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Thin-Film Flow Influenced by Thermal Noise

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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.

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References

  1. A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69:931 (1997).

    Article  ADS  Google Scholar 

  2. G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87:113 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Grün, On the convergence of entropy consistent for lubrication-type equations in multiple space dimensions. Math. Comp. 72:1251 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. K. R. Mecke, Integral geometry in statistical physics. Int. J. Mod. Phys. B 12: 861 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  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).

    Article  PubMed  ADS  Google Scholar 

  6. R. Konrad, Master's thesis, Universität Ulm (2003).

  7. L. D. Landau and E. M. Lifšic, Hydrodynamik, vol. VI of Lehrbuch der Theoretischen Physik 5th ed. (Akademie Verlag, 1991).

  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).

    Article  ADS  MathSciNet  Google Scholar 

  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).

    Article  ADS  Google Scholar 

  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).

    Article  ADS  MathSciNet  Google Scholar 

  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).

    Article  PubMed  ADS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

  13. M. Moseler and U. Landman, Formation, stability, and breakup of nanojets. Science 289:1165 (2000).

    Google Scholar 

  14. D. G. A. L. Aarts, M. Schmidt and H. N. W. Lekkerkerker, Direct visual observation of thermal capillary waves. Science 304:847 (2004).

    Article  PubMed  ADS  Google Scholar 

  15. S. Dietrich and M. Napiórkowski, Microscopic derivation of the effective interface Hamiltonian for liquid-vapor interfaces. Physica A 177:437 (1991).

    Article  ADS  Google Scholar 

  16. D. Blömker, S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation. Comm. Math. Phys. 223:553 (2001).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hillard models. Preprint, RWTH Aachen (2004).

  18. C. Cardon-Weber, Cahn-Hilliard stochastic equation: existence of the solution and of its density. Bernoulli 7:777 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation. Nonlinear Anal. TMA 26:241 (1995).

    Article  MathSciNet  Google Scholar 

  20. H. Risken, The Fokker-Planck Equation, vol. 18 of Springer Series in Synergetics (Springer, Berlin, 1984).

  21. C. W. Gardiner, Handbook of Stochastic Methods, vol. 13 of Springer Series in Synergetics (Springer, Berlin, 1983).

  22. F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83:179-206 (1990).

    Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  25. G. Grün, Droplet spreading under weak slippage: existence for the Cauchy problem. Comm. Partial Diff. Equations 29:1697–1744 (2004).

    Article  MATH  Google Scholar 

  26. A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Kortweg-de Vries equations. J. Funct. Anal. 169:532 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  27. G. Tessitore and J. Zabczyk, Strict positivity for stochastic heat equations. Stochastic Processes Appl. 77:83 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  28. J. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Num. Anal. 38:681 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  29. A. Bertozzi, G. Grün and T. Witelski, Dewetting films: bifurcations and concentrations. Nonlinearity 14:1569 (2001).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. N. Dirr and G. Grün, On the stochastic thin-film equation – an existence result, in preparation.

  31. K. R. Mecke, R. Fetzer and M. Rauscher (2005), to be published.

  32. G. Grün and M. Rumpf, Simulation of singularities and instabilities arising in thin film flow. Eur. J. Appl. Math. 12:293 (2001).

    Article  MATH  Google Scholar 

  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).

    Article  PubMed  ADS  Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Beringstr. 6, 53115, Bonn, Germany

    Günther Grün

  2. Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Staudtstr. 7/B3, 91058, Erlangen, Germany

    Klaus Mecke

  3. Max-Planck-Institut für Metallforschung, Heisenbergstr. 3, 70569, Stuttgart, Germany

    Markus Rauscher

  4. ITAP, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Markus Rauscher

Authors
  1. Günther Grün
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  2. Klaus Mecke
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  3. Markus Rauscher
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Correspondence to Günther Grün.

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Grün, G., Mecke, K. & Rauscher, M. Thin-Film Flow Influenced by Thermal Noise. J Stat Phys 122, 1261–1291 (2006). https://doi.org/10.1007/s10955-006-9028-8

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  • DOI: https://doi.org/10.1007/s10955-006-9028-8

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