Two-phase flow problem coupled with mean curvature flow

We prove the existence of generalized solution for incompressible and viscous non-Newtonian twophase fluid flow for spatial dimension d = 2 and 3. Separating two shear thickening fluids with power law viscosity strictly above critical growth p =(d+2)/2, the phase boundary moves along with the fluid flow plus its mean curvature while exerting surface tension force to the fluid. An approximation scheme combining the Galerkin method and the phase field method is adopted.