Thermal instability stationary convection

Let us first recall briefly the classical Benard-Rayleigh problem of thermal convection in an isotropic liquid. When a horizontal layer of isotropic liquid bounded between two plane parallel plates spaced d apart is heated from below, a steady convective flow is observed when the temperature difference between the plates exceeds a critical value A 7. The flow has a stationary cellular character with a spatial periodicity of about 2d. The mechanism for the onset of convection may be looked upon as follows. A fluctuation T in temperature creates warmer and cooler regions, and due lO buoyancy effects the former tends to move upwards and the latter downwards. When AT AT, the fluctuation dies out in time because of viscous effects and heat loss due to conductivity. At the threshold the energy loss is balanced exactly and beyond it instability develops. Assuming a one-dimensional model in which T and the velocity (normal to the layer) vary as exp (i j,y) with x ji/rf, the threshold is given by the dimensionless Rayleigh number 

The situation is altered profoundly in the case of a nematic because of its anisotropic transport properties. Dubois-Violette was the first to give an approximate theoretical treatment of thermal convection in a planar (homogeneously aligned) nematic and to show by consideration of torques that such a system will be unstable against cellular flow when the film is heated from below if 0, or when it is heated from above if 0, where = K —K is the anisotropy of thermal conductivity (which is positive for all known nematics )- Dubois-Violette also showed that the critical temperature gradient fi (=ATJd) should be much less than that 

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The studies outlined here are the most important ones that established the fundamental of principles of thermal instability in nematics. A number of theoretical and experimental investigations on these and other geometries have since been reported. A particularly interesting study is that of Lekkerkerker who predicted that a homeotropic nematic heated from below (which, it will be recalled, is stable against stationary convection) should become unstable with respect to oscillatory convection. The phenomenon was demonstrated experimentally by Guyon et... 

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