Compressible phasefield models for phase transition

在本研究项目的第一阶段，我们主要考虑将Navier-Stokes-Korteweg系统作为具有相变的两相流的数学模型。给出了该系统的二维和三维数值处理算法，并研究了尖锐界面极限下的界面跳跃条件。由于计算成本的限制，计算域的实际直径必须非常(不切实际)很小。因此，在新的时期，我们将考虑一个相场模型，它应该能够克服这个问题。在文献中，已经在两相体积密度恒定的情况下这样做了，这两个相的体积密度是由靠近界面的这两个值的平滑内插给出的。插值法是以满足热力学性质的方式设计的。在新的时期，我们想要得到一个热力学上一致的模型，在这个模型中，两相体积中的密度是可压缩的。随后将开发相应的数字代码。这个程序还应该能够--如果有必要的话，经过一些修改--也能够解决Dreyer/Kraus项目中开发的相场问题。此外，我们还将研究在全动力学情况下相场模型中尖锐界面极限的跳跃条件。这个项目与A3(Dreyer/Kraus)密切相关。主要区别如下：此外，在A3中，将考虑经典的Navier-Stokes Korteweg模型以及温度和问题的非局部版本。A2和A3中建立相场模型的方法不同，这将导致相场方程的不同。特别是在A2中，体积密度之间的内插将针对1进行，而不是针对p。因此，我们预计A2和A3中的调节系统将不同，并且锐化界面极限的研究涉及不同的系统，希望可以用类似的方法来处理。此外，在A2中，我们将为A2和A3中相应的相场方法开发一个数值代码。

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.