2 results
Motion of a sphere in a viscous density stratified fluid
- Arun Kumar Varanasi, Ganesh Subramanian
-
- Journal:
- Journal of Fluid Mechanics / Volume 949 / 25 October 2022
- Published online by Cambridge University Press:
- 29 September 2022, A29
-
- Article
- Export citation
-
We examine the translation of a sphere in a stratified ambient in the limit of small Reynolds numbers ($Re \ll 1$) and viscous Richardson numbers ($Ri_v \ll 1$); here, $Re = {\rho Ua}/{\mu }$ and $Ri_v = {\gamma a^3 g}/{\mu U}$, with $a$ being the sphere radius, $U$ the translation speed, $\rho$ and $\mu$ the density and viscosity of the stratified ambient, $g$ the acceleration due to gravity, and $\gamma$ the density gradient characterizing the ambient stratification. In contrast to most earlier efforts, our study considers the convection-dominant limit corresponding to $Pe = {Ua}/{D} \gg 1$, $D$ being the diffusivity of the stratifying agent. We characterize in detail the velocity and density fields around the particle in what we term the Stokes stratification regime, defined by $Re \ll Ri_v^{{1}/{3}} \ll 1$, and corresponding to the dominance of buoyancy over inertial forces. Buoyancy forces associated with the perturbed stratification fundamentally alter the viscously dominated fluid motion at large distances of order the stratification screening length that scales as $a\,Ri_v^{-{1}/{3}}$. The motion at these distances transforms from the familiar fore–aft symmetric Stokesian form to a fore–aft asymmetric pattern of recirculating cells with primarily horizontal motion within, except in the vicinity of the rear stagnation streamline. At larger distances, the motion is vanishingly small except within (a) an axisymmetric horizontal wake whose vertical extent grows as $O(r_t^{{2}/{5}})$, $r_t$ being the distance in the plane perpendicular to translation, and (b) a buoyant reverse jet behind the particle that narrows as the inverse square root of distance downstream. As a result, for $Pe = \infty$, the motion close to the rear stagnation streamline starts off pointing in the direction of translation, in the inner region, and decaying as the inverse of the downstream distance; the motion reverses beyond distance $1.15a\,Ri_v^{-{1}/{3}}$, with the eventual reverse flow in the far-field buoyant jet again decaying as the inverse of the distance downstream. For large but finite $Pe$, the narrowing jet is smeared out beyond a distance of $O(a\,Ri_v^{-{1}/{2}}\, Pe^{{1}/{2}})$, leading to an exponential decay of the aforementioned reverse flow.
The rotation of a sedimenting spheroidal particle in a linearly stratified fluid
- Arun Kumar Varanasi, Navaneeth K. Marath, Ganesh Subramanian
-
- Journal:
- Journal of Fluid Mechanics / Volume 933 / 25 February 2022
- Published online by Cambridge University Press:
- 24 December 2021, A17
-
- Article
- Export citation
-
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.