An immersed isogeometric Navier-Stokes-Cahn-Hilliard method based on a dissipation analysis as a guiding principle
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In this talk, we present the immersed isogeometric analysis diffuse interface multi-phase flow framework proposed in [1]. This framework enables the simulation of binary-fluid flows through complex geometries such as porous media. The key features of the proposed framework are the use of optimal-regularity B-splines for a computationally efficient higher-order method, a generalized Navier-slip boundary condition for the tangential velocity components, Nitsche's method for the convective impermeability boundary condition, and skeleton- and ghost-penalties to guarantee stability. Throughout this talk, we shall use a dissipation analysis as a guiding principle: for canonical linear examples, both Nitsche's method and the ghost-penalty strategy are well established and understood. Their forms and the choices of penalty factors rely on coercivity proofs. Yet, for the severely more complex case of the Navier-Stokes-Cahn-Hilliard equations (which are non-linear, non-coercive, non-steady and 4-field mixed), such an analysis does not apply. Rather, it is a bound on the free energy evolution that provides a handle on the stability of the system. This is thus also the viewpoint that we will take as we motive the choices for the finite element formulation for our immersed NSCH framework.
