The Rayleigh problem for nematic liquid crystals. Short time solution

Abstract

The exact solution of the full Navier–Stokes equations for the problem of an infinite plate in Newtonian fluid given an impulsive motion in its plane (the Rayleigh problem) is extended here to the situation, where the fluid is that of nematic liquid crystal. If the initial director direction is parallel to the plate in the x-direction with the surface prepared in order to maintain that initial alignment and the plate is given an impulsive motion in the y-direction, then the classical solution goes through with a constant viscosity. However, in the more realistic situation in which the plate has been prepared so directors are parallel or perpendicular to it and the initial director direction is parallel or perpendicular to the plate, then the velocity fields do not satisfy a diffusion equation. To solve these more complicated situations, we first investigate the consequences of this imposed plate motion for very short times by means of the assumption that the variable coefficients in the full balance and constitutive equations of the nematic liquid crystal are replaced by constant values they take originally. In principle this problem can be solved with fairly general impulsive velocity conditions imposed on the plate. Particular attention is however given to the simplest case, where the plate is given a specified velocity which decays with time. Measurements of the subsequent velocity profiles can be used to infer properties of the nematic. In the analysis given here, specific attention is given to the effect of director inertia.