Two-Phase Flows with Bulk–Surface Interaction: Thermodynamically Consistent Navier–Stokes–Cahn–Hilliard Models with Dynamic Boundary Conditions

• Published in: Journal of mathematical fluid mechanics Vol. 25; no. 3
• Main Authors: Giorgini, Andrea, Knopf, Patrik
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2023; Springer Nature B.V
• Subjects: Binary fluids; Boundary conditions; Classical and Continuum Physics; Conservation laws; Contact angle; Energy dissipation; Fluid flow; Fluid mechanics; Fluid- and Aerodynamics; Incompressible flow; Lagrange multiplier; Mass transfer; Mathematical Methods in Physics; Navier-Stokes equations; Physics; Physics and Astronomy; Surface diffusion; Theoretical mathematics; Two phase flow; Bulk–surface interaction; Two-phase flows; 76T99; 35Q35; 35D30; 76D03; Navier–Stokes–Cahn–Hilliard system; 76D05; Dynamic boundary conditions; 76T06
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-023-00811-w

We derive a novel thermodynamically consistent Navier–Stokes–Cahn–Hilliard system with dynamic boundary conditions. This model describes the motion of viscous incompressible binary fluids with different densities. In contrast to previous models in the literature, our new model allows for surface diffusion, a variable contact angle between the diffuse interface and the boundary, and mass transfer between bulk and surface. In particular, this transfer of material is subject to a mass conservation law including both a bulk and a surface contribution. The derivation is carried out by means of local energy dissipation laws and the Lagrange multiplier approach. Next, in the case of fluids with matched densities, we show the existence of global weak solutions in two and three dimensions as well as the uniqueness of weak solutions in two dimensions.