Singularities and nonuniqueness in cylindrical flow of nematic liquid crystals

Abstract

The subject of this paper is the behavior of the director field of a nematic liquid crystal in flow through a tube with circular cross-section. Both the flow and the director field are assumed to have cylindrical symmetry. The requirement of finite Frank-Oseen energy forces ``admissible" director fields to be axially directed at the location of the symmetry axis. Thus, the angle between the axis and the director field at the location of the axis amounts to $k\pi$, $k$ being an integer. In the steady case, it is shown that $k$ is largely (but not uniquely) determined by the (Dirichlet) boundary conditions. In particular, this may give rise to line singularities with finite energy density. Moreover, the associated Dirichlet problem may have several distinct solutions with identical value of $k$. Analogously to the heat flow of harmonic mappings, finite-time blow-up phenomena for the associated parabolic problem are established.