Settling spheroids, and spheroids in linear flow fields, in a fluid with spatial viscosity variations

A generalized reciprocal theorem is used to relate the force and torque induced on an arbitrary particle by an arbitrary viscosity variation in an inertia-less fluid to integrals involving Stokes flow fields and the spatial dependence of viscosity. These expressions are analytically evaluated using spheroidal harmonics and then used to obtain the mobility of the particle during sedimentation, and in a linear flow, of a fluid with linear viscosity stratification. The coupling between the rotational and translational motion induced by stratification causes a particle to fall under gravity along a curved trajectory and rotates the spheroid’s centerline, creating a variety of rotational and translational dynamics. The settling dynamics are independent of the initial orientation but depend on the particle’s aspect ratio and the alignment of gravity with the stratification direction. Spheroids with an aspect ratio between 0.55 and 2.0 exhibit the largest variety of settling behaviors. Interestingly, this range covers most microplastics and typical microorganisms. In a simple shear flow, cross-streamline migration occurs due to the stratification-induced force generated on a rotating particle. Similarly, a particle no longer stays at the stagnation point of a uniaxial extensional flow and its axis does not align with the extensional axis. While fully analytical results are obtained, numerical simulations provide a source of validation. These simulations also provide additional insights into the stratification-induced force- and torque-producing mechanisms through the stratification-induced stress, which is not accessed in the analytical calculations based on the reciprocal theorem.