Thermal Convection with LTNE

Abstract

The object of this chapter is to present the equations of thermal convection with local thermal non-equilibrium (LTNE) effects and analyse stability properties of their solutions. The linear operator which arises in the LTNE equations often belongs to a special class of operators, known as symmetric operators, and this class of linear operator has special mathematical properties in the context of stability. Before we briefly discuss an important general nonlinear stability result we include an example of a symmetric linear operator which occurs in the classical theory of thermal convection in a fluid, a phenomenon usually referred to as Bénard convection. After discussing symmetric operators we analyses stability of thermal convection in a local thermal non-equilibrium porous material. We analyse the cases where the porous medium is of Darcy type with an isotroipc peremability, of Darcy type with an anisotropic permeability, of Forchheimer type, and of Brinkman type. In all cases both linear instability and global nonlinear stability is discussed.