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 center-of-mass trajectory during sedimentation. Depending on the viscosity gradient direction and particle shape, a steady orientation may arise at long times or the particle may tumble continuously. These results are significantly different from the case when no viscosity gradient is present, where the particle stays at its initial orientation for all times. The particle's center of mass trajectory can also be altered depending on the particle's orientation behavior, for example giving rise to diagonal motion or zig-zagging motion. We summarize the observations for prolate and oblate spheroids for different viscosity gradient directions and provide phase plots delineating different dynamical regimes. We also provide guidelines to extend the analysis when the viscosity gradient exhibits a more complicated spatial behavior.