  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.
