The drag of a freely sendimentating sphere in a sheared weakly viscoelastic fluid

Abstract

We study the steady, creeping viscoelastic flow around a freely rotating rigid sphere in a stream with uniform velocity which is subject to simple shear flow far from the sphere. The sphere moves along the vorticity axis of the shear flow. The viscoelasticity of the stream is modelled using the Oldroyd-B and Upper Convected Maxwell constitutive equations. A spherical coordinate system with origin at the centre of the sphere is used to describe the flow field. The solution of the governing equations is expanded as a series for small values of the Deborah number and the resulting sequence of three-dimensional partial differential equations is solved analytically up to second order in the Deborah number. The solution is used to deduce the resulting drag force on the sphere in terms of the dimensionless parameters of the flow. The expression for the drag shows that there is a significant increase of the drag due to the shearing, except in the Newtonian case.