Convective instabilities in cholesteric and smectic-Aliquid crystals

The convective instabilities occurring in cholesteric and smectic-$A$ liquid crystals are investigated by linear stability analysis. We find oscillatory instabilities not predicted previously and we present a thorough examination of the stationary instabilities which have been discussed previously in a simplified version by Dubois-Violette and Parsons. The differences in the convective instabilities in cholesterics and smectics $A$ due to the differences in the underlying hydrodynamic equations and to the different orders of magnitude of the corresponding transport coefficients and static susceptibilities are considered in some detail. We discuss the mechanisms of the instabilities which are different from those of nematics and we obtain the threshold conditions for cholesterics and smectics $A$ for the critical temperature difference and for the frequency by energy-balance considerations. The influence of a temperature-dependent pitch is investigated in a separate section.