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The rotation of a sedimenting anisotropic particle in a linearly stratified ambient

We derive the torque on a spheroid of an arbitrary aspect ratio $κ$ sedimenting in a linearly stratified ambient. The analysis demarcates regions in parameter space corresponding to broadside-on and edgewise (longside-on) settling in the limit $Re, Ri_v \ll 1$, where $Re = ρ_0UL/μ$ and $Ri_v =γL^3g/μU$, 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, $μ$ the fluid viscosity, and $γ\,(>0)$ the (constant)\,density gradient characterizing the stably stratified ambient, with $ρ_0$ being the fluid density taken to be a constant within the Boussinesq framework. A reciprocal theorem formulation identifies three contributions to the torque: (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 linear stratification that also orients the spheroid broadside-on; and (3) a hydrodynamic contribution arising from the perturbation of the ambient stratification by the spheroid whose nature depends on $Pe$; $Pe = UL/D$ being the Peclet number with $D$ the diffusivity of the stratifying agent. For $Pe \gg 1$, the hydrodynamic contribution is $O(Ri_v^{\frac{2}{3}}$) in the Stokes stratification regime characterized by $Re \ll Ri_v^{\frac{1}{3}}$, and orients the spheroid edgewise regardless of $κ$. The differing orientation dependencies of the inertial and large-$Pe$ hydrodynamic stratification torques imply that the broadside-on and edgewise settling regimes are separated by two distinct $κ$-dependent critical curves in the $Ri_v/Re^{\frac{3}{2}}-κ$ plane. The predictions are consistent with recent experimental observations.