The Rayleigh problem for nematic liquid crystals.: Similarity solutions

The exact solution of the full Navier-Stokes equations for the problem of an infinite plate on z = 0 in a 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 complicated situation in which the plate has been prepared so that the directors are aligned or perpendicular to it and the directors for z > 0 have an initial alignment given by 0(0) (the tilt angle of the directors out of the plate) with 0 less than or equal to theta (0) < <pi>/2, assuming the same uniform twist initial alignment phi (z, 0) = phi (0) as the preferred alignment for phi on the surface plate, then the velocity field does not satisfy a diffusion equation. Here the situation when the plate is given an impulsive motion of the form nu (x) = u(0)/t(1/2) and nu (y) = nu (0)/t(1/2) for t > 0 is considered. When the approximation of neglecting fluid inertia is made, the equation for the variation of the tilt angle of the directors in time and position becomes a non linear diffusion equation with the diffusion coefficient depending on theta. The twist angle, phi, of the directors also satisfies a diffusion equation with a diffusion coefficient varying in space and time, this being determined by the behaviour of the tilt angle. However, the inertia is needed to satisfy the initial static condition of the nematic and it is included by means of a singular perturbation method. An approximate solution of the problem when the initial alignment for phi is different from the preferred direction for phi on the surface, that is, when phi (z,0) = phi (0) is different from phi (0,t) = phi (1) is also given. The effect of the director inertia is neglected throughout this paper, in common with approximations usually made in the nematic literature. (C) 2001 Elsevier Science Ltd. All rights reserved.