Lamb’s solution and the stress moments for a sphere in Stokes flow

Abstract

Abstract This paper describes a method to generate expressions for the moments of the fluid stress integrated over a sphere in a general Stokes flow based on Lamb’s general solution of the Stokes equations. Explicit results up to moments of order four are given together with a general recurrence relation for moments of higher order. The connection of the results to earlier expressions for the stress moments leading, in particular, to generalized Faxen-type relations is demonstrated and the connection with representations of the rotation group is addressed. This study is motivated by the central role played by the Lamb coefficient in the physalis method for the resolved simulation of particulate flows. An example of the application of this method to the shear flow of a suspension of 3200 equal spheres at finite Reynolds number is given. In addition, the present theory is applied to the analysis of the particle contribution to the mixture stress in particulate flows at both finite and vanishing Reynolds numbers. It is shown that, in the case of spatial non-uniformity, the mixture stress acquires new contributions, including a new viscosity parameter, which are explicitly calculated for the case of a dilute suspension with negligible inertia.