Convection in fluid and porous media

Abstract

The subject of convection in fluid and porous media is investigated. Particular attention is paid to penetrative convection. The first two chapters are devoted to penetrative convection when fluid overlies and saturates a porous medium. Penetrative convection is described by a quadratic equation of state in the first instance and via internal heating in the second. Linear instability analyses are performed in both cases. A surprising and striking array of streamlines are presented at the onset of convection. The streamlines exhibit novel behaviour when physical parameters of the problem are varied. Penetrative convection in a horizontally isotropic porous layer is discussed next. Again penetrative convection is described by a quadratic equation of state and internal heating. The internally heated model is dealt with primarily as it yields a global nonlinear stability bound. The two models are shown to be mathematically adjoint and the nonlinear stability results compared with previously published linear ones. Good agreement between the two is seen. The effect of convection on the evolution of under-ice meltponds is investigated next. Linear and nonlinear analyses are employed to yield instability and global stability results respectively. Discrepancy between the two is found and the region of possible subcritical instabilities is presented. Finally convection in a porous medium is investigated via a cubic equation of state. It is found that unconditional nonlinear stability results can be established if Forchheimer theory is introduced. The results are compared to previously published linear ones and it is shown that the linear theory essentially captures the physics involved.