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

The Doi-Hess theory coupled with an anisotropic Marrucci-Greco distortional elasticity potential provides a kinetic, mean field description of the coupling between hydrodynamics, molecular orientation by excluded volume, and elastic distortions of flowing nematic liquid crystalline polymers (LCPs). In this paper we provide the first numerical algorithm and implementation of kinetic-scale models for structure formation in confined, planar Couette cells. The model and algorithm extend kinetic simulations of Larson and Ottinger [18] and others [7, 16, 10, 11] for homogeneous nematic polymers in imposed shear flows. The focus here is one-dimensional flow and LCP structures that form between oppositely moving, parallel plates, a classical model problem that has been studied in detail with continuum models [3, 22, 23] and a variety of mesoscopic, second-moment orientation tensor models [26, 20, 25, 12, 2]. We benchmark the kinetic simulations against these coarse-grained predictions with a focus on delineating closure-sensitive structure phenomena. 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 integration of these ordinary differential equations, we employ the newly developed spectral deferred correction algorithm. We close with an application of the kinetic theory and numerical code to explore steady state 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 [12] and distinct [2] Frank elasticity constants, in the low Deborah and low Ericksen limits.