Skip to main content
SpringerLink
Log in

Sharp Interface Limits for Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids

  • Chapter
  • First Online:
Transport Processes at Fluidic Interfaces

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

We consider the mathematical relation between diffuse interface and sharp interface models for the flow of two viscous, incompressible Newtonian fluids like oil and water. In diffuse interface models a partial mixing of the macroscopically immiscible fluids on a small length scale ɛ > 0 and diffusion of the mass particles are taken into account. These models are capable to describe such two-phase flows beyond the occurrence of topological singularities of the interface due to collision or droplet formation. Both for theoretical and numerical purposes a deeper understanding of the limit ɛ → 0 in dependence of the scaling of the mobility coefficient m ɛ is of interest. Here the mobility is the inverse of the Peclet number and controls the strength of the diffusion. We discuss several rigorous mathematical results on convergence and non-convergence of solutions of diffuse interface to sharp interface models in dependence of the scaling of the mobility.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abels, H.: On the notion of generalized solutions of two-phase flows for viscous incompressible fluids. RIMS Kôkyûroku Bessatsu B1, 1–15 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Abels, H.: On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Rat. Mech. Anal. 194(2), 463–506 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abels, H., Lengeler, D.: 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 

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

    Google Scholar 

  5. Abels, H., Röger, M.: 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, 2403–2424 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Abels, H., Schaubeck, S.: Nonconvergence of the capillary stress functional for solutions of the convective Cahn-Hilliard equation. In: Mathematical Fluid Dynamics, Present and Future. Springer Proceedings in Mathematics & Statistics, vol. 183. Springer, Cham (2016)

    Google Scholar 

  7. Abels, H., Wilke, M.: Well-posedness and qualitative behaviour of solutions for a two-phase Navier-Stokes-Mullins-Sekerka system. Interfaces Free Bound. 15(1), 39–75 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22(3), 1150013, 40 pp. (2012)

    Google Scholar 

  9. Alikakos, N.D., Bates, P.W., Chen, X.: 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 

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

    Google Scholar 

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

    Google Scholar 

  12. Chen, X.: Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Commun. Partial Differ. Equ. 19(7–8), 1371–1395 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, X., Hilhorst, D., Logak, E.: Mass conserving Allen-Cahn equation and volume preserving mean curvature flow. Interfaces Free Bound. 12(4), 527–549 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)

    MATH  Google Scholar 

  17. Giga, Y., Novotny, A. (ed.): Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham (2018)

    MATH  Google Scholar 

  18. Gurtin, M.E., Polignone, D., Viñals, J.: 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 

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

    Article  Google Scholar 

  20. Jacqmin, D.: Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 155(1), 96–127 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, C., Shen, J.: 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 

  22. Liu, C., Sato, N., Tonegawa, Y.: Two-phase flow problem coupled with mean curvature flow. Interfaces Free Bound. 14(2), 185–203 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A: Math. Phys. Eng. Sci. 454(1978), 2617–2654 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Málek, J., Nečas, J., Rokyta, M., Ružička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation, vol. 13, vii, 317 pp. Chapman & Hall, London (1996)

    Google Scholar 

  25. Modica, L.: 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 

  26. Modica, L., Mortola, S.: Un esempio di Γ −-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977)

    Google Scholar 

  27. Moser, M.: Lokale Wohlgestelltheit für ein Navier-Stokes/mittleren Krümmungsfluss-System. Master thesis, Regensburg (2016)

    Google Scholar 

  28. Schaubeck, S.: Sharp interface limits for diffuse interface models. PhD thesis, University Regensburg, urn:nbn:de:bvb:355-epub-294622 (2014)

    Google Scholar 

  29. Starovoĭtov, V.N.: A model of the motion of a two-component fluid taking into account capillary forces. Prikl. Mekh. Tekhn. Fiz. 35(6), 85–92 (1994)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Regensburg, Universitätsstraße 31, 93053, Regensburg, Germany

    Helmut Abels & Andreas Schöttl

  2. NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200061, China

    YuNing Liu

Authors
  1. Helmut Abels
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YuNing Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andreas Schöttl
    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. Fachbereich Mathematik, Technische Universität Darmstadt , Darmstadt, Germany

    Dieter Bothe

  2. Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Arnold Reusken

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Abels, H., Liu, Y., Schöttl, A. (2017). Sharp Interface Limits for Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids. In: Bothe, D., Reusken, A. (eds) Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-56602-3_9

Download citation

Publish with us

Policies and ethics

Search

Navigation