On the Navier-Stokes Cahn-Hilliard Model with Non-Matching Densities

Prof. Dr.-Ing. Dominik Schillinger, TU Darmstadt, Institut für Mechanik

Modeling the interaction of multiple fluids and the evolution of the corresponding interfaces has been the subject of active research since the time of Raleigh, van der Waals and Korteweg. Today, several classes of multiphase flow models exist that - backed by more recent progress in numerical methods and scientific computing - have started to make an impact in engineering and sciences applications. Navier-Stokes Cahn-Hilliard (NSCH) models represent one such class, coupling the momentum balance and conservation of mass of a Newtonian fluid with spinodal decomposition.
We first discuss the fundamental physical mechanisms that govern multiphase flows and derive (phenomenologically and via the theory of mixtures) the classical NSCH model [Hohenberg, Halperin, 1977]. Its key limit is the assumption of constant density, making it not applicable to problems with (large) density ratios. To remedy this deficiency, a number of NSCH models with non-matching densities have been proposed, see e.g. [Lowengrub, Truskinovsky, 1998, Aki, Dreyer, Giesselmann and Kraus, 2023]. Although aiming to represent the same physics, they seem to differ at first sight. We resolve this contradiction by introducing a general formulation that unites all of these models [ten Eikelder, van der Zee, Akkerman and Schillinger, 2023]. Our development is based on three unifying principles: (1) there is only one system of balance laws based on continuum mixture theory that describes the physical model, (2) there is only one natural energy-dissipation law that leads to quasi-incompressible NSCH models, (3) variations between the models only appear in the constitutive choices. We complete our talk by briefly discussing a divergence-conforming isogeometric finite element discretization and by demonstrating the versa-tility of NSCH models via relevant benchmark computations such as a rising air bubble in water and the contraction of a liquid filament.