Orientational dynamics of weakly inertial axisymmetric particles in steady viscous flows

J Einarsson,J.R Angilella,B Mehlig

doi:10.1016/j.physd.2014.04.002

Abstract

The orientational dynamics of weakly inertial axisymmetric particles in a steady flow is investigated. We derive an asymptotic equation of motion for the unit axial vector along the particle symmetry axis, valid for small Stokes number St, and for any axisymmetric particle in any steady linear viscous flow. This reduced dynamics is analysed in two ways, both pertain to the case of a simple shear flow. In this case inertia induces a coupling between precession and nutation. This coupling affects the dynamics of the particle, breaks the degeneracy of the Jeffery orbits, and creates two limiting periodic orbits. We calculate the leading-order Floquet exponents of the limiting periodic orbits and show analytically that prolate objects tend to a tumbling orbit, while oblate objects tend to a log-rolling orbit, in agreement with previous analytical and numerical results. Second, we analyse the role of the limiting orbits when rotational noise is present. We formulate the Fokker–Planck equation describing the orientational distribution of an axisymmetric particle, valid for small St and general Péclet number Pe. Numerical solutions of the Fokker–Planck equation, obtained by means of expansion in spherical harmonics, show that stationary orientational distributions are close to the inertia-free case when Pe St≪1, whereas they are determined by inertial effects, though small, when Pe≫1/St≫1.

Highlights

Suspensions of rigid particles are abundant in both nature and technology, and are studied in many disciplines of science
The goal of the present study is to investigate the orientational dynamics of axisymmetric particles with small inertia, to describe and quantify the attracting orbits, and to analyse their effect on the orientational distribution when noise is present
We have derived an approximate equation of motion for the unit axial vector along the particle symmetry axis, assuming that the Stokes number is small

Summary

Introduction

Suspensions of rigid particles are abundant in both nature and technology, and are studied in many disciplines of science. The goal of the present study is to investigate the orientational dynamics of axisymmetric particles with small inertia, to describe and quantify the attracting orbits, and to analyse their effect on the orientational distribution when noise is present To accomplish this goal we first derive an equation of motion for the orientation of an axisymmetric particle in steady flows in the absence of noise, valid for small values of the Stokes number St. We define the Stokes number by St = ms/bμ, where m is the particle mass, s is a typical flow-gradient rate, b is the minor particle length and μ the dynamic viscosity of the fluid (see Sec. 2 for details). The new equation of motion allows us to derive a Fokker-Planck equation determining the combined effects of particle inertia and noise upon the orientational distribution of the particles This equation describes the evolution of an ensemble of weakly inertial particles in a steady flow, subject to random Brownian rotations.

Equation of motion

Simple shear flow: stability analysis

Competition between noise and inertia

Conclusions