Modeling and sharp interface limits of local and non-local generalized Navier-Stokes-Korteweg Systems

水中汽泡的等温形成和随后的演化可以用经典的Navier-Stokes-Korteweg(NSK)系统描述。然而，众所周知，温度的变化在动力学中起着重要的作用，特别是在液-汽相变附近。因此，这个项目的第一个目标是使NSK系统与控制温度演变的内部能量平衡在热力学上保持一致。从液体到气相的相变通常发生在厚度较小的区域。相应密度的比率取决于温度。例如，在345℃时，这一比率为6，而在2℃时，这一比率为180000。显然，这些事实给经典NSK系统的数值处理带来了困难。此外，范德瓦尔斯提出的非凸自由能并不能恰当地代表真实水的性质。因此，该项目第二部分的一个主要目标是开发新的模式，以避免NSK系统的这两个缺点。在这里，我们考虑引入一个人工相场和相应的Allen-Cahn型耦合到NSK系统的演化方程。这类模型允许界面厚度的变化不影响表面张力，而且范德华自由能可以用液体和气相的两个自由能来代替。本项目的第三部分重点介绍了尖锐的接口限制。相场模型包含作为特定限制的系统的相应尖锐界面描述。对这些限制的研究是强制性的，因为尖锐的界面模型对于定性描述更容易理解，因此在物理上更合理。换句话说，必须检查所提出的相场模型是否导致物理上允许的尖锐界面系统。

The isothermal formation and subsequent evolution of vapor bubbles in water can be described by the classical Navier–Stokes–Korteweg (NSK) system. However, it is well known that the variation of temperature plays an import- ant role in the dynamics, in particular in the vicinity of the liquid–vapor phase transition. Thus the first objective of this project is the thermodynamical consistent coupling of the NSK system with the balance of internal energy that controls the evolution of the temperature. Phase transitions from the liquid to the vapor phase usually take place in a region of small thickness. The ratio of the corresponding densities depends on temperature. For example, that ratio is 6 at 345° C, but 180000 at 2° C.Obviously, these facts make difficulties in the numerical treatment of the classical NSK system. Moreover, the non–convex free energy according to van der Waals does not appropriately represent the properties of real water. Therefore a major aim in the second part of this project is the development of new models which avoid both shortcomings of the NSK system. Here, we think of introducing an artificial phase field and a corresponding evolution equation of Allen–Cahn type coupled to the NSK system. Such kind of models allow that changes of the interface thickness does not influence the surface tension and furthermore the van der Waals free energy can be substituted by the two free energies of the liquid and vapor phase. The third part of this project focuses on sharp interface limits. Phase field models contain corresponding sharp interface descriptions of the system as a particular limit. The study of these limits are mandatory, because sharp interface models are for qualitative descriptions easier accessible and therefore physically better justifiable. In other words, it must be checked whether a proposed phase field model leads to a physically admissible sharp interface system.