Mixing of a two-phase fluid by cavity flow

Interface stretching during mixing of a two-phase fluid in shear flow is investigated numerically by introducing a mesoscopic description of the fluid. The classical infinitely thin boundary of separation between the two phases is replaced by a transition region of small but finite width, across which the order parameter of the two-phase fluid changes continuously. We consider the case of a conserved scalar order parameter and a fluid velocity that satisfies a modified Navier-Stokes equation that includes an explicit coupling term to the order parameter. In the macroscopic limit of a very thin interface, this coupling term gives rise to capillary forces. We focus on the limit of low Reynolds number flow and compute the interface stretching as a function of time for a range of parameters of the fluid. At early times and small coupling, our calculation agrees with the classical case of a material line passively advected by the flow. At later times, the interface stretching is seen to reach a maximum as capillary forces and diffusive relaxation of the order parameter become dominant.