ABSTRACT
Two-phase incompressible flows with surface tension forces are usually modeled by either a diffusive interface or a sharp interface model. In this chapter, we restrict to the numerical simulation of the latter class of models. For numerical simulations based on a diffusive interface model, we refer to the literature, for example, [1-4]. In systems with incompressible fluids, a sharp interface model for the fluid dynamics typically consists of the Navier-Stokes equations for the bulk fluids with an interfacial surface tension force term on the right-hand side in the momentum equation. This model is combined with a convection-diffusion equation for mass transport (of a solute). We consider flow regimes with a low Reynold’s number (laminar flow), a small capillary number (significant surface tension forces), and a moderate Schmidt number, so that concentration boundary layers can be resolved. We explain why these models typically have a very high numerical complexity. Such flow models cannot be solved reliably and accurately by the commercial codes that are available today. There is a need for more efficient and reliable numerical techniques for this class of models. The development, analysis, and application of such tailor-made numerical simulation methods are an important field in numerical analysis and computational engineering science. In this chapter, we restrict to a certain class of finite element discretization methods for two-phase incompressible flows, which has been developed in recent years. We treat three important and rather general finite element techniques that can be used for an accurate discretization of the mass transport
17.1 Introduction .......................................................................................................................... 353 17.2 Mathematical Model ............................................................................................................. 354
17.2.1 Fluid Dynamics and Mass Transport ....................................................................... 354 17.2.2 Example of a Two-Phase Flow with Mass Transport ............................................... 356 17.2.3 One-Fluid Model ...................................................................................................... 358
17.3 Numerical Challenges...........................................................................................................360 17.4 XFEM for Nonaligned Discontinuities ................................................................................ 361 17.5 Nitsche Method for Interface Conditions .............................................................................364 17.6 Space-Time Formulation of Parabolic Problems .................................................................. 365 17.7 Space-Time Nitsche-XFE Method ........................................................................................ 367 17.8 Numerical Simulation of Mass Transport Problem .............................................................. 368 17.9 Conclusion and Outlook ....................................................................................................... 369 Acknowledgment ........................................................................................................................... 370 References ...................................................................................................................................... 370
equation: (1) the extended finite element method (XFEM) for the approximation of discontinuities; (2) Nitsche-method for a convenient handling of interface conditions (e.g., Henry condition); and (3) the space-time finite element technique. We restrict ourselves to an explanation of the main ideas of these methods and refer to the literature for more detailed information. Some results of numerical experiments with these methods that were obtained using the DROPS solver [5] are presented. This finite element code is specifically developed for testing, improving, and validating (new) tailor-made numerical methods for the simulation of sharp interface models for two-phase incompressible flows.
