Phase-field descriptions of two-phase compressible fluid flow: Interstitial working and a reduction to Korteweg theory
Authors:
Heinrich Freistühler and Matthias Kotschote
Journal:
Quart. Appl. Math. 77 (2019), 489-496
MSC (2010):
Primary 35Q35
DOI:
https://doi.org/10.1090/qam/1504
Published electronically:
April 20, 2018
MathSciNet review:
3962577
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The Navier-Stokes-Allen-Cahn (NSAC), the Navier-Stokes-Cahn-Hilliard (NSCH), and the Navier-Stokes-Korteweg (NSK) equations have been used in the literature to model the dynamics of two-phase fluids. In their previous article Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids, Arch. Rational Mech. Anal. 224 (2017), 1–20, the authors showed that both NSAC and NSCH reduce to versions of NSK, when one makes the (unphysical) assumption that microforces are absent. The present paper shows that the same reduction property holds without that assumption.
References
- S. Benzoni-Gavage, R. Danchin, S. Descombes, and D. Jamet, Structure of Korteweg models and stability of diffuse interfaces, Interfaces Free Bound. 7 (2005), no. 4, 371–414. MR 2191693, DOI https://doi.org/10.4171/IFB/130
- T. Blesgen, A generalisation of the Navier-Stokes equations to two-phase-flows, J. Phys. D: Appl. Phys. 32 (1999), 1119–1123.
- J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal. 88 (1985), no. 2, 95–133. MR 775366, DOI https://doi.org/10.1007/BF00250907
- H. Freistühler, A relativistic version of the Euler-Korteweg equations, submitted.
- H. Freistühler, Spatiotemporally objective descriptions of two-phase fluid flows, manuscript.
- Heinrich Freistühler and Matthias Kotschote, Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids, Arch. Ration. Mech. Anal. 224 (2017), no. 1, 1–20. MR 3609243, DOI https://doi.org/10.1007/s00205-016-1065-0
- Heinrich Freistühler and Blake Temple, Causal dissipation for the relativistic dynamics of ideal gases, Proc. A. 473 (2017), no. 2201, 20160729, 20. MR 3668121, DOI https://doi.org/10.1098/rspa.2016.0729
- Morton E. Gurtin, Debra Polignone, and Jorge Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci. 6 (1996), no. 6, 815–831. MR 1404829, DOI https://doi.org/10.1142/S0218202596000341
- 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 Sér. II 6 (1901), 1–24.
- J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1978, 2617–2654. MR 1650795, DOI https://doi.org/10.1098/rspa.1998.0273
- A. B. Roshchin and L. M. Truskinovskiĭ, A model of a weakly nonlocal compressible medium with relaxation, Prikl. Mat. Mekh. 53 (1989), no. 6, 904–910 (Russian); English transl., J. Appl. Math. Mech. 53 (1989), no. 6, 715–720 (1991). MR 1037486, DOI https://doi.org/10.1016/0021-8928%2889%2990075-0
- M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), no. 4, 301–315. MR 683192, DOI https://doi.org/10.1007/BF00250857
- C. Truesdell and W. Noll, The nonlinear field theories of mechanics, 2nd ed., Springer-Verlag, Berlin, 1992. MR 1215940
- L. Truskinovsky, Kinks versus shocks, Shock induced transitions and phase structures in general media, IMA Vol. Math. Appl., vol. 52, Springer, New York, 1993, pp. 185–229. MR 1240340, DOI https://doi.org/10.1007/978-1-4613-8348-2_11
References
- S. Benzoni-Gavage, R. Danchin, S. Descombes, and D. Jamet, Structure of Korteweg models and stability of diffuse interfaces, Interfaces Free Bound. 7 (2005), no. 4, 371–414. MR 2191693, DOI https://doi.org/10.4171/IFB/130
- T. Blesgen, A generalisation of the Navier-Stokes equations to two-phase-flows, J. Phys. D: Appl. Phys. 32 (1999), 1119–1123.
- J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal. 88 (1985), no. 2, 95–133. MR 775366, DOI https://doi.org/10.1007/BF00250907
- H. Freistühler, A relativistic version of the Euler-Korteweg equations, submitted.
- H. Freistühler, Spatiotemporally objective descriptions of two-phase fluid flows, manuscript.
- Heinrich Freistühler and Matthias Kotschote, Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids, Arch. Ration. Mech. Anal. 224 (2017), no. 1, 1–20. MR 3609243, DOI https://doi.org/10.1007/s00205-016-1065-0
- Heinrich Freistühler and Blake Temple, Causal dissipation for the relativistic dynamics of ideal gases, Proc. A. 473 (2017), no. 2201, 20160729, 20. MR 3668121, DOI https://doi.org/10.1098/rspa.2016.0729
- Morton E. Gurtin, Debra Polignone, and Jorge Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci. 6 (1996), no. 6, 815–831. MR 1404829, DOI https://doi.org/10.1142/S0218202596000341
- 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 Sér. II 6 (1901), 1–24.
- J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1978, 2617–2654. MR 1650795, DOI https://doi.org/10.1098/rspa.1998.0273
- A. B. Roshchin and L. M. Truskinovskiĭ, A model of a weakly nonlocal compressible medium with relaxation, Prikl. Mat. Mekh. 53 (1989), no. 6, 904–910 (Russian); English transl., J. Appl. Math. Mech. 53 (1989), no. 6, 715–720 (1991). MR 1037486, DOI https://doi.org/10.1016/0021-8928%2889%2990075-0
- M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), no. 4, 301–315. MR 683192, DOI https://doi.org/10.1007/BF00250857
- C. Truesdell and W. Noll, The nonlinear field theories of mechanics, 2nd ed., Springer-Verlag, Berlin, 1992. MR 1215940
- L. Truskinovsky, Kinks versus shocks, Shock induced transitions and phase structures in general media, IMA Vol. Math. Appl., vol. 52, Springer, New York, 1993, pp. 185–229. MR 1240340, DOI https://doi.org/10.1007/978-1-4613-8348-2_11
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35Q35
Retrieve articles in all journals
with MSC (2010):
35Q35
Additional Information
Heinrich Freistühler
Affiliation:
Department of Mathematics, University of Konstanz, 78457 Konstanz, Germany
Email:
heinrich.freistuehler@uni-konstanz.de
Matthias Kotschote
Affiliation:
Department of Mathematics, University of Konstanz, 78457 Konstanz, Germany
MR Author ID:
829072
Email:
matthias.kotschote@uni-konstanz.de
Received by editor(s):
January 10, 2018
Received by editor(s) in revised form:
February 25, 2018
Published electronically:
April 20, 2018
Article copyright:
© Copyright 2018
Brown University