Development, analysis and simulation of laboratory scale models of some problems in astrophysical convection

Abstract

This dissertation addresses some open questions in the linear and non-linear theories of thermal convection in regions that are unbounded in the direction of gravity. The first part of the dissertation seeks to model and analyze an instability, the occurrence of which requires the existence of a thin unstably stratified region in an otherwise stably stratified environment that is common in some situations involving astrophysical convection. Convective motion that takes place in stars and planets is characterized by the absence of horizontal boundaries and unstable density stratification of small thickness compared to the full extent of the fluid region. The instability is of buoyancy nature and typically induced by a density stratification that is unstable over a thin part of a vertically unbounded region that is otherwise stably stratified. We model this situation by considering a fluid region having a very large or infinite vertical extent and put forth a mathematical model that can be described as a Rayleigh-B\'{e}nard instability between two stably stratified layers. Thus, we attempt to uncover the instability threshold conditions and corresponding flow patterns when the base state consists of a step-function density profile. This case is investigated in both the Rayleigh-B\'{e}nard (horizontal fluid layer of infinite extent) and Ostroumov (vertical channel) geometries. Our analysis also puts forth the dependence of the threshold instability conditions on the location of the density jump. It predicts new patterns consisting of either symmetric or antisymmetric lens shape instead of the oval shape typically observed in the continuous stratification case. The appearance of the lens shape is attributed to the discontinuity. The linear analysis is extended to the weakly nonlinear regime where we show that the bifurcation from the conduction state is supercritical. Hence, our results are testable experimentally. Furthermore, we derive expressions for the convective thermal flux at the locus of the density jump and put forth a small scale laboratory experimental set-up that can be used to test our theoretical predictions. The experimental set-up describes the situation of a mass of cold fluid that is suddenly made to overly a mass of warm same fluid. The experiment itself serves to quantify the heat transfer between the two masses of fluid upon mixing when they are in direct contact. Do they mix by diffusion alone or by diffusion and convection; and how does the mixing evolve? The second part of the dissertation deals with the three-dimensional Ostroumov problem undergoing rigid body rotation. We carry out both the linear and non-linear theories and derive the threshold instability conditions and corresponding flow patterns for a variety of cells. We consider both closed and open cells either at the top/bottom or in one of the horizontal directions. Our analysis leads to the derivation of non-linear evolution equation for the amplitude of motion, the solution of which yields the stable non-linear solutions as functions of the Taylor and Prandtl numbers. We also examine the question of pattern formation as functions of the main physical parameters.