Sedimentation of spheroids in Newtonian fluids with spatially varying viscosity

Abstract

This paper examines the rigid body motion of a spheroid sedimenting in a Newtonian fluid with a spatially varying viscosity field. The fluid is at zero Reynolds number, and the viscosity varies linearly in space in an arbitrary direction with respect to the external force. First, we obtain the correction to the spheroid's rigid body motion in the limit of small viscosity gradients, using a perturbation expansion combined with the reciprocal theorem. Next, we determine the general form of the particle's mobility tensor relating its rigid body motion to an external force and torque. The viscosity gradient does not alter the force/translation and torque/rotation relationships, but introduces new force/rotation and torque/translation couplings that are determined for a wide range of particle aspect ratios. Finally, we discuss results for the spheroid's rotation and centre-of-mass trajectory during sedimentation. A steady orientation arises at long time whose value depends on the viscosity gradient direction and particle shape. These results are significantly different than when no viscosity gradient is present, where the particle stays at its initial orientation for all times. We summarize the observations for prolate and oblate spheroids for different viscosity gradient directions and provide plots for the orientation and centre-of-mass trajectory versus time. We also provide guidelines to extend the analysis when the viscosity gradient exhibits a more complicated spatial behaviour.