A diffused interface model describing the evolution of two conterminous incompressible fluids in a porous medium is discussed. The system consists of the Cahn-Hilliard equation with Flory-Huggins logarithmic potential, coupled via surface tension term with the evolutionary Stokes equation at the pore scale. An evolving diffused interface of finite thickness, depending on the scale parameter ε separates the fluids. The model is studied in a bounded domain Ω with a sufficiently smooth boundary ∂Ω in R d for d = 2, 3. At first, we investigate the existence of the system at the micro-scale and derive the essential a-priori estimates. Then, using the two-scale convergence approach and unfolding operator technique, we obtain the homogenized model for the microscopic one.
A phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects is considered. The pore-scale model consists of a strongly coupled system of Stokes-Cahn-Hilliard equations. The fluids are separated by an evolving diffuse interface of a finite width depending on the scale parameter ε in the considered model. At first the well-posedness of a coupled system of partial differential equations at micro scale is investigated. We obtained the homogenized equations for the microscopic model via unfolding operator and two-scale convergence approach.
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