Effective viscosity of a two dimensional passive suspension in a liquid crystal solvent

TL;DR

This paper derives an explicit formula for the effective shear viscosity of a dilute two-dimensional suspension of particles in a liquid crystal solvent, showing how it depends on particle concentration and material parameters.

Contribution

It provides the first asymptotic analysis of the effective viscosity in a 2D passive suspension within a liquid crystal solvent, including explicit formulas and parameter dependence.

Findings

Effective viscosity increases linearly with particle area fraction.

Derived explicit formulas for viscosity dependence on material parameters.

Identified conditions where increasing shear-flow alignment reduces viscosity.

Abstract

Suspension of particles in a fluid solvent are ubiquitous in nature, for example, water mixed with sugar or bacteria self-propelling through mucus. Particles create local flow perturbations that can modify drastically the effective (homogenized) bulk properties of the fluid. Understanding the link between the properties of particles and the fluid solvent, and the effective properties of the medium is a classical problem in fluid mechanics. Here we study a special case of a two dimensional model of a suspension of undeformable particles in a liquid crystal solvent. In the dilute regime, we calculate asymptotic solutions of the perturbations of the velocity and director fields and derive an explicit formula for an effective shear viscosity of the liquid crystal medium. Such effective shear viscosity increases linearly with the area fraction of particles, similar to Einstein formula but with a different prefactor. We provide explicit asymptotic formulas for the dependence of this prefactor on the material parameters of the solvent. Finally, we identify a case of decreasing the effective viscosity by increasing the magnitude of the shear-flow alignment coefficient of the liquid crystal solvent.