The flow field due to a sphere moving in a viscous, density-stratified fluid

Abstract

In this paper, we study the disturbance velocity and density fields induced by a sphere translating vertically in a viscous density-stratified ambient. Specifically, we consider the limit of a vanishingly small Reynolds number $(Re = \rho U a/\mu \ll 1)$, a small but finite viscous Richardson number $(Ri_v = \gamma a^3g/\mu U\ll 1)$ and large Péclet number $(Pe = Ua/D\gg 1)$. Here, $a$ is the sphere's radius, $U$ its translational velocity, $\rho$ an appropriate reference density within the framework of the Boussinesq approximation, $\mu$ the ambient viscosity, $\gamma$ the absolute value of the background density gradient, g is acceleration due to gravity and $D$ the diffusivity of the stratifying agent. For the scenario where buoyancy forces first become comparable to viscous forces at large distances, corresponding to the Stokes-stratification regime defined by $Re \ll Ri_v^{1/3} \ll 1$ for $Pe \gg 1$, important flow features have been identified by Varanasi & Subramanian (J. Fluid Mech., vol. 949, 2022, A29) – these include a vertically oriented reverse jet, and a horizontal axisymmetric wake, on scales larger than the primary (stratification) screening length of ${O}(aRi_v^{-1/3})$. Here, we study the reverse-jet region in more detail, and show that it is only the central portion of a columnar structure with multiple annular cells concentric about the rear stagnation streamline. In the absence of diffusion, corresponding to $Pe = \infty$ $( \beta _\infty = Ri_v^{1/3}Pe^{-1} = 0)$, this columnar structure extends to downstream infinity with the number of annular cells diverging in this limit. We provide expressions for the boundary of the structure, and the number of cells within, as a function of the downstream distance. For small but finite $\beta _\infty$, two length scales emerge in addition to the primary screening length – a secondary screening length of ${O}(aRi_v^{-1/2}Pe^{1/2})$ where diffusion starts to smear out density variations across cells, leading to exponentially decaying density and velocity fields; and a tertiary screening length, $l_t \sim {O}(aRi_v^{-1/2}Pe^{1/2}[\zeta + \frac {13}{4}\ln {\zeta } + ({13^2}/{4^2})({\ln \zeta }/{\zeta })])$ with $\zeta = \frac {1}{2}\ln ({\sqrt {{\rm \pi} }Ri_v^{-1}Pe^3}/{2160})$, beyond which the columnar structure ceases to exist. The latter causes a transition from a vertical to a predominantly horizontal flow, with the downstream disturbance fields reverting from an exponential to an eventual algebraic decay, analogous to that prevalent at large distances upstream.