Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. I: The Algorithm and Benchmarks

Poiseuille flows of nematic polymers, SIAM J. Appl. Math., submitted). The model consists of a Smoluchowski equation for the space-time evolution of the orientational probability distribution function, coupled with a momentum flow balance equation, a constitutive equation for the extra stress, and the continuity equation. The Smoluchowski equation is first reduced to a finite set of partial differential equations in time and space for spherical harmonic amplitudes. Then we discretize the spatial variables (by the method of lines) using high-order finite differences, which reduces the full system to a large set of ordinary differential equations. Adaptive grid generation techniques are implemented. To provide an accurate and stable time integration, we employ the newly developed spectral deferred correction algorithm. We close with an application of the kinetic theory and nu- merical code to explore steady state flow-molecular structures in slow planar Couette experiments (low Deborah number) and low Ericksen number, where distortional elasticity dominates short-range excluded volume effects. We confirm recent analytical results based on mesoscopic closure models derived from this kinetic model, both with equal (M. G. Forest et al., J. Rheol., 48 (2004), pp. 175- 192) and distinct (Z. Cui, M. G. Forest, and Q. Wang, On weak plane Couette and Poiseuille flows of nematic polymers, SIAM J. Appl. Math., submitted) Frank elasticity constants, in the dual limit of low Deborah and low Ericksen numbers.