  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.
