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Abstract

The formation of convective patterns in Rayleigh-Bénard and related systems, in particular the interaction between orthogonal roll patterns in the neighbourhood of a lateral boundary, is studied analytically and numerically. The effects on the patterns of convection of forcing at the boundary, the Prandtl number of the fluid, the wavelength perpendicular to the boundary and the wavelength parallel to the boundary are also included. Numerical solutions are found using an explicit finite difference scheme and appear to be in good agreement with theoretical predictions. The evolution of the roll pattern with time is illustrated by contour plots.

Weakly nonlinear theory is used to determine finite-amplitude patterns of convection near a lateral boundary x=0, leading to a coupled pair of equations for the amplitudes of x-rolls and y-rolls parallel and perpendicular to the boundary respectively. The solution for real amplitude functions independent of y is described in detail in chapter 2. This identifies a steady-state structure consisting of two main regions, one containing a combination of x-rolls and y-rolls and the other containing x-rolls only. The transition zone which occurs between the two regions is considered in chapter 3. Complex amplitude functions are then considered, allowing variations in the wavelength of the x-roll pattern to be considered in chapter 4 and of the y-roll pattern in chapter 5. Finally, phase-winding effects associated with variations in the wavelength of the x-roll pattern when there is no lateral forcing are investigated in chapter 6.

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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