Models of Two-Phase Fluid Dynamics à la Allen-Cahn, Cahn-Hilliard, and ... Korteweg!
Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 57-67.

One purpose of this paper on the Navier-Stokes-Allen-Cahn (NSAC), the Navier-Stokes-Cahn-Hilliard (NSCH), and the Navier-Stokes-Korteweg (NSK) equations consists in surveying solution theories that one of the authors, M. K., has developed for these three evolutionary systems of partial differential equations. All three theories start from a Helmholtz free energy description of the compressible two-phase fluids whose dynamics they describe in various ways. While a diphasic fluid composed from two constituents of individually constant density is still compressible as long as these two densities are different from each other, the abovementioned solution theories for NSAC and NSCH do not apply in this “quasi-incompressible” case, as the Helmholtz-energy framework degenerates. The second purpose of the paper is to present an observation made by both authors together that shows how to fill these gaps. As ‘by-products’ one obtains (a) in the case that the phases can transform into each other, a justification of NSK, and (b) in the case that they cannot, a new Korteweg type system with non-local ‘viscosity’.

Reçu le : 2015-03-09
Accepté le : 2015-02-10
Publié le : 2016-02-15
DOI : https://doi.org/10.5802/cml.24
Classification : 76N10,  35Q35,  82B24,  82B26,  35M10
@article{CML_2015__7_2_57_0,
     author = {Heinrich Freist\"uhler and Matthias Kotschote},
     title = {Models of Two-Phase Fluid Dynamics \`a la  Allen-Cahn, Cahn-Hilliard, and ... Korteweg!},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {7},
     number = {2},
     year = {2015},
     pages = {57-67},
     doi = {10.5802/cml.24},
     language = {en},
     url = {cml.centre-mersenne.org/item/CML_2015__7_2_57_0/}
}
Heinrich Freistühler; Matthias Kotschote. Models of Two-Phase Fluid Dynamics à la  Allen-Cahn, Cahn-Hilliard, and ... Korteweg!. Confluentes Mathematici, Tome 7 (2015) no. 2, pp. 57-67. doi : 10.5802/cml.24. https://cml.centre-mersenne.org/item/CML_2015__7_2_57_0/

[1] H. Abels, H. Garcke, G. Grün: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Meth. Appl. Sci. 22 (2012), 1150013.

[2] S. M. Allen, J. W. Cahn : A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica 27 (1979), 1085–1095.

[3] S. Benzoni-Gavage: Stability of subsonic planar phase boundaries in a van der Waals fluid Arch. Rational Mech. Anal. 150 (1999) 23–55.

[4] S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness for the Euler-Korteweg model in several space dimensions. Indiana Univ. Math. J. 56 (2007), 1499–1579.

[5] S. Benzoni-Gavage, R. Danchin, S. Descombes, D. Jamet: Structure of Korteweg models and stability of diffuse interfaces. Interfaces Free Bound. 7 (2005), 371–414.

[6] T. Blesgen: A generalisation of the Navier-Stokes equations to two-phase-flows. J. Phys. D: Appl. Phys. 32 (1999), 1119–1123.

[7] J. W. Cahn, J. E. Hilliard: Free energy of a non-uniform system I. Interfacial free energy. J. Chem. Phys. 28 (1958), 258–267.

[8] R. Danchin, B. Desjardins : Existence of solutions for compressible fluid models of Korteweg type. Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 97–133.

[9] J. E. Dunn, J. Serrin: On the thermomechanics of interstitial working. Arch. Rational Mech. Anal. 88 (1985), 95–133.

[10] J. L. Ericksen: Liquid crystals with variable degree of orientation Arch. Rational Mech. Anal. 113 (1990), 97–120.

[11] H. Freistühler: Phase transitions and traveling waves in compressible fluids. Arch. Rational Mech. Anal. 211 (2014), 189–204.

[12] H. Freistühler, M. Kotschote: Internal structure of dynamic phase-transition fronts in a fluid with two compressible or incompressible phases, Bulletin Institute of Mathematics Academia Sinica (2015).

[13] H. Freistühler, M. Kotschote: Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids. Submitted.

[14] H. Hattori, D. Li: The existence of global solutions to a fluid dynamic model for materials of Korteweg type. J. Partial Differential Equations 9 (1996), 323–342.

[15] D.J. Korteweg: Sur la forme que prennent les équations des mouvements des fluides si l’on tient compte des forces capillaires par des variations de densité. Arch. Néer. Sci. Exactes II 6 (1901), 1–24.

[16] M. Kotschote: Spectral analysis for traveling waves in compressible two-phase fluids of Navier-Stokes-Allen-Cahn type. Submitted.

[17] M. Kotschote: Mixing rules and the Navier-Stokes-Cahn-Hilliard equations for compressible heat-conductive fluids. Bulletin Brazilian Mathematical Society (2015).

[18] M. Kotschote: Strong solutions of the Navier-Stokes equations for a compressible fluid of Allen-Cahn type. Arch. Ration. Mech. Anal. 206 (2012), 489–514.

[19] M. Kotschote: On compressible non-isothermal fluids of non-Newtonian Korteweg-type, SIAM J. Math. Anal. 44 (2012), 74–101.

[20] M. Kotschote: Strong solutions to the compressible Navier-Stokes-Cahn-Hilliard system for heat-conductive fluids, in preparation.

[21] M. Kotschote, R. Zacher: Strong solutions in the dynamical theory of compressible fluid mixtures, Math. Models Methods Appl. Sci. 25, no. 7 (2015), 1217–1256.

[22] J. Lowengrub, L. Truskinovsky: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454 (1998), 2617–2654.

[23] H. Triebel: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam - New York - Oxford, 1978.

[24] H. Triebel: Theory of Function Spaces. Geest & Portig K.-G., Leipzig, 1983.