Two-phase flows with bulk-surface interaction described by a Navier–Stokes–Cahn–Hilliard model with dynamic boundary conditions

• Applicant: Dr. Patrik Knopf
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 524694286

We are planning to investigate a thermodynamically consistent diffuse interface model for two-phase flows with possibly unmatched densities that was recently derived by A. Giorgini and the applicant in [arXiv:2208.09695]. The model consists of a Navier-Stokes equation, which describes the time evolution of the velocity field associated with the two fluids, and a convective Cahn-Hilliard system describing the location of the two fluids in the considered domain via a phase-field function. These two subsystems are coupled in a nonlinear way by means of a phase-field depending source term in the Navier-Stokes equation as well as convection terms in the Cahn-Hilliard subsystem. In the case of unmatched densities, meaning that the individual constant densities of the two fluids are different, there is an additional nonlinear coupling as the density function appearing in the Navier-Stokes equation does also depend on the phase-field. In contrast to previous phase-field models for two-phase flows in the literature (e.g., the well-known Abels-Garcke-Grün model), our new model introduced in [arXiv:2208.09695] allows for dynamic changes of the contact angle between the diffuse interface separating the fluids and the boundary, and is also capable of describing the situation of absorption of material by the boundary. This is achieved by the usage of a class of dynamic boundary conditions for the Cahn-Hilliard subsystem that has previously been introduced for the Cahn-Hilliard equation by K.F. Lam, C. Liu, S. Metzger and the applicant in [ESAIM: Mathematical Modelling and Numerical Analysis 55(1): 229-282, 2021]. In order to make the model (thermodynamically) consistent, we impose a certain Navier slip boundary condition on the velocity field. The model derivation for our new Navier-Stokes-Cahn-Hilliard model as well as some first analytical results in the case of matched densities were presented in [arXiv:2208.09695]. The goal of the proposed project is to further investigate the model in terms of mathematical analysis first in the case of matched densities but later also in the much more difficult case of unmatched densities where the two fluids may have a different individual density.

• DFG Programme: Research Grants