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

TL;DR

The paper analyzes inertial effects on the interphase drag force and rheology of dilute, buoyant droplets at low but finite Reynolds number using a reciprocal theorem. It derives explicit $O(Re)$ corrections to the drag and to the first and second moments of hydrodynamic force acting on translating droplets, revealing how velocity variance of the dispersed phase enters ensemble-averaged interphase exchange and the continuous-phase stress. The results show that, in dilute emulsions, the effective stress includes terms quadratic in the relative velocity and in velocity fluctuations, yielding non-Newtonian, second-gradient rheology; density ratio effects vanish for the leading force moments at $Re\ll1$, while velocity-variance closures remain as the main open ingredient. The framework connects isolated-droplet solutions with averaged two-phase models, providing a systematic pathway to incorporate inertia-driven force moments into closures and informing potential extensions to higher-$Re$ regimes via Oseen corrections. Overall, the work clarifies how inertia and velocity fluctuations shape interphase momentum exchange and suspension rheology in dilute buoyant droplet suspensions, with practical implications for designing and modeling emulsions and flotation processes.

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, $Re = a U ρ_f / μ_f$, 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.

Figures (1)

• Figure 1: Sketch of the problem. $\textbf{u}_{o/i}[\textbf{x}|\textbf{y},\textbf{w}]$ is the disturbance velocity field evaluated at x, generated by a test droplet positioned at y with centre of mass velocity w. n is the surface normal (outward) vector of the droplet. The exterior domain of the test droplet is denoted $\Omega_o$, and the interior domain $\Omega_i$.