Compressible phasefield models for phase transition

在本研究计划的第一阶段，我们主要考虑Navier-Stokes-Korteweg系统作为有相变的两相流的数学模型。在二维和三维的数值处理算法已经开发了这个系统和跳跃条件的界面上的尖锐的界面限制进行了研究。由于计算成本的限制，事实证明，计算域的实际直径必须非常小（不切实际）。因此，在新的时期，我们将考虑相场模型，这应该能够克服这个问题。在文献中，对于两相体积密度恒定的情况已经做过了，这两个值在界面附近的平滑插值给出。插值设计的方式，满足热力学性质。在新时期，我们希望得到一个两相体积密度可压缩的热力学相容模型。随后将开发相应的数字代码。这段代码也应该能够-如果必要的一些修改后-以解决也相领域的问题，这是发展中国家的项目德雷尔/克劳斯。此外，我们将研究在完全动态的情况下，相场模型的尖锐的界面极限跨越界面的跳跃条件。该项目与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模型以及温度和非局部问题。在A2和A3中开发相场模型的方法将是不同的，这将导致用于相场的不同方程。特别是在A2中，堆积密度之间的插值将针对1而不是针对p进行。因此，我们预计A2和A3中的控制系统将是不同的，并且尖锐界面极限的研究涉及不同的系统，这些系统有望用类似的方法处理。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.