A Smoluchowski equation for a sheared suspension of frictionally interacting rods

TL;DR

This paper develops a new theoretical framework for dense sheared suspensions of frictional rods, deriving constitutive equations and a Smoluchowski equation that extend Doi's model to include frictional interactions.

Contribution

It introduces a novel mean field approach to incorporate frictional forces into the Smoluchowski equation for rod suspensions, generalizing existing models for liquid crystals.

Findings

Derived dissipation functions for frictional contacts.

Formulated a generalized contact number equation.

Extended Doi's model to include friction effects.

Abstract

In this work we develop constitutive equations for a dense, sheared suspension of frictionally interacting rods by applying Onsager's variational method as formulated by Doi. We treat both solid friction, of the Amontons-Coulomb form; and lubricated friction, which scales with relative tangential velocity at the contact point. Dissipation functions in terms of the rod angular velocity are derived via a mean field approach for each form of friction, and from these, a Rayleighian for dense suspensions of rigid rods under shear constructed. Derivatives of this Rayleighian with respect to rod angular velocity and velocity gradient give a Smoluchowski equation and stress tensor, respectively. We show that these are representable as perturbations to Doi's model for a sheared liquid crystal. We also suggest a form for the average number of contacts between rods as a function of volume fraction, aspect ratio, and nematic order parameter, generalizing Philipse's random contact equation for disordered packings.