Skip to main content
SpringerLink
Log in

Weak Solutions and Diffuse Interface Models for Incompressible Two-Phase Flows

  • Living reference work entry
  • First Online:
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Abstract

In two-phase flows typically a change of topology arises. This happens when two drops merge or when one bubble splits into two. In such a case the concept of classical solutions to two-phase flow problems, which describe the interface as a smooth hypersurface, breaks down.This contribution discusses two possible approaches to deal with this problem. First of all weak formulations are discussed which allow for topology changes during the evolution. Such weak formulations involve either varifold solutions, so-called renormalized solutions or viscosity solutions.A second approach replaces the sharp interface by a diffuse interfacial layer which leads to a phase field-type representation of the interface. This approach leads typically to quite smooth solutions even when the topology changes.This contribution introduces the solution concepts, discusses modeling aspects, gives an account of the analytical results known, and states how one can recover the sharp interface problem as an asymptotic limit of the diffuse interface problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  1. H. Abels, Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids (Habilitation thesis, Leipzig, 2007)

    MATH  Google Scholar 

  2. H. Abels, On the Notion of Generalized Solutions of Two-Phase Flows for Viscous Incompressible Fluids (RIMS Kôkyûroku Bessatsu B1, Kyoto, 2007), pp. 1–15

    Google Scholar 

  3. H. Abels, On generalized solutions of two-phase flows for viscous incompressible fluids. Interfaces Free Bound. 9, 31–65 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Commun. Math. Phys. 289, 45–73 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194, 463–506 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Abels, Double obstacle limit for a Navier-Stokes/Cahn-Hilliard system, in Parabolic Problems: The Herbert Amann Festschrift. Progress in Nonlinear Differential Equations and Their Applications, vol. 80 (Birkhaĺuser, Basel, 2011), pp. 1–15, 1–20

    Google Scholar 

  7. H. Abels, Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow. SIAM J. Math. Anal. 44(1), 316–340 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Abels, D. Breit, Weak solutions for a non-Newtonian diffuse interface model with different densities. Nonlinearity 29, 3426–3453 (2016)

    Article  MATH  Google Scholar 

  9. H. Abels, D. Depner, H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(6), 1175–1190 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Abels, D. Depner, H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities. J. Math. Fluid Mech. 15(3), 453–480 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Abels, L. Diening, Y. Terasawa, Existence of weak solutions for a diffuse interface model of non-Newtonian two-phase flows. Nonlinear Anal. Real World Appl. 15, 149–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Abels, H. Garcke, G. Grün, Thermodynamically consistent diffuse interface models for incompressible two-phase flows with different densities. Preprint Nr. 20/2010 University Regensburg (2010)

    Google Scholar 

  13. H. Abels, H. Garcke, G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22(3), 1150013 (2012)

    Google Scholar 

  14. H. Abels, D. Lengeler, On sharp interface limits for diffuse interface models for two-phase flows. Interfaces Free Bound. 16(3), 395–418 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Abels, Y. Liu, Sharp interface limit for a Stokes/Allen-Cahn system. (2016), Preprint arXiv:/1611.04422

  16. H. Abels, M. Röger, Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6), 2403–2424 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Abels, S. Schaubeck, Nonconvergence of the capillary stress functional for solutions of the convective Cahn-Hilliard equation. Accepted for publication for the Proceedings of International Conference on Mathematical Fluid Dynamics, Present and Future, Springer Proceedings in Mathematics and Statistics (2016)

    Google Scholar 

  18. H. Abels, M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67, 3176–3193 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. N.D. Alikakos, P.W. Bates, X. Chen, Convergence of the Cahn–Hilliard equation to the Hele–Shaw model. Arch. Ration. Mech. Anal. 128(2), 165–205 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. H.W. Alt, The entropy principle for interfaces. Fluids and solids. Adv. Math. Sci. Appl. 2, 585–663 (2009)

