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

• Applicant: Professor Dr. Wolfgang Dreyer
• Subject Area: Mathematics
• Term: from 2010 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 170465967

• Final Report Year: 2015

Final Report Abstract

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 important role in the dynamics, in particular in the vicinity of liquid–vapor phase transitions. Thus the first objective of this project has been the thermodynamical consistent coupling of the NSK systems with the balance of internal energy that controls the evolution of the temperature. In addition, we have studied for this system the asymptotic behavior of various smallness parameters, i.e. letting the Mach number to zero with fixed viscosity and capillarity or letting the viscosity and capillarity tending to zero with fixed Mach number. 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 of the second part of this project has been the development of new models which avoid both shortcomings of the Navier–Stokes–Korteweg system. For that reason, we have introduced an artificial phase field and a corresponding evolution equation of Allen–Cahn type which couples to the NS 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 focused 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 important 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. For the generalized NSK–system and the proposed Navier–Stokes–Allen–Cahn type systems for quasi–incompressible and compressible fluids, we have studied the asymptotic behavior of various smallness parameters. For instance, for the proposed Navier–Stokes–Allen–Cahn type systems for compressible fluids with reactions we have investigated two physically scaling regimes, i.e. a non–dissipative and a dissipative regime, where we have recovered in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satisfy, for instance, a Young–Laplace and a Stefan type law.

Publications

• An anisotropic, inhomogeneous, elastically modified Gibbs– Thomson law as singular limit of a diffuse interface model, Adv. Math. Sci. Appl., 20, pp. 511–545, 2010
  H. Garcke and C. Kraus
  
• A diffuse a interface model for quasi–incompressible flows: Sharp interface limits and numerics, ESAIM Proceedings, 38, pp. 54–77, 2012
  G. Aki, J. Daube, W. Dreyer, J. Giesselmann, M. Kränkel, C. Kraus
  
• Asymptotic analysis for Korteweg models, Interfaces Free Bound., 14, pp. 105–143, 2012
  W. Dreyer, J. Giesselmann, C. Kraus and C. Rohde
  
• Existence results for diffuse interface models describing phase separation and damage, European J. Appl. Math., 24, pp. 179–211, 2013
  C. Heinemann and C. Kraus
  (See online at https://doi.org/10.1017/S095679251200037X)
• Sharp interface limit of a diffuse interface model of Navier–Stokes–Allen–Cahn type for mixtures, Oberwolfach Reports, 29, pp. 1719-1722, 2013
  C. Kraus
  
• A compressible mixture model with phase transition, Phys. D, 273-274, pp. 1–13, 2014
  W. Dreyer, J. Giesselmann and C. Kraus
  (See online at https://doi.org/10.1016/j.physd.2014.01.006)
• A quasi-incompressible diffuse model with phase transition – modeling and sharp interface limits, Math. Models Methods Appl. Sci., 24 , pp. 827–86, 2014
  G. Aki, W. Dreyer, J. Giesselmann and C. Kraus
  (See online at https://doi.org/10.1142/S0218202513500693)
• Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model, Mathematical Modeling and Numerical Analysis M2AN, 49(1), pp. 275–301, 2015
  J. Giesselmann and T. Pryer
  (See online at https://doi.org/10.1016/j.physd.2014.01.006)
• Low Mach asymptotic preserving scheme for the Euler-Korteweg model, IMA Journal of Numerical Analysis, 32, pp. 802–832, 2015
  J. Giesselmann
  (See online at https://doi.org/10.1093/imanum/dru022)