Effect of weak fluid inertia upon Jeffery orbits

TL;DR

This paper investigates how weak fluid inertia influences the rotation of axisymmetric particles in shear flow, revealing stability characteristics of Jeffery orbits and resolving previous numerical simulation puzzles.

Contribution

It provides a perturbative analysis of inertial effects on Jeffery orbits for spheroidal particles at small Reynolds numbers, including stability analysis.

Findings

Inertial effects lift Jeffery orbit degeneracy.

Log-rolling orbit is unstable for prolate spheroids.

Both unsteady and nonlinear Navier-Stokes terms are significant.

Abstract

We consider the rotation of small neutrally buoyant axisymmetric particles in a viscous steady shear flow. When inertial effects are negligible the problem exhibits infinitely many periodic solutions, the "Jeffery orbits". We compute how inertial effects lift their degeneracy by perturbatively solving the coupled particle-flow equations. We obtain an equation of motion valid at small shear Reynolds numbers, for spheroidal particles with arbitrary aspect ratios. We analyse how the linear stability of the \lq log-rolling\rq{} orbit depends on particle shape and find it to be unstable for prolate spheroids. This resolves a puzzle in the interpretation of direct numerical simulations of the problem. In general both unsteady and non-linear terms in the Navier-Stokes equations are important.