Title:

Linear and nonlinear convective instabilities in porous media

After the first studies on the onset of convection in porous medium by Horton & Rogers and Lapwood, the HortonRogersLapwood problem, also wellknown as DarcyBénard problem, this problem has attracted much attention from researchers to study the onset of convection in porous medium due to its industrial and environmental applications in fields such as thermal insulation engineering, the growth of crystalline materials, patterned ground formation under water and application to oceanic and planetary mantle. By being derived from these early studies of the DarcyBénard problem, this thesis extends those studies in different directions and it consists of three main studies on convection in a porous medium. Thus, it is important in this chapter (from section 1.1 to 1.5) to introduce, explain and define all the terms and conditions used in studies of DarcyBénard problems to provide better understanding in the subsequent chapters. The first study on the onset of convection in porous medium is presented in Chapter 2. The given title is "the onset of convection in the unsteady thermal boundary layer above a sinusoidally heated surface embedded in a porous medium." This study investigates the instability of the unsteady thermal boundary layer which is induced by varying the temperature of the horizontal boundary sinusoidally in time about the ambient temperature of the porous medium. This study has application in the diurnal heating and cooling from above in subsurface groundwater. The investigation of the occurrence of the instability is undertaken by finding the critical DarcyRayleigh number that marks the onset of convection. In order to do that, a linear stability analysis is applied by perturbing the basic state using a disturbance with a small amplitude. An unsteady solver is used together with NewtonRaphson iteration to find marginal instability. One finding is that the disturbance has a period which is twice that of the underlying basic state. Cells which rotate clockwise at first tend to rise upwards from the surface and weaken, but they induce an anticlockwise cell near the surface at the end of one forcing period which is otherwise identical to the corresponding clockwise cell found at the start of that forcing period. The second study is presented in chapter 3 entitled the linear stability of the unsteady thermal boundary layer in a semiinfinite layered porous medium. The general aim of this study is to examine the stability criteria of two dimensional unsteady thermal boundary layer that is bounded from below by an impermeable surface which is induced by suddenly raising the temperature of the lower horizontal boundary of the two layers semiinfinite porous domain. Due to the sudden temperature increase, it is suspected that an evolving thermal boundary layer formed is potentially unstable. A full linear stability analysis is performed using the smallamplitude disturbance to perturb the basic state of the temperature profile and the parabolic equations are solved using the Keller box method to mark the onset of convection. The growth or decay of the disturbances is monitored by the computation of the thermal energy of the disturbance. This study results in the finding of the locus in parameter space where two modes with different critical wave numbers have simultaneous onset, and also find cases where the two minima in the neutral curve and the intermediate maximum merge to form a quartic minimum. Still concerning about the layering effect, the third subsequent study is presented in "The effect of layering on unsteady conduction: an analytical solution method" which is in chapter 4. We considered a semiinfinite solid domain which exhibits layering and the thermal conductivity and diffusivity of each layer is different, therefore the nondimensional parameters are conductivity ratios and diffusivity ratios. The boundary of that domain is suddenly raised to a new temperature and detailed study is performed to the 2layer and 3layer system over a wide range of variation of the governing nondimensional parameters. The analytical solution of the governing equation is obtained by employing the Laplace transform. It is concluded that the thermal diffusivity ratio and the thermal conductivity ratio are the coefficients that play the important role in determining the manner in which conduction occurs. In particular, the thermal diffusivity ratio affects how quickly the temperature field evolves in time. The first three studies in this thesis considers the case of local thermal equilibrium (LTE), therefore we are keen to study the effect of local thermal nonequilibrium (LTNE) to the onset of DarcyBénard convection in porous medium which is presented in the last chapter; chapter 5. This work is the extension of work by Banu and Rees into the weakly nonlinear regime. The aim of this chapter is achieved by employing a weakly nonlinear analysis to determine whether the convection pattern immediately postonset is two dimensional (rolls) or three dimensional (square cells) which will be decided based on the coupling coefficients value set at 1. On those occasions where the coupling coefficient of the amplitude equation is above 1 then roll solutions are stable i.e two dimensional. Alternatively, if the coupling coefficient is below 1, then the roll solutions are unstable and threedimensional square cells form the stable pattern. It is found in this study that the roll solutions are stable.
