Stability Studies of Porous Media Including Surface Reactions

Abstract

We investigate the onset of thermal convection in a number of porous models, with a focus on the influence of a boundary reaction. The models that we consider are:
the Darcy porous model; the Darcy model with inclusion of the Soret effect and the Brinkman model, all with an exothermic surface reaction on the lower boundary. Numerical results are presented for each of these models and we show that the Darcy and Brinkman models with a surface reaction are structurally stable. Finally we derive stability results for a vertical porous channel that is in thermal non-equilibrium.

In Chapter 2 we investigate how the parameters of an exothermic reaction on the lower boundary of a horizontal Darcy porous layer affect the linear instability boundary. We show that for low Lewis numbers stationary convection is dominant and for larger Lewis number oscillatory convection dominates. We use a non-linear analysis to and stability boundaries for this model in Chapter 3, showing how some
of the reaction parameters affect this boundary. It is shown that the two boundaries do not coincide and there is a region in which sub-critical instabilities may occur. Structural stability on the reaction parameters is established for this model in Chapter 4.

The impact of including the Soret effect on the stability of the Darcy model with a surface reaction on the lower boundary is considered in Chapter 5. When stationary convection dominates we find that increasing the Soret effect increases the critical Rayleigh number that defines the instability boundary.

Chapter 6 discusses instabilities in a highly porous layer with an exothermic surface reaction on the lower boundary. The Brinkman model is used to take into account the impact of higher level derivatives of the fluid velocity. We show that this model is structurally stable on the parameters of the reaction in Chapter 7.

Finally, in Chapter 8 we analytically derive two stability results for a vertical porous channel in thermal non-equilibrium. The first is that the model is stable for any initial data provided the Rayleigh number is below a given threshold. The second is that there is stability for any Rayleigh number given restrictions on the initial data.