ANALYSIS OF A SIMPLIFIED LIQUID-CRYSTAL SHEAR-FLOW MODEL

This paper is concerned with the analysis of the nonlinear two-point boundary value problem phi'' = lambda xcos(2) phi, phi(-1/2) = 0, phi(1/2) = m pi. This system models the preferred direction of orientation of liquid crystal molecules flowing in a channel. It is derived by simplification of a formulation in terms of the Ericksen-Leslie equations. The problem has interesting bifurcation and singular-perturbation phenomena. There are infinitely many distinct solution branches, the number growing roughly quadratically with the parameter lambda, and individual solutions typically possess O(1/root lambda) boundary layers at each endpoint and an O(1/(3) root lambda) interior layer at s = 0. Analytical techniques (involving differential inequalities of Nagumo type) can be used to rigorously prove the existence of the infinite family of stable solutions; while numerical investigations (using the COLSYS/COLNEW and AUTO86 packages) are used to explore the unstable solutions and bifurcation behavior, which involves limit/turning points and simple bifurcation points.