Elongational perturbations on nematic liquid crystal polymers under a weak shear

Abstract

The two-dimensional Smoluchowski equation is employed to study the effect of elongational perturbations on nematic liquid crystal polymers under a weak shear. We use the multiscale asymptotic analysis to show that (1) when the elongational perturbation is small relative to the weak shear, the orientational probability density function (pdf) tumbles periodically only in an intermediate range of polymer concentration; outside this intermediate range (i.e., for very small and very large polymer concentration) the orientational pdf converges to a steady state and there is no tumbling. (2) When the elongational perturbation is about 20% of the shear rate or larger, the intermediate range of tumbling disappears and the orientational pdf always converges to a steady state regardless of the polymer concentration. Our theoretical predictions are consistent with various earlier results based on the Leslie–Ericksen theory [C. V. Chaubal and L. G. Leal, J. Non-Newtonian Fluid Mech. 82, 22 (1999)] or analogous 3D numerical simulations [M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett. 93, 088301 (2004); M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newtonian Fluid Mech. 118, 17 (2004)].