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, $Re=aU{\rho }_f\mathrm{/}{\mu }_f$, 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.