The rotation of a sedimenting spheroidal particle in a linearly stratified fluid

Abstract

We derive analytically the angular velocity of a spheroid, of an arbitrary aspect ratio, sedimenting in a linearly stratified fluid. The analysis demarcates regions in parameter space corresponding to broadside-on and edgewise settling in the limit, where and, the Reynolds and viscous Richardson numbers, respectively, are dimensionless measures of the importance of inertial and buoyancy forces relative to viscous ones. Here, is the spheroid semi-major axis, an appropriate settling velocity scale, the fluid viscosity and 0)]]> the (constant) density gradient characterizing the stably stratified ambient, with the fluid density taken to be a constant within the Boussinesq framework. A reciprocal theorem formulation identifies three contributions to the angular velocity: (1) an inertial contribution that already exists in a homogeneous ambient, and orients the spheroid broadside-on; (2) an hydrostatic contribution due to the ambient stratification that also orients the spheroid broadside-on; and (3) a hydrodynamic contribution arising from the perturbation of the ambient stratification whose nature depends on; being the Péclet number with the diffusivity of the stratifying agent. For, this contribution is and orients prolate spheroids edgewise for all 1)]]>. For oblate spheroids, it changes sign across a critical aspect ratio, orienting oblate spheroids with <![CDATA[\kappa _c < \kappa edgewise and those with <![CDATA[\kappa broadside-on. For, the hydrodynamic component is always smaller in magnitude than the hydrostatic one, so a sedimenting spheroid in this limit always orients broadside-on. For, the hydrodynamic contribution is dominant, being) in the Stokes stratification regime characterized by, and orients the spheroid edgewise regardless of. Consideration of the inertial and large- stratification-induced angular velocities leads to two critical curves which separate the broadside-on and edgewise settling regimes in the - plane, with the region between the curves corresponding to stable intermediate equilibrium orientations. The predictions for large are broadly consistent with observations