Liquid fraction profile in a liquid foam under an applied voltage

A liquid foam, a dense assembly of gas bubbles in a surfactant solution, is a deformable porous material. As classically observed in divided systems, electrokinetic transport can be induced, for example, when an electric field is applied from either side of the foam sample. We determine here the liquid fraction profile obtained when a foam is submitted to an electro-osmotic flow, when the surfactant induces so-called rigid or mobile interfaces. We show that the main governing equation for the liquid fraction repartition in space and time is diffusivelike and similar to the one describing pressure-induced drainage only. The electric field however significantly affects the general profile by modifying the boundary conditions. A capillary number that compares electro-osmotic stress and capillary pressure in the foam geometry is introduced and characterizes the magnitude of electrically induced flow in a macroscopic foam. In particular, the ability of a foam to capture liquid from a reservoir is quantitatively estimated.