Inertial effects on the interphase drag force and rheology of dilute suspensions of buoyant droplets at low Reynolds number

In this work, we compute the hydrodynamic force and the first and second moments of force acting on a translating spherical droplet immersed in a uniform flow using the reciprocal theorem. We consider the low but finite Reynolds number regime, $Re = a U ρ_f / μ_f$, and the dilute limit of small droplet volume fraction $ϕ$. Here, $U$ denotes the magnitude of the relative velocity between the phases, $a$ the droplet radius, and $ρ_f$ and $μ_f$ the density and viscosity of the continuous phase, respectively. We show that the $O(Re)$ inertial corrections to the first and second moments of force scale as $O(ρ_f ϕU^2)$ and $O(aρ_fϕU^2)$, respectively. Moreover, the ensemble average of the drag force and the higher-order force moments over the distribution of droplet velocities introduces additional contributions proportional to the velocity variance of the dispersed phase, both in the interphase momentum exchange and in the effective stress of the continuous phase. As a consequence, in dilute emulsions of buoyant droplets, the effective stress depends quadratically on the relative velocity between the phases, on the velocity variance of the dispersed phase, and on the spatial gradients of these quantities.

💡 Research Summary

This paper presents a rigorous theoretical investigation of the hydrodynamic forces acting on a translating spherical droplet (or buoyant bubble) immersed in a uniform flow at low but finite Reynolds number, with particular emphasis on dilute suspensions where the droplet volume fraction ϕ ≪ 1. The authors adopt a two‑fluid framework and begin by writing the ensemble‑averaged mass and momentum equations for the continuous (fluid) phase and the dispersed (droplet) phase. In these equations the interphase momentum exchange term F (the drag force) and higher‑order force moments appear as unresolved closures that must be expressed in terms of macroscopic variables.

To obtain these closures, the paper introduces a conditional‑averaging procedure. The probability density P