Dilute emulsions with surface tension

We consider an emulsion formed by two newtonian fluids in which one fluid is dispersed under the form of droplets of arbitrary shape in the presence of surface tension. We consider both cases of droplets with fixed centers of mass and of convected droplets. In the non-dilute case, for spherical droplets of radius $a_\epsilon$ of the same order as the period length $\epsilon$, the two models were studied by Lipton-Avellaneda (1990) and Lipton-Vernescu (1994). Here we are interested in the time-dependent, dilute case when the characteristic size of the droplets $a_\epsilon$, of arbitrary shape, is much smaller than $\epsilon$. We study the limit behavior when $\epsilon \to 0$ in each of these two models. We establish a Brinkman type law for the critical size $a_\epsilon = O(\epsilon ^3)$ in the first case, whereas in the second case there is no “strange” term, and in the limit the flow is unperturbed by the droplets.