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

Published online by Cambridge University Press:  24 December 2021

Arun Kumar Varanasi ,
Navaneeth K. Marath  and
Ganesh Subramanian Open the ORCID record for Ganesh Subramanian [Opens in a new window]
Arun Kumar Varanasi*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore560064, India
Navaneeth K. Marath*
Affiliation:
Mechanical Engineering, Indian Institute of Technology, Ropar140001, India
Ganesh Subramanian*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore560064, India
*
†Email addresses for correspondence: [email protected], [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected], [email protected]
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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.

JFM classification

Geophysical and Geological Flows: Stratified flows Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A17
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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