Linear and nonlinear Rayleigh-Bénard convection in the absence of horizontal boundaries

Abstract

In the first part of the thesis, we will investigate the linear and weakly non- linear solutions to a convection problem that was first studied by Ostroumov in 1947. The problem pertains to the stability of the equations governing convective motion in an infinite vertical fluid layer that is heated from below. Ostroumov's linear stability analysis yields instability threshold conditions that are characterized by zero wave number for the Fourier mode in the vertical direction and by eigenfunctions that are independent of the vertical coordinate. Thus, any undertaking at determining the super critical nonlinear solutions and their stability through a small amplitude expansion fails. This failure is due to the fact that the terms induced by the nonlinear interaction of the linear modes vanish identically. Here, we put forth exact and stable solutions to the Ostroumov problem. These solutions are characterized by the same critical conditions for linear instability as the Ostroumov solutions. Moreover, we are able to use a small amplitude analysis to extend the analysis to the super critical regime and obtain the nonlinear steady stable solutions. Furthermore, when the analysis is extended to the case where the fluid layer thickness is also allowed to be infinite, we found that the infinite fluid region becomes linearly unstable through a Batchelor-Nitsche instability mechanism. The nonlinear solutions as well as similarity type solutions are then provided. Finally, numerical solutions of the full nonlinear problem are also presented which shed light on the flow patterns and temperature distribution induced by these new solutions. In the second part we consider Rayleigh-Bénard convection with a static density distribution whose unstably stratifed part occupies a very thin central layer. We get asymptotic relations for the critical Rayleigh number for small and large values of the thickness control parameter. Some limiting cases corresponding to the linear eigenvalue problem are treated analytically and the results confirmed by a detailed numerical investigation. For the moderate values of the thickness control parameter an analytical nonlinear stability three-dimensional study is under- taken in the case of poorly conducting boundaries. A weakly nonlinear evolution equation for the leading order temperature perturbation is also derived and solved numerically as function of thickness control parameter " and Prandtl number.