Magnetic Field Effect on Stability of Convection in Fluid and Porous Media

Abstract

We investigate the linear instability and nonlinear stability for some convection models, and present results and details of their computation in each case. The convection models we consider are: convection in a variable gravity field with magnetic field effect; magnetic effect on instability and nonlinear stability in a reacting fluid; magnetic effect on instability and nonlinear stability of double diffusive convection in a reacting fluid; Poiseuille flow in a porous medium with slip boundary conditions.
The structural stability for these convection models is studied. A priori bounds are derived. With the aid of these a priori bounds we are able to demonstrate continuous dependence of solutions on some coefficients. We further show that the solution depends continuously on a change in the coefficients.
Chebyshev collection, finite element, finite difference, high order finite difference methods are also developed for the evaluation of eigenvalues and eigenfunctions inherent in stability analysis in fluid and porous media, drawing on the experience of the implementation of the well established techniques in the previous work. These generate sparse matrices, where the standard homogeneous boundary conditions for both porous and fluid media problems are contained within the method.
When the difference between the linear (which predicts
instability) and nonlinear (which predicts stability) thresholds is very large, the validity of the linear instability threshold to capture the onset of the instability is unclear. Thus, we develop a three dimensional simulation to test the validity of these thresholds. To achieve this we transform the problem into a velocity-vorticity formulation and utilise second order finite difference schemes. We use both implicit and explicit schemes to enforce the free divergence equation.