Natural convection between concentric spheres in a slightly-thermally stratified medium

Abstract

In the present paper, analytical solutions are obtained using perturbation expansion in powers of Grashof number for steady, axisymmetric flow of a viscous fluid contained between two concentric spheres. A uniform gravity field acts vertically downward. The outer sphere is assumed to be maintained at a variable temperature such that conditions for vertical stratification are satisfied. Analysis is presented for two cases: when a constant-heat-flux condition on the inner sphere surface is imposed or when its surface temperature is kept constant. Streamlines, isotherms and velocity components are shown graphically in an axial plane for each case. For the case of isothermal inner sphere, a dimensionless stratification parameter S governs the flow. Solutions for S = 0 correspond to the unstratified case. When S tends to infinity, the flow pattern has both vertical and horizontal symmetry. But when the inner sphere surface is kept at constant heat flux, the flow and temperature fields are governed by another dimensionless parameter Q. The case Q = 0 corresponds to thermally insulated inner sphere. For this case, flow is similar to that occurring when S tends to infinity, but the directions of the streamlines are reversed.