside the channel.