Free convective Darcy flow induced by a temperature gradient in a hemispherical porous medium

Abstract

Assuming the validity of the Darcy model, the steady-state development of the flow and heat transfer induced by a temperature gradient in a hemispherical porous medium with a free surface is investigated using a semi-analytical method. The streamlines are in the form of eddies with expanding vortices on either side of the axis of symmetry with the intensity of the flow field increasing with an increase in the Rayleigh number. The rate of momentum transfer is relatively higher than in a non-Darcy regime. The axial velocity is monotonic and is found to increase with an increase in the Rayleigh number. The behavior of the fluid motion on the free surface is similar to that of the axial flow. The thermal field exhibits two distinctive regions, one with a lower temperature gradient and the other with a higher but constant temperature gradient. Around the deepest end of the hemisphere, the local heat transfer rate remains unaffected by heat convection and a singularity in the graph of the Nusselt number is noticed for which a theoretical justification is given.