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We investigate a new diffuse-interface model that describes creepingtwo-phase flows (i.e., flows exhibiting a low Reynolds number), especiallyflows that permeate a porous medium. The system of equations consists of aBrinkman equation for the volume averaged velocity field as well as aconvective Cahn--Hilliard equation with dynamic boundary conditions for thephase-field, which describes the location of the two fluids within the domain.The dynamic boundary conditions are incorporated to model the interaction ofthe fluids with the wall of the container more precisely. In particular, theyallow for a dynamic evolution of the contact angle between the interfaceseparating the fluids and the boundary, and also for a convection-inducedmotion of the corresponding contact line. For our model, we first prove theexistence of global-in-time weak solutions in the case where regular potentialsare used in the Cahn--Hilliard subsystem. In this case, we can further show theuniqueness of the weak solution under suitable additional assumptions.Moreover, we further prove the existence of weak solutions in the case ofsingular potentials. Therefore, we regularize such singular potentials by aYosida approximation, such that the results for regular potentials can beapplied, and eventually pass to the limit in this approximation scheme.

Two-phase flows through porous media described by a Cahn--Hilliard--Brinkman model with dynamic boundary conditions