 # Nonlinear Stability Analyses for Variable Viscosity and Compressible Convection Problems

Richardson, Lorna Logan (1993) Nonlinear Stability Analyses for Variable Viscosity and Compressible Convection Problems. PhD thesis, University of Glasgow.

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## Abstract

In this thesis we present nonlinear energy stability analyses of a variety of convection problems, specifically concentrating on compressible convection and convection where the viscosity of the fluid depends on the temperature. To begin we introduce the energy method, which we shall employ in later chapters in order to derive nonlinear stability criteria, and illustrate its effectiveness with a simple one-dimensional example. We then study the stability of a system containing a generalized incompressible fluid, employing linear theory, the energy method and an asymptotic analysis. Next we study a variable viscosity fluid, first establishing continuous dependence of the solution to the Oberbeck-Boussinesq equations, both forward and backward in time, on the viscosity. We then look at convection where the viscosity is first a linear function of the temperature and then a quadratic one. In both cases we employ a generalized energy in order to derive a nonlinear stability boundary. To conclude our analysis of a variable viscosity fluid we also introduce a temperature dependent conductivity and carry out both linear and heuristic nonlinear analyses. We then turn our attention to the phenomenon of convection within a porous medium, concentrating first on penetrative convection. We examine the stability of convection in a porous medium containing an internal heat source and a salt field, using linear theory, the energy method and a weighted energy. Then, returning to the topic of variable viscosity, we demonstrate that, in a porous medium, the solution to the equations of motion depends continuously on the viscosity both forward and backward in time. Finally we study a porous medium saturated with a variable viscosity fluid, employing the Brinkman equation, coupled with the energy method, in order to derive a nonlinear stability boundary.

Item Type: Thesis (PhD) Doctoral Adviser: B Straughan Applied mathematics, Fluid mechanics, Thermodynamics 1993 Enlighten Team glathesis:1993-76413 Copyright of this thesis is held by the author. 19 Nov 2019 14:32 19 Nov 2019 14:32 http://theses.gla.ac.uk/id/eprint/76413 View Item