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The Efffect of Non-Uniform Temperature Gradients on Thermal Instabilities in Nematic Liquid Crystals

This paper employs continuum theory to investigate the onset of convective instability in a thin sample of nematic liquid crystal contained between two large horizontal flat plates. Whereas earlier investigations assume an initial uniform temperature gradient across the sample, we present an initial investigation to consider the effect of a basic non-uniform temperature gradient upon the critical threshold. To this end, a variational method of solution is developed and results are obtained for two experimental arrangements of particular interest when the initial temperature profile is piecewise linear. In addition, we derive analytic expressions which yield fairly accurate results for the critical thresholds. Introduction The onset of stationary convective instabilities in thin layers of nematic liquid crystal confined between horizontal flat plates has been the subject of several investigations [1—4]. These analyses examine two simple experimental arrangements of particular interest. In one the anisotropic axis of the nematic material is initially aligned everywhere parallel to the plates (planar configuration) and in the other it is uniformly aligned perpendicular to the plates (homeotropic configuration). Of these investigations only that by Barratt and Sloan [3] attempts to find an analytic solution capable of yielding accurate quantitative results for all nematic materials, without the necessity of integrating the governing differential equations numerically. They obtain the solution in the form of a consistency condition involving an infinite series and demonstrate that a simple computation, using only the first term in the series, yields a result that is fairly close to the accurate solution. In the above analyses it is assumed that the temperature gradient is uniform across the layer. Of course this assumption may not always be valid and the effect of an initial non-uniform temperature gradient upon the onset of convective 172 P.J· Barrattetal. instability could be of some importance. An example of how such a non-uniform profile could arise is illustrated by the transient heating or cooling of one of the boundaries. There is a considerable literature on the effects of initial non-uniform temperature gradients in layers of Newtonian fluids and it is found that instability may occur at markedly lower thresholds than those predicted by analyses of the classical Benard problem. Rather than considering the basic temperature profile to depend explicitly on time, it is common (cp. [5-8]) to introduce the simplification that it is 'frozen' at some instant in time. This quasi-static approximation appears reasonable provided any disturbances grow significantly faster than the rate at which the basic temperature profile evolves. Although there are arguments against the use of 'frozen' initial profiles (see for example reference [9]), Nield [6] indicates that the use of such profiles is justifiable provided one is only interested in obtaining the critical threshold and not the growth rate of disturbances beyond this threshold. 1.