Simple shear flow round a rigid sphere: inertial effects and suspension rheology

TLDR

The study analyzes the flow field around a neutrally buoyant rigid sphere in a compressible Newtonian fluid subjected to simple shear at large distances. It aims to use the velocity and pressure fields to quantify inertial effects on suspension rheology. By enforcing stress continuity, zero net torque, and zero net force on the sphere, the authors compute the velocity and pressure fields to \(O(R^{3/2})\) (\(R=a^{2}G/\nu\)). The computed streamlines show loss of fore‑aft symmetry, and the results provide an \(O(R^{3/2})\) correction to Einstein’s viscosity law for dilute suspensions and reveal a non‑isotropic normal stress due to inertia.

Abstract

An analysis is presented of the flow field near a neutrally-buoyant rigid spherical particle immersed in an in compressible Newtonian fluid which, at large distances from the particle, is undergoing simple shear flow. Subject to conditions of continuity of stress at the particle surface and to conditions of zero net torque and zero net force on the sphere, the effect of fluid inertia on the velocity and pressure fields in the vicinity of the particle has been computed to $O(R^{\frac{3}{2}})$ , where R = a 2 G /ν is a shear Reynolds number, a being the sphere radius, G the velocity gradient in the free stream (taken to be a positive number), and ν the kinematic viscosity. Some streamlines have been computed and plotted. These illustrate how the fore–aft symmetry of the creeping-motion solution is destroyed when one includes inertial effects. Knowledge of the velocity and pressure fields enables one to compute the effect of inertial forces in suspension rheology. The results include a correction to the Einstein viscosity law to $O(R^{\frac{3}{2}})$ for a dilute (non-interacting) suspension of spheres. In addition it is found that inertial effects give rise to a non-isotropic normal stress.