| Considered herein are three problems related to convective instabilities in which buoyancy is a fluid driving force. This bouyancy is produced by imposed temperature gradients across the fluid and is measured by a Rayleigh number.; The first of these problems considers a convection layer heated from below in a time-periodic manner with fluctuations of small amplitude (delta). The nonlinear theory is studied using perturbation theory in the neighborhood of the onset of convection. It is shown, for this type of boundary condition that the critical Rayleigh number for the onset of convection is increased by an amount (delta)('2). The preferred mode for the weakly nonlinear convective motion takes the form of hexagonal cells instead of roll cells which are preferred in layers subject to steady heating.; The second problem considers the flow of a fluid in thermosyphons, thin closed loops. These loops are thermally coupled and we investigate the effects of coupling on the steady and unsteady dynamics of the loops. The Biot number, L, governs the flow of heat from one loop to the other. For L = 0, the loops decouple and the fluid flow is governed by the Lorenz equations. Here static, dynamic and periodic flows appear. Chaotic flows can be attained under certain conditions. For nonzero L, this situation is altered. Analytical and numerical methods are used to examine the coupled problem. It is found that the coupling can delay instabilities and the onset of chaos but can give rise to new states that alter the symmetries of the flow.; Finally we study a two-dimensional problem which exhibits free surface driving forces in addition to buoyancy. The non-dimensional surface tension gradient is called the Marangoni number. The linear problem has been studied by Nield, who showed that the two driving forces reinforce each other to cause instability. Amplitude equations for the nonlinear problem are generated by series truncation methods generalizing to free surface problems, the Lorenz system.; Specifically, for the case M << 0 the resulting dynamical system is analyzed analytically for steady states and bifurcation. Numerical integration of this system is performed to find that chaos in this system is enhanced over that in the Lorenz system. The dimension of the system, however, is the same as the Lorenz system. |
