A bulk-surface continuum theory for fluid flows and phase segregation with finite surface thickness

In this continuum theory, we propose a mathematical framework to study the mechanical interplay of bulk-surfaces materials undergoing deformation and phase segregation. To this end, we devise a principle of virtual powers with a bulk-surface dynamics, which is postulated on an arbitrary part $\mathcal{P}$ where the boundary $\partial\mathcal{P}$ may lose smoothness, that is, the normal field may be discontinuous at an edge $\partial^2\mathcal{P}$. The final set of equations somewhat resemble the Navier--Stokes--Cahn--Hilliard equation for the bulk and the surface. Aside from the systematical treatment based on a specialized version of the virtual power principle and free-energy imbalances for bulk-surface theories, we consider two additional ingredients: an explicit dependency of the apparent surface density on the surface thickness and mixed boundary conditions for the velocity, chemical potential, and microstructure.