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

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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.