Extrapolation theory for Stokes flow past a deformed sphere

TL;DR

This paper develops an extrapolation method for calculating Stokes flow around highly deformed spheres, enabling precise computation of forces and torques for complex shapes using basis functions and matrix algebra.

Contribution

It introduces an explicit extrapolation operator based on spherical harmonics for Stokes flow around deformed spheres, with a numerical framework for arbitrary deformation amplitudes.

Findings

Explicit formulas for flow, force, and torque corrections to first order in deformation.

A matrix algebra approach for numerical implementation of the extrapolation operator.

Application to rotating and translating spheres with arbitrary surface velocities.

Abstract

We formulate a method for computing Stokes flow past a highly deformed sphere with arbitrarily defined surface velocity. The fundamental ingredient is an explicit extrapolation operator extending a velocity field from the surface of a sphere, which is expressed in terms of a complete set of basis Stokes fields for the pressure and velocity derived from scalar and vector spherical harmonics. We present a matrix algebra packaging suitable for numerical computation to arbitrary order in the deformation amplitude (deviation from sphericity). The hydrodynamic force and torque on a deformed sphere with arbitrary surface velocity are expressed in terms of basis field amplitudes, and for the classic problem of a rotating and translating rigid body, we compute explicitly the first order in deformation corrections to the flow field as well as the hydrodynamic force and torque.