A thermodynamically consistent model for bulk-surface viscous fluid mixtures: Model derivation and mathematical analysis

Abstract: We derive and analyze a new diffuse interface model for incompressible, viscous fluid mixtures with bulk-surface interaction. Our system consists of a Navier--Stokes--Cahn--Hilliard model in the bulk that is coupled to a surface Navier--Stokes--Cahn--Hilliard model on the boundary. Compared with previous models, the inclusion of an additional surface Navier--Stokes equation is motivated, for example, by biological applications such as the seminal \textit{fluid mosaic model} (Singer & Nicolson, \textit{Science}, 1972) in which the surface of biological cells is interpreted as a thin layer of viscous fluids. We derive our new model by means of local mass balance laws, local energy dissipation laws, and the Lagrange multiplier approach. Moreover, we prove the existence of global weak solutions via a semi-Galerkin discretization. The core part of the mathematical analysis is the study of a novel bulk-surface Stokes system and its corresponding bulk-surface Stokes operator. Its eigenfunctions are used as the Galerkin basis to discretize the bulk-surface Navier--Stokes subsystem.