Skip to main content
Log in

Gradient flow perspective on thin-film bilayer flows

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air phase. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well-known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding reduced thin-film model and formally identify solutions with those of a PDE problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using \(\Gamma \)-convergence. For time-dependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69(3):931

    Article  ADS  Google Scholar 

  2. Huh C, Scriven LE (1971) Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J Colloid Interface Sci 35(1):85–101

    Article  Google Scholar 

  3. Hervet H, de Gennes PG (1984) The dynamics of wetting: precursor films in the wetting of dry solids. Comptes Rendus de l’Académie des Sci 299:499–503

    Google Scholar 

  4. de Gennes PG, Brochard-Wyart F, Quéré D (2004) Capillarity and wetting phenomena: drops, bubbles, pearls, waves. Springer, Berlin

    Book  Google Scholar 

  5. Joanny JF (1987) Wetting of a liquid substrate. Physicochem Hydrodyn 9(1–2):183–196

    Google Scholar 

  6. Brochard-Wyart F, Martin P, Redon C (1993) Liquid/liquid dewetting. Langmuir 9(12):3682–3690

    Article  Google Scholar 

  7. Kriegsmann JJ, Miksis MJ (2003) Steady motion of a drop along a liquid interface. SIAM J Appl Math 64(1):18–40

    Article  MATH  MathSciNet  Google Scholar 

  8. Pototsky A, Bestehorn M, Merkt D, Thiele U (2004) Alternative pathways of dewetting for a thin liquid two-layer film. Phys Rev E 70(2):025201

    Article  ADS  Google Scholar 

  9. Craster RV, Matar OK (2006) On the dynamics of liquid lenses. J Colloid Interface Sci 303(2):503–516

    Article  Google Scholar 

  10. Karapetsas G, Craster RV, Matar OK (2011) Surfactant-driven dynamics of liquid lenses. Phys Fluids 23(12):122106–122106

    Article  ADS  Google Scholar 

  11. Danov KD, Paunov VN, Alleborn N, Raszillier H, Durst F (1998) Stability of evaporating two-layered liquid film in the presence of surfactant. Chem Eng Sci 53(15):2809–2822

    Article  Google Scholar 

  12. Kriegsmann JJ (1999) Spreading on a liquid film. PhD thesis, Northwestern University

  13. Jachalski S, Kitavtsev G, Taranets R (2014) Weak solutions to lubrication systems describing the evolution of bilayer thin films. Commun Math Sci 12(3):527–544

    Article  MATH  MathSciNet  Google Scholar 

  14. Escher J, Matioc BV (2014) Non-negative global weak solutions for a degenerated parabolic system approximating the two-phase Stokes problem. J Differ Equ 256(8):2659–2676

  15. Merkt D, Pototsky A, Bestehorn M, Thiele U (2005) Long-wave theory of bounded two-layer films with a free liquid-liquid interface: short-and long-time evolution. Phys Fluids 17:064104

    Article  MathSciNet  ADS  Google Scholar 

  16. Otto F (2001) The geomety of dissipative evolution equations: the porous medium equation. Commun Partial Differ Equ 26(1–2):101–174

    Article  MATH  Google Scholar 

  17. Rumpf M, Vantzos O (2013) Numerical gradient flow discretization of viscous thin films on curved geometries. Math Models Methods Appl Sci 23(05):917–947

    Article  MATH  MathSciNet  Google Scholar 

  18. von Helmholtz H (1868) Theorie der stationären Ströme in reibenden Flüssigkeiten. Verh Naturh-Med Ver Heidelb 11:223

    Google Scholar 

  19. Rayleigh JWS (1913) On the motion of a viscous fluid. Philos Mag 6(26):621–628

    Google Scholar 

  20. Neumann FE (1894) Vorlesung über die Theorie der Capillarität. BG Teubner, Leipzig, pp 113–116

    Google Scholar 

  21. Otto F (1998) Lubrication approximation with prescribed nonzero contact angle. Commun Partial Differ Equ 23(11–12):2077–2164

    MATH  Google Scholar 

  22. Jachalski S, Huth R, Kitavtsev G, Peschka D, Wagner B (2013) Stationary solutions of liquid two-layer thin-film models. SIAM J Appl Math 73(3):1183–1202

    Article  MATH  MathSciNet  Google Scholar 

  23. Bernis F, Peletier LA, Williams SM (1992) Source type solutions of a fourth order nonlinear degenerate parabolic equation. Nonlinear Anal Theory Methods Appl 18(3):217–234

    Article  MATH  MathSciNet  Google Scholar 

  24. Giacomelli L, Gnann MV, Otto F (2013) Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3. Eur J Appl Math 24:735–760

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

SJ and DP thank the German Research Foundation DFG for financial support through the project Structure formation in thin liquid–liquid films in the SPP 1506 and DFG Research Center MATHEON through the project C10. The work of GK was supported by the postdoctoral scholarship at the Max-Planck-Institute for Mathematics in the Sciences in Leipzig. We also thank Maciek Korzec (Technical University Berlin) for useful discussions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to D. Peschka.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huth, R., Jachalski, S., Kitavtsev, G. et al. Gradient flow perspective on thin-film bilayer flows. J Eng Math 94, 43–61 (2015). https://doi.org/10.1007/s10665-014-9698-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-014-9698-1

Keywords

Navigation