Stability Analyses of Multi-Component Convection-Diffusion Problems

Tracey, John (1997) Stability Analyses of Multi-Component Convection-Diffusion Problems. PhD thesis, University of Glasgow.

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Abstract

This thesis employs analytical and numerical techniques to investigate the stability of multi-component convection-diffusion problems. In chapter 1 the physical importance of these problems is discussed and a brief review of the relevant literature is given. The ideas of linear and nonlinear stability are introduced here via a simple one-dimensional example. The second chapter presents linear and classical energy stability analyses for a system consisting of an infinite layer of a fluid-saturated porous medium with two salt fields present, using the Darcy-Oberbeck-Boussinesq scheme of equations. The linear stability analysis produces highly unusual neutral curves and it is the investigation of these curves that motivates the rest of this thesis. The nonlinear stability analysis in chapter 2 produces a nonlinear boundary that may be far from the linear boundary, highlighting a weakness of the energy method when the onset of linear instability is by an oscillatory mode. However, this problem is somewhat ameliorated in the third chapter by a generalised energy analysis which produces far more satisfactory results. In the fourth chapter a non-Boussinesq buoyancy law is employed in this multi- component porous problem, introducing the phenomenon of penetrative convection. A numerical investigation of the linear stability of the problem is given, using a Chebyshev tau method, and the effect of the non-Boussinesq buoyancy law on the neutral curves is shown. A weighted energy method is used to obtain an unconditional nonlinear stability boundary. In the fifth chapter rather than the fluid-saturated porous medium of previous chapters attention is turned to an infinite layer of viscous fluid. An internal heat source is introduced as an alternative model of penetrative convection. The linearised version of this problem is shown to be the adjoint of previous work by Straughan and Walker (1997). The linear stability of the problem is again investigated numerically. Finally an appendix gives details of the Chebyshev tau method employed in chapters 4 and 5.

Item Type: Thesis (PhD)
Qualification Level: Doctoral
Additional Information: Adviser: Brian Straughan
Keywords: Applied mathematics, Fluid mechanics
Date of Award: 1997
Depositing User: Enlighten Team
Unique ID: glathesis:1997-76478
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 19 Nov 2019 14:17
Last Modified: 19 Nov 2019 14:17
URI: http://theses.gla.ac.uk/id/eprint/76478

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