On the stability of a steady convective flow due to nonlinear heat sources in a magnetic field

Abstract

Consider a layer of a viscous incompressible fluid bounded by two vertical planes. There exists a steady flow in the vertical layer caused by internal heat generation. The heat sources are distributed within the fluid in accordance with the Arrhenius law. A magnetic field of constant strength is applied in the direction perpendicular to the planes. The flow is characterized by four dimensionless parameters: the Grashof number, the Prandtl number, the Hartmann number and the Frank-Kamenetsky parameter. This problem is important in applications such as biomass thermal conversion. The objective of the study is to determine the factors that enhance mixing and lead to more efficient energy conversion. The problem is described by a system of magnetohydrodynamic equations under the Boussinesq approximation. The nonlinear system of ordinary differential equations describing the steady flow is solved numerically. Linear stability of the steady flow is investigated using the method of normal modes. The corresponding linear stability problem is solved numerically by means of a collocation method. The solution is found for different values of the parameters characterizing the problem. It is found that the increase of the Frank-Kamenetsky parameter destabilizes the flow. On the other hand, the increase of the Hartmann number stabilizes the flow.