Numerical modeling of the magneto convection of a Newtonian fluid confined between two vertically eccentric hemispheres

Abstract

This work is a contribution to the numerical study of the phenomenon of heat transfer by laminar natural convection of an electrically conductive Newtonian fluid subjected to a uniform horizontal magnetic field. The study focused on a hemispherical cavity delimited by two vertically eccentric hemispheres. A constant flux density is imposed on the inner hemisphere while the outer hemisphere is maintained at a constant temperature. The combination of thermal and electrical boundary conditions is exploited to obtain the critical values of the parameters marking the onset of instability. The Boussinesq approximation is used to study the equations governing this fluid instability. The projection of these equations in the bispheric coordinate system as well as the discretization by the finite difference method facilitated the development of a computer code in Fortran. The exploitation of this code made it possible to determine the growth rates for Hartmann values equal to 1; 10 and 100, from Rayleigh equal to 103; 104; 105 and 106, with eccentricity equal to ± 0.2; ± 0.5 and 0 and a radius ratio equal to 2. The aim is to highlight the effect of the magnetic field on the heat transfer. At the end of the study, the results obtained are consistent and revealing: they are in good agreement with those of references drawn from the literature.