    MathSciNet  MATH  Google Scholar 

  21. D.M. Ambrose, L. Filho, C. Milton, H.J. Nussenzveig Lopes, W.A. Strauss, Transport of interfaces with surface tension by 2D viscous flows. Interfaces Free Bound. 12(1), 23–44 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, vol. xviii (Clarendon Press, Oxford, 2000), p. 434

    MATH  Google Scholar 

  23. D.-M. Anderson, G.B. McFadden, A.A. Wheeler, Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)

    Article  MathSciNet  Google Scholar 

  24. J.T. Beale, The initial value problem for the Navier-Stokes equations with a free surface. Commun. Pure Appl. Math. 34, 359–392 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  25. J.T. Beale, Large-time regularity of viscous surface waves. Arch. Ration. Mech. Anal. 84, 307–352 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Bosia, Analysis of a Cahn-Hilliard-Ladyzhenskaya system with singular potential. J. Math. Anal. Appl. 397(1), 307–321 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20(2), 175–212 (1999)

    MathSciNet  MATH  Google Scholar 

  28. F. Boyer, Nonhomogeneous Cahn-Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(2), 225–259 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31, 41–68 (2002)

    Article  MATH  Google Scholar 

  30. D. Breit, L. Diening, S. Schwarzacher, Solenoidal Lipschitz truncation for parabolic PDEs. Math. Models Methods Appl. Sci. 23(14), 2671–2700 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Caginalp, X. Chen, Convergence of the phase field model to its sharp interface limits. Eur. J. Appl. Math. 9(4), 417–445 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differ. Geom. 44(2), 262–311 (1996)

    MathSciNet  MATH  Google Scholar 

  33. S.P. Chen, R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems. Pac. J. Math. 136(1), 15–55 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Danchin, P.B. Mucha, Incompressible flows with piecewise constant density. Arch. Ration. Mech. Anal. 207(3), 991–1023 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. P. De Mottoni, M. Schatzman, Geometrical evolution of developed interfaces. Trans. Am. Math. Soc. 347(5), 1533–1589 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids. Arch. Ration. Mech. Anal. 137(2), 135–158 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. H. Ding, P.D.M. Spelt, C. Shu, Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 22, 2078–2095 (2007)

    Article  MATH  Google Scholar 

  38. R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  39. L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics, vol. viii+268 (CRC Press, Boca Raton, 1992)

    Google Scholar 

  40. P.C. Fife, O. Penrose, Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter. Electron. J. Differ. Equ. 16, 1–49 (1995)

    MathSciNet  MATH  Google Scholar 

  41. H. Garcke, B. Stinner, Second order phase field asymptotics for multi-component systems. Interfaces Free Bound. 8(2), 131–157 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Y. Giga, S. Takahashi, On global weak solutions of the nonstationary two phase Stokes flow. SIAM J. Math. Anal. 25, 876–893 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  43. E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80 (Birkhäuser Verlag, Basel, 1984), pp.  xii+240

    Google Scholar 

  44. M. Grasselli, D. Pražák, Longtime behavior of a diffuse interface model for binary fluid mixtures with shear dependent viscosity. Interfaces Free Bound. 13(4), 507–530 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. M.E. Gurtin, D. Polignone, J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6(6), 815–831 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  46. P.C. Hohenberg, B.I. Halperin, Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977)

    Article  Google Scholar 

  47. D.D. Joseph, Fluid dynamics of two miscibel liquids with diffusion and gradient stresses. J. Mech. B/Fluids 9, 565–596 (1990)

    Google Scholar 

  48. N. Kim, L. Consiglieri, J.F. Rodrigues, On non-Newtonian incompressible fluids with phase transitions. Math. Methods Appl. Sci. 29(13), 1523–1541 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. H. Poincaré Anal. Nonlinéaire 25(4), 679–696 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Liu I-S, Continuum Mechanics. Advanced Texts in Physics (Springer, Berlin, 2002), pp. xii+297

