Extended theory of the Be?nard convection problem
The onset of convection in a horizontal fluid layer, heated from below, is, examined by means of perturbation analysis. The resulting eigenvalue system of equations is solved by means of a new extension of a Fourier series technique. Two sets of coupled effects are investigated:
(i) thermal buoyancy and surface-tension effects, and
(ii) thermal buoyancy and solute buoyancy effects.
For the first set of effects the magnetohydrodynamic problem is also studied.
For the surface-tension problem, attention is focussed on the case where the lower boundary is a rigid conductor and the upper free surface is subject to a general thermal condition. It is found that for this care the surface-tension and buoyancy forces reinforce each other and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. The stream line patterns produced by the two agencies acting separately are again similar.
When the fluid is electrically conducting and is in the presence of a vertical magnetic field, it is found that the field always has a stabilizing effect. When convection cells are formed in the presence of such a field, their horizontal dimensions are less than for cells formed in the absence of the field. The magnetic field accentuates the difference between the cells induced by surface tension and those by buoyancy, and thus reduces the coupling between the destabilizing forces. Increase of magnetic field causes the buoyancy cell pattern to become more symmetrical, but causes the streamlines in surface-tension cells to become bunched near the surface. When the magnetic field is large, the transition from one type of cell to the other type is extremely sudden, at least when the upper surface is a good thermal conductor.
It has been found that, on the model considered, there can be no oscillatory for this problem. However, dimensional analysis reveals that, for a sufficiently flexible upper surface, oscillatory Instability might in fact occur.
Finally the thermohaline problem, where the density varies with both temperature and the concentration of some solute, is studied. The eigenvalue equation is now found for general boundary conditions. The degree of coupling between the thermal and the solute effects again depends on the similarity between convection cells caused by the two agencies acting separately. (For one extreme case studied the coupling is zero for a certain range of parameters.) In this problem both monotonic and oscillatory instability can now occur.