No abstract

No abstract

SynopsisOne of the confounding issues in laminar flow processing of nematic polymers is the generation of molecular orientational structures on length scales that remain poorly characterized with respect to molecular and processing control parameters. For plane Couette flow within the Leslie-Ericksen continuum model, theoretical results since the 1970s yield two fundamental predictions about the length scales of nematic distortion: a power law scaling behavior, Er Ϫp , 1 4 р p р 1, where Er is the Ericksen number ͑ratio of viscous to elastic stresses͒; the exponent p varies according to whether the structure is a localized boundary layer or an extended structure. Until now, comparable results which incorporate molecular elasticity ͑i.e., distortions in the shape of the orientational distribution͒ have not been derived from mesoscopic Doi-Marrucci-Greco ͑DMG͒ tensor models. In this paper, we derive asymptotic, one-dimensional gap structures, along the flow-gradient direction, in ''slow'' Couette cells, which reflect self-consistent coupling between the primary flow, in-plane director ͑nematic͒ and order parameter ͑molecular͒ elasticity, and confinement conditions ͑plate speeds, gap height, and director anchoring angle͒. We then read off the small Deborah number, viscoelastic structure predictions: The flow is simple shear. The orientation structures consist of: two molecular-elasticity boundary layers with the Marrucci scaling Er Ϫ1/2 , which are amplified by tilted plate anchoring; and a nonuniform, director-dominated structure that spans the entire gap, with Er Ϫ1 average length scale, present for any anchoring angle. We close with direct numerical simulations of the DMG steady, flow-nematic boundary-value problem, first to benchmark the small Deborah number structure formulas, and then to document onset of new flow-orientation structures as the asymptotic expansions become disordered.

Spectral deferred correction is a flexible technique for constructing high-order, stiffly-stable time integrators using a low order method as a base scheme. Here we examine their use in conjunction with splitting methods to solve initial-boundary value problems for partial differential equations. We exploit their close connection with implicit Runge-Kutta methods to prove that up to the full accuracy of the underlying quadrature rule is attainable. We also examine experimentally the stability properties of the methods for various splittings of advection-diffusion and reaction-diffusion equations.

The purpose of this paper is to connect two critical aspects of nanocomposite materials engineering: the knowledge of the orientational distribution of quiescent or flowing anisotropic macromolecules, and homogenization theory of composites with spheroidal inclusions at low volume fractions. The nano‐elements considered herein are derived from the class of high‐aspect‐ratio nematic polymers, either rod‐like or platelet spheroids. By combining the two features, we derive the effective electrical conductivity tensor in closed form. Scaling properties of enhanced conductivity versus volume fraction and weak shear rate become explicit. The most dramatic effect is that the effective conductivity tensor inherits hysteresis, bi‐stability, and discontinuous jumps from the isotropic–nematic first‐order phase transition. These formulas reveal finer estimates that depend on a competition between two inherently extreme parameters in nematic polymer nanocomposites: the molecular aspect ratio and the conductivity ratio of the inclusions and matrix. Herein, we confine our attention to steady monodomain orientational distributions at rest and in weak shear flows, which serve as benchmarks and guides for future extensions and numerical approaches.