Compressible phasefield models for phase transition

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During the first period of this research project we have mainly considered the Navier-Stokes Korteweg system as a mathematical model for two-phase flows with phase transition. An algorithm for the numerical treatment of this system has been developed in 2D and 3D and the jump condition across the interface for the sharp interface limit has been studied. Due to the limitation of the computational costs it turned out that the realistic diameter of the computational domain has to be very (unrealistic) small. Therefore in the new period we will consider a phase field model which should be able to overcome this problem. In the literature it has been done already for the case of constant densities in the bulks of the two phases that are given by a smooth interpolation of these two values next to the interface. The interpolation is designed in such a way that the thermodynamic properties are satisfied. In the new period we would like to derive a thermodynamically consistent model in which the densities in the bulks of the two phases are compressible. Subsequently the corresponding numerical code will be developed. This code should also be able – if necessary after some modifications – to solve also the phase field problem which is developed in the project of Dreyer/Kraus. Furthermore we will study the jump condition across the interface for the sharp interface limit in the fully dynamical case for the phase field model. This project is closely related to A3 (Dreyer/Kraus). The main differences are the following: Additionally in A3 the classical Navier-Stokes Korteweg model together with temperature and the nonlocal versions of the problems will be considered. The approach to develop the phase field model in A2 and A3 will be different, which will lead to different equations for the phase field. In particular in A2 the interpolation between the bulk densities will be done for 1 , not for p. Therefore we expect that the governing system will be different in A2 and A3 and that the study of the sharp interface limit concerns different systems which can hopefully be treated with similar methods. Furthermore in A2 we will develop a numerical code for the corresponding phase field approaches in A2 and A3.