Inertial drag on a sphere settling in a stratified fluid
R. Mehaddi, F. Candelier, B. Mehlig
TL;DR
The study addresses the inertial drag on a sphere settling in a linearly stratified fluid under small ${\rm Re}$ and ${\rm Ri}$. It develops a uniformly valid perturbation theory via asymptotic matching that introduces the scales $\ell_o = a/{\rm Re}$ and $\ell_s = (\nu\kappa/N^2)^{1/4}$ with $\epsilon = a/\ell_s$, yielding a compact drag correction $f_3 = -6\pi u_3 (1+\epsilon M_{33})$, where $M_{33}$ is given by a two-dimensional integral (Eq. for $M_{33}$). The analysis reveals three distinct regimes—diffusion-dominated, advection-dominated, and inertia-dominated—characterized by explicit scalings $(0.6621\epsilon)$, $(1.060\, {\rm Ri}^{1/3})$, and $(3/8){\rm Re}$, with a crossover near ${\rm Fr} \sim 1/{\rm Re}$. Comparisons with recent DNS at small ${\rm Re}$ show good agreement in the diffusion and advection regimes and highlight finite-size and unsteady effects as potential sources of discrepancy at larger ${\rm Fr}$. The framework provides a unified, quantitative tool for predicting inertial drag in stratified media, with implications for particulate transport and microorganism motility in oceans and lakes.
Abstract
We compute the drag force on a sphere settling slowly in a quiescent, linearly stratified fluid. Stratification can significantly enhance the drag experienced by the settling particle. The magnitude of this effect depends on whether fluid-density transport around the settling particle is due to diffusion, to advection by the disturbance flow caused by the particle, or due to both. It therefore matters how efficiently the fluid disturbance is convected away from the particle by fluid-inertial terms. When these terms dominate, the Oseen drag force must be recovered. We compute by perturbation theory how the Oseen drag is modified by diffusion and stratification. Our results are in good agreement with recent direct-numerical simulation studies of the problem at small Reynolds numbers and large (but not too large) Froude numbers.