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$\kappa$, sedimenting in a linearly stratified fluid. The analysis demarcates regions in parameter space corresponding to broadside-on and edgewise settling in the limit$Re, Ri_v \ll 1$, where$Re = \rho _0UL/\mu$and$Ri_v =\gamma L^3\,g/\mu U$, the Reynolds and viscous Richardson numbers, respectively, are dimensionless measures of the importance of inertial and buoyancy forces relative to viscous ones. Here,$L$is the spheroid semi-major axis,$U$an appropriate settling velocity scale,$\mu$the fluid viscosity and$\gamma \ (>0)$the (constant) density gradient characterizing the stably stratified ambient, with the fluid density$\rho_0$taken to be a constant within the Boussinesq framework. A reciprocal theorem formulation identifies three contributions to the angular velocity: (1) an$O(Re)$inertial contribution that already exists in a homogeneous ambient, and orients the spheroid broadside-on; (2) an$O(Ri_v)$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$Pe$;$Pe = UL/D$being the Péclet number with$D$the diffusivity of the stratifying agent. For$Pe \ll 1$, this contribution is$O(Ri_v)$and orients prolate spheroids edgewise for all$\kappa \ (>1)$. For oblate spheroids, it changes sign across a critical aspect ratio$\kappa _c \approx 0.41$, orienting oblate spheroids with$\kappa _c < \kappa < 1$edgewise and those with$\kappa < \kappa _c$broadside-on. For$Pe \ll 1$, the hydrodynamic component is always smaller in magnitude than the hydrostatic one, so a sedimenting spheroid in this limit always orients broadside-on. For$Pe \gg 1$, the hydrodynamic contribution is dominant, being$O(Ri_v^{{2}/{3}}$) in the Stokes stratification regime characterized by$Re \ll Ri_v^{{1}/{3}}$, and orients the spheroid edgewise regardless of$\kappa$. Consideration of the inertial and large-$Pe$stratification-induced angular velocities leads to two critical curves which separate the broadside-on and edgewise settling regimes in the$Ri_v/Re^{{3}/{2}}$–$\kappa$plane, with the region between the curves corresponding to stable intermediate equilibrium orientations. The predictions for large$Pe$are broadly consistent with observations.