Journal of Statistical Physics

, Volume 122, Issue 6, pp 1261–1291 | Cite as

Thin-Film Flow Influenced by Thermal Noise

Article

Abstract

We study the influence of thermal fluctuations on the dewetting dynamics of thin liquid films. Starting from the incompressible Navier-Stokes equations with thermal noise, we derive a fourth-order degenerate parabolic stochastic partial differential equation which includes a conservative, multiplicative noise term—the stochastic thin-film equation. Technically, we rely on a long-wave-approximation and Fokker–Planck-type arguments. We formulate a discretization method and give first numerical evidence for our conjecture that thermal fluctuations are capable of accelerating film rupture and that discrepancies with respect to time-scales between physical experiments and deterministic numerical simulations can be resolved by taking noise effects into account.

Key Words

wetting microfluidics thin film flow stochastic hydrodynamics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69:931 (1997).CrossRefADSGoogle Scholar
  2. 2.
    G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87:113 (2000).CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    G. Grün, On the convergence of entropy consistent for lubrication-type equations in multiple space dimensions. Math. Comp. 72:1251 (2003).CrossRefMATHADSMathSciNetGoogle Scholar
  4. 4.
    K. R. Mecke, Integral geometry in statistical physics. Int. J. Mod. Phys. B 12: 861 (1998).CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    J. Becker, G. Grün, R. Seemann, H. Mantz, K. Jacobs, K. R. Mecke and R. Blossey, Complex dewetting scenarios captured by thin film models. Nature Materials 2: 59 (2003).CrossRefPubMedADSGoogle Scholar
  6. 6.
    R. Konrad, Master's thesis, Universität Ulm (2003).Google Scholar
  7. 7.
    L. D. Landau and E. M. Lifšic, Hydrodynamik, vol. VI of Lehrbuch der Theoretischen Physik 5th ed. (Akademie Verlag, 1991).Google Scholar
  8. 8.
    K. T. Mashiyama and H. Mori, Origin of the Landau-Lifshitz hydrodynamic fluctuations in nonequilibrium systems and a new method for reducing the Boltzmann equation. J. Stat. Phys. 18:385 (1978).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    D. Forster, D. R. Nelson and M. J. Stephen, Long-time tails and the large-eddy behavior of a randomly stirred fluid. Phys. Rev. Lett. 36:867 (1976).CrossRefADSGoogle Scholar
  10. 10.
    D. Forster, D. R. Nelson and M. J. Stephen, Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16:732 (1977).CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    P. C. Hohenberg and J. B. Swift, Effects of additive noise at the onset of Rayleigh-Bénard convection. Phys. Rev. A 46:4773 (1992).CrossRefPubMedADSGoogle Scholar
  12. 12.
    J. B. Swift, K. L. Babcock and P. C. Hohenberg, Effects of thermal noise in Taylor-Couette flow with corotation and axial through flow. Physica A 204:625 (1994).CrossRefADSGoogle Scholar
  13. 13.
    M. Moseler and U. Landman, Formation, stability, and breakup of nanojets. Science 289:1165 (2000).Google Scholar
  14. 14.
    D. G. A. L. Aarts, M. Schmidt and H. N. W. Lekkerkerker, Direct visual observation of thermal capillary waves. Science 304:847 (2004).CrossRefPubMedADSGoogle Scholar
  15. 15.
    S. Dietrich and M. Napiórkowski, Microscopic derivation of the effective interface Hamiltonian for liquid-vapor interfaces. Physica A 177:437 (1991).CrossRefADSGoogle Scholar
  16. 16.
    D. Blömker, S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation. Comm. Math. Phys. 223:553 (2001).CrossRefMATHADSMathSciNetGoogle Scholar
  17. 17.
    D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hillard models. Preprint, RWTH Aachen (2004).Google Scholar
  18. 18.
    C. Cardon-Weber, Cahn-Hilliard stochastic equation: existence of the solution and of its density. Bernoulli 7:777 (2001).MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation. Nonlinear Anal. TMA 26:241 (1995).CrossRefMathSciNetGoogle Scholar
  20. 20.
    H. Risken, The Fokker-Planck Equation, vol. 18 of Springer Series in Synergetics (Springer, Berlin, 1984).Google Scholar
  21. 21.
    C. W. Gardiner, Handbook of Stochastic Methods, vol. 13 of Springer Series in Synergetics (Springer, Berlin, 1983).Google Scholar
  22. 22.
    F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83:179-206 (1990).Google Scholar
  23. 23.
    A. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: regularity and long time behaviour of weak solutions. Comm. Pure Appl. Math. 49:85–123 (1996).CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation. Arch. Ration. Mech. Anal. 129:175–200 (1995).CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    G. Grün, Droplet spreading under weak slippage: existence for the Cauchy problem. Comm. Partial Diff. Equations 29:1697–1744 (2004).CrossRefMATHGoogle Scholar
  26. 26.
    A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Kortweg-de Vries equations. J. Funct. Anal. 169:532 (1999).CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    G. Tessitore and J. Zabczyk, Strict positivity for stochastic heat equations. Stochastic Processes Appl. 77:83 (1998).CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Num. Anal. 38:681 (2000).CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    A. Bertozzi, G. Grün and T. Witelski, Dewetting films: bifurcations and concentrations. Nonlinearity 14:1569 (2001).CrossRefMATHADSMathSciNetGoogle Scholar
  30. 30.
    N. Dirr and G. Grün, On the stochastic thin-film equation – an existence result, in preparation.Google Scholar
  31. 31.
    K. R. Mecke, R. Fetzer and M. Rauscher (2005), to be published.Google Scholar
  32. 32.
    G. Grün and M. Rumpf, Simulation of singularities and instabilities arising in thin film flow. Eur. J. Appl. Math. 12:293 (2001).CrossRefMATHGoogle Scholar
  33. 33.
    B. Davidovitch, E. Moro and H.A. Stone, Spreading of viscous fluid drops on a solid substrate assistd by thermal fluctuations. Phys. Rev. Lett. 95:244505 (2005).CrossRefPubMedADSGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Institut für Theoretische PhysikUniversität Erlangen-NürnbergErlangenGermany
  3. 3.Max-Planck-Institut für MetallforschungStuttgartGermany
  4. 4.ITAPUniversität StuttgartStuttgartGermany

Personalised recommendations