Poiseuille flow and apparent viscosity of nematic liquid-crystals

In this work, we study the stationary Poiseuille flow of nematic liquidcrystals through a tube under a constant gradient pressure. We first examine the dynamics of steady-state system of Poiseuille flow with strong anchoring boundary conditions. For zero gradient pressure (equilibrium situation), the steady-state system can be converted essentially to a Hamiltonian system but it is not structurally stable in its natural form. Interestingly, by revealing a special structure of the problem, the system can be converted to one with normal hyperbolicity so that the limiting dynamics is actually structurally stable. In the second part, based on the structures of the system established in the first part, we provide an approximation for the so-called apparent viscosity η. As discovered by Atkin [Arch. Ration. Mech. Anal. 38 (1970), pp. 224-240] and verified experimentally by Fisher and Fredrickson [Mol. Cryst. Liq. Cryst. 8 (1969), pp. 267-284], the apparent viscosity η is in fact a function of the ratio between the efflux and the radius of the tube. But the function is unknown, and it has been approximated numerically for a couple of cases. The approximation procedure provided in this paper allows one to obtain the Taylor expansion of the function to any order and the coefficients in the expansion can be expressed as integrals involving structural functions and physical parameters of the problem only.