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

We derive the torque on a spheroid of an arbitrary aspect ratio $\kappa$ sedimenting in linearly stratified ambient. The analysis demarcates regions parameter space corresponding to broadside-on and edgewise (longside-on) settling limit $Re, Ri_v \ll 1$, where $Re = \rho_0UL/\mu$ $Ri_v =\gamma L^3g/\mu U$, Reynolds viscous Richardson numbers, respectively, are dimensionless measures importance inertial buoyancy forces relative ones. Here, $L$ is semi-major axis, $U$ appropriate velocity scale, $\mu$ fluid viscosity, $\gamma\,(>0)$ (constant)\,density gradient characterizing stably ambient, with $\rho_0$ being density taken be constant within Boussinesq framework. A reciprocal theorem formulation identifies three contributions torque: (1) $O(Re)$ contribution that already exists homogeneous orients broadside-on; (2) $O(Ri_v)$ hydrostatic due ambient linear stratification also (3) hydrodynamic arising from perturbation by whose nature depends $Pe$; $Pe UL/D$ Peclet number $D$ diffusivity stratifying agent. For \gg $O(Ri_v^{\frac{2}{3}}$) Stokes regime characterized Ri_v^{\frac{1}{3}}$, regardless $\kappa$. differing orientation dependencies large-$Pe$ torques imply regimes separated two distinct $\kappa$-dependent critical curves $Ri_v/Re^{\frac{3}{2}}-\kappa$ plane. predictions consistent recent experimental observations.