Stability Analyses for Porous Convection Including Second Sound Effects

Abstract

We investigate various models of thermal convection in a fluid saturated porous medium of both Darcy and Brinkman types. The linear instability and global (unconditional) nonlinear stability thresholds are analysed. Analytical solutions and numerical solutions are obtained by employing the Chebyshev tau and compound matrix techniques, and we investigate the effect that the inertia term and other physical parameters have on the stability of the system. The thesis is split into two parts. In PartI we consider a coupled model of thermal convection in a fluid saturated porous material and theories of viscous fluid motion
which allows heat to travel as a wave. This is discussed in the first three chapters.

In Chapter 2 the instability mechanism is investigated in complete detail and it is shown that stationary convection is likely to prevail under normal terrestrial conditions, but if the thermal relaxation time is sufficiently large there is a possible parameter range which allows for oscillatory convection. However, the presence of the Guyer-Krumhansl terms has the effect of damping the oscillatory convection and returning the instability mechanism to one of stationary convection.

In Chapter 3 the constitutive equation for the heat flux is
governed by a couple of the Guyer-Krumhansl equations and the Cattaneo-Fox law. In particular, we study the effects of the Guyer-Krumhansl terms on oscillatory convection. It is found that for a certain range of the Guyer-Krumhansl coefficient stationary convection occurs while changing the range results in oscillatory convection. Numerical results quantify this effect.

The thermal instability in a Brinkman porous medium incorporating fluid inertia for both free--free and fixed--fixed boundaries is considered in Chapter 4. We have incorporated the Cattaneo--Christov theory in the onstitutive equation for the heat flux. For fixed surfaces, the results are generated by using the Chebyshev tau method. The results reveal that employing the Cattaneo--Christov theory has a pronounced effect in determining the convection instability threshold.

Part II concerns the effect of an anisotropic permeability on thermal instability in the modelling problems of thermal
convection of Darcy type with and without the inclusion of an inertia term, which represented the last three chapters.

In Chapter 5 we allow a non-zero inertia term and also allow the permeability to be an anisotropic tensor. For particular numerical results we consider the case when the vertical component of the permeability tensor is variable. Linear instability results are calculated numerically and it is proved that the nonlinear energy stability bound is the same as the linear one. We perform the linear instability and nonlinear stability analysis, in the case
where the inertial term vanishes, to investigate the effect of anisotropy with rotation on the stability thresholds in Chapter 6, showing that the nonlinear critical Rayleigh numbers coincide with those of the linear analysis. The results reveal that the inclusion of the inertial term for this model can play an important role on the onset of convection in Chapter 7.