Asymptotic of the velocity of a dilute suspension of droplets with interfacial tension

Abstract

In this paper we derive the asymptotic expansion of the velocity field of a small deformable droplet immersed in an incompressible Newtonian fluid. Using an appropriate physical scaling of the surface tension with respect to the droplet volume, we show that the first order of the asymptotic can be expressed in terms of the velocity field in the absence of the droplet and a new kind of moment tensor, called the curvature moment tensor. Our asymptotic formula extends those already derived for rigid droplets and aimed to obtain simplified macroscale properties of a dilute suspension composed of identical droplets dispersed in an incompressible Newtonian fluid from knowledge of its microscopic properties. We finally determine explicitly the curvature moment tensor for ellipses and ellipsoids.