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Abstract

Rayleigh's (1916) attempt to describe the experimental observations of Benard (1900) is the foundation of a large number of theoretical studies on thermal convection. Many of these investigations are based on the assumptions that the horizontal fluid layer is confined between stress-free upper and lower boundaries and is unbounded in the horizontal directions. Here, three-dimensional thermal convection in an idealistic infinite channel of rectangular cross-section with no-slip sidewalls and stress-free upper and lower boundaries, in which an adverse temperature gradient is maintained by heating the underside is investigated using both linear and nonlinear techniques. The amplitude equation for nonlinear disturbances is derived and the variation of its coefficients with both aspect ratio (the width of the channel) and Prandtl number ascertained. The results show that the amplitude of the motion undergoes a supercritical bifurcation as the Rayleigh number passes through the critical value for instability predicted by linear theory.

The effect of introducing distant endwalls is investigated using the technique developed by Daniels (1977) for the related two-dimensional problem. If the endwalls of the long three-dimensional box are rigid and the thermal conditions there are consistent with a basic state of no motion a supercritical bifurcation occurs at a new critical Rayleigh number. By determining the higher order amplitude equation the question of wavenumber selection, as developed by Cross et al (1980) for two-dimensional rolls, is extended to the three-dimensional case for the long box with finite length and aspect ratio, for Rayleigh numbers slightly above critical. The results indicate a physical behaviour similar to that observed in experiments.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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