    Google Scholar 

  51. C. Liu, J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D 179(3–4), 211–228 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  52. J. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454, 2617–2654 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  53. L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98(2), 123–142 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  54. L. Modica, S. Mortola, Un esempio di \(\Gamma ^{-}\)-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977)

    Google Scholar 

  55. I. Müller, Thermodynamics (Pitman, Boston, 1985)

    MATH  Google Scholar 

  56. A. Nouri, F. Poupaud, An existence theorem for the multifluid Navier-Stokes problem. J. Differ. Equ. 122, 71–88 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  57. A. Nouri, F. Poupaud, Y. Demay, An existence theorem for the multi-fluid Stokes problem. Q. Appl. Math. 55(3), 421–435 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  58. P.I. Plotnikov, Generalized solutions to a free boundary problem of motion of a non-Newtonian fluid. Sib. Math. J. 34(4), 704–716 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  59. J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics, vol. 105 (Birkhäuser, Cham, 2016)

    Google Scholar 

  60. M. Röger, Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation. Interfaces Free Bound. 6(1), 105–133 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  61. R. Salvi, On the existence of free surface problem for viscous incompressible flow. G.P. Galdi (ed. et al): Topics in mathematical fluid mechanics. Aracne. Quad. Mat. 10, 247–275 (2002)

    Google Scholar 

  62. R. Schätzle, Hypersurfaces with mean curvature given by an ambient Sobolev function. J. Differ. Geom. 58(3), 371–420 (2001)

    MathSciNet  MATH  Google Scholar 

  63. S. Schaubeck, Sharp interface limits for diffuse interface models. PhD thesis (2014), http://epub.uni-regensburg.de/29462/. Accessed 24 Feb 2016

  64. Y. Shibata, S. Shimizu, L p -L q maximal regularity and viscous incompressible flows with free surface. Proc. Jpn Acad. Ser. A Math. Sci. 81(9), 151–155 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  65. Y. Shibata, S. Shimizu, Report on a local in time solvability of free surface problems for the Navier-Stokes equations with surface tension. Appl. Anal. 90(1), 201–214 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  66. V.A. Solonnikov, Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations. Boundary value problems of mathematical physics and related questions in the theory of functions 9. Zap. Naučn. Sem. Leningrad. Otdel Mat. Inst. Steklov. 59, 178–254 (1976)

    Google Scholar 

  67. V.N. Starovoĭtov, On the motion of a two-component fluid in the presence of capillary forces. Mat. Zametki 62(2), 293–305 (1997)

    Article  MathSciNet  Google Scholar 

  68. V.A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible fluid. Math. USSR Izvestiya 31(2), 381–405 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  69. S. Takahasi, On global weak solutions of the nonstationary two phase Navier-Stokes flow. Adv. Math. Sci. Appl. 5(1), 321–342 (1995)

    MathSciNet  Google Scholar 

  70. A. Tani, N. Tanaka, Large-time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Ration. Mech. Anal. 130, 303–314 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  71. A. Wagner, On the Bernoulli free-boundary problem. Differ. Int. Equ. 14(1), 51–58 (2001)

    MathSciNet  MATH  Google Scholar 

  72. A. Wagner, On the Bernoulli free boundary problem with surface tension. Free boundary problems: theory and applications (Crete, 1997). Chapman Hall/CRC Res. Notes Math. 409, 246–251 (1999)

    Google Scholar 

  73. K. Yeressian, On varifold solutions of two-phase incompressible viscous flow with surface tension. J. Math. Fluid Mech. 17(3), 463–494 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, UniversitätsstraSSe 31, 93040, Regensburg, Germany

    Helmut Abels & Harald Garcke

Authors
  1. Helmut Abels
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Harald Garcke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Helmut Abels .

Editor information

Editors and Affiliations

  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this entry

Cite this entry

Abels, H., Garcke, H. (2016). Weak Solutions and Diffuse Interface Models for Incompressible Two-Phase Flows. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_29-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10151-4_29-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Cham

  • Online ISBN: 978-3-319-10151-4

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation