WIAS Preprint No. 1883, (2013)

Stability analysis of non-constant base states in thin film equations



Authors

  • Dziwnik, Marion
  • Korzec, Maciek D.
  • Münch, Andreas
  • Wagner, Barbara

2010 Mathematics Subject Classification

  • 35B40 35C20 76M45

2008 Physics and Astronomy Classification Scheme

  • 81.07.-b

Keywords

  • multiple-scale methods, stability analysis, rim instability, free boundaries, dewetting film

Abstract

We address the linear stability of non-constant base states within the class of mass conserving free boundary problems for degenerate and non-degenerate thin film equations. Well-known examples are the finger-instabilities of growing rims that appear in retracting thin solid and liquid films. Since the base states are time dependent and do not have a simple travelling wave or self-similar form, a classical eigenvalue analysis fails to provide the dominant wavelength of the instability. However, the initial fronts evolve on a slower time-scale than the typical perturbations. We exploit this time-scale separation and develop a multiple-scale approach for this class of stability problems. We show that the value of the dominant wavelength is rapidly attained once the base state has entered an approximately self-similar scaling. We note that this value is different from the one obtained by the linear stability analysis with "frozen modes", frequently found in the literature. Furthermore we show that for the present class of stability problems the dispersion relation behaves linear for large wavelengths, which is in contrast to many other instability problems in thin film flows.

Appeared in

  • with the title : ``Stability analysis of unsteady, nonuniform base states in thin film equations'', Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014) pp. 755--780.

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