Convection with Chemical Reaction, and Waves in Double Porosity Materials

Abstract

We consider two cases of solid skeleton of porous materials: fixed skeleton saturated with fluid in motion and deformed skeleton.

In the first case, we study a problem involving the onset of thermosolutal convection in a fluid saturated porous media when the solute concentration is subject to a chemical reaction in which the solubility of the dissolved mineral is a function of temperature, particularly the effect of a reaction rate on the stability of the systems. We consider the Darcy model, the Brinkman model, and the Darcy model with anisotropic permeability and thermal diffusivity. Moreover, in all models the systems are subjected to heat on the lower boundary and salt on the upper or lower boundary.

In chapter 2 we show that the solutions to the Darcy and the Brinkman thermosolutal convection depend continuously on the reaction term when the chemical equilibrium is a linear function in temperature by establishing a priori bounds. While in chapter 3 we show continuous dependence of the solution to the Brinkman thermosolutal convection on reaction using a priori bounds for the solution when the chemical equilibrium function is an arbitrary function of temperature.

In chapter 4 we investigate the effect of the reaction terms on the onset of stability in a Darcy type porous medium using the energy method. We use the D^2 Chebyshev Tau technique to solve the associated system of equations and the corresponding boundary conditions. We obtain the energy stability boundaries for different values of the reaction terms and compare them with the linear instability boundaries obtained by Pritchard & Richardson(2007). We find that the two boundaries do not coincide when there is reaction and a region of potential sub-critical instability occur.

In chapter 5 we use the energy method to obtain the non-linear stability boundaries for thermosolutal convection porous medium of a Brinkman type with reaction. We implement the compound matrix technique to solve the associated system of equations with the corresponding boundary conditions. We compare the non-linear stability boundaries for different values of the reaction terms and the Brinkman coefficient with the relevant linear instability boundaries obtained by Wang & Tan(2009). Our investigation shows that a region of potential sub-critical instability may appear as we increase the reaction rate.

We study the effect of the mechanical anisotropy parameter and the thermal anisotropy parameter on the stability of a Darcy reactive thermosolutal porous medium in chapter 6 using the energy method. Particularly, we restrict consideration to horizontal isotropy in mechanical and thermal properties of the porous skeleton. We find that the anisotropic permeability has opposite effect to that of the thermal anisotropy parameter on the stability on the system.

In the second case, deformed solid skeleton, we study wave motion in elastic materials of double porosity structure.

In chapter 7 we derive the amplitude and describe the behaviour of a one-dimensional acceleration wave based on an internal strain energy function. The overall situation is complicated as a wave moves in a three-dimensional body, therefore in chapter 8 we investigate the propagation of an acceleration wave in three-dimensional fully non-linear model.