Effects of transpiration on free convection in an annulus between concentric porous spheres

Abstract

The research reported here deals with the effects of transpiration on free convection in an annulus between concentric porous spheres. An incompressible Newtonian fluid fills the annulus between two concentric porous spheres which are maintained at uniform temperatures. The spheres are stationary and a uniform gravitational field acts vertically downward, parallel to the fixed axis of the spheres. The temperature difference between the spheres gives rise to a free convection flow pattern in the annulus. By uniformly injecting or sucking fluid through the porous spherical boundaries, a radial flow-field is created which interacts with the free convection flow pattern. The purpose of this investigation is to study the interaction of this radial flow-field with the free convection flow pattern, since heat transfer at the porous surfaces is significantly influenced by varying the injection and suction rates. An analytical solution of the governing steady-state Navier-Stokes equation of motion and the energy equation is obtained by employing a regular perturbation technique. Solutions for the stream-function and temperature are obtained in the form of power series expansions in terms of the Rayleigh number Ra, and injection/suction Reynolds number Re. The analytical solution is valid for all values of the Prandtl number Pr, and relatively small values of Ra and Re. A finite-difference solution of the governing steady-state equations of motion and energy is also provided. The range of validity of the analytical solution is determined by comparison with the numerical solution. Results for the flow patterns, velocity distributions, temperature profiles, isotherms, local and average Nusselt numbers are presented for various values of Ra, Re, Pr, and the radius ratio $\lambda$