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, italic Re equals a upper U rho Subscript f Baseline divided by mu Subscript f Baseline Re = a U ρ f / μ f $\textit{Re} = a U \rho _{\!f} / \mu _{\!f}$ , and the dilute limit of small droplet volume fraction phi ϕ $\phi$ . Here, upper U U $U$ denotes the magnitude of the relative velocity between the phases, a a $a$ the droplet radius and rho Subscript f ρ f $\rho _{\!f}$ and mu Subscript f μ f $\mu _{\!f}$ the density and viscosity of the continuous phase, respectively. We show that the upper O left parenthesis italic Re right parenthesis O ( Re ) $O(\textit{Re})$ inertial corrections to the first and second moments of force scale as upper O left