Theory of Rayleigh-Bénard convection in planar nematic liquid crystals

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Abstract
A rigorous three-dimensional linear and weakly nonlinear analysis of Rayleigh-Bénard convection in planarly aligned nematic liquid crystals is presented. We use realistic boundary conditions and allow for a stabilizing magnetic field. The analysis includes a determination of all parameters of the amplitude equation in different regimes of the magnetic field where normal, oblique, and parallel rolls occur. It turns out that there exists a range at intermediate fields were the primary transition is subcritical for the standard material (4-methoxybenzylidene-4’-n-butylaniline). The (‘‘nonlinear’’) stability of rolls is determined to second order in the amplitude so that nonpotential effects (especially mean flow) are included. In the low-field supercritical range the roll solutions are destabilized at a secondary threshold via a skewed-varicose instability leading presumably to spatiotemporal chaos. In the high-field supercritical normal-roll range a secondary transition can lead to oblique rolls that are ultimately destabilized by a short-wavelength instability.