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

TL;DR

This paper investigates how inertial effects influence the drag force and rheology of dilute suspensions of buoyant droplets at low Reynolds numbers, revealing quadratic dependencies on velocity and its variance.

Contribution

It provides a detailed analysis of inertial corrections to force moments and their impact on the effective stress in dilute droplet suspensions at low Reynolds numbers.

Findings

Inertial corrections scale as O(ρ_f φ U^2) and O(a ρ_f φ U^2).

Drag force and force moments depend on velocity variance.

Effective stress depends quadratically on relative velocity and its variance.

Abstract

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, $Re = a U \rho_f / \mu_f$, and the dilute limit of small droplet volume fraction $\phi$. Here, $U$ denotes the magnitude of the relative velocity between the phases, $a$ the droplet radius, and $\rho_f$ and $\mu_f$ 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 $O(\rho_f \phi U^2)$ and $O(a\rho_f\phi U^2)$, 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.