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Preprint 34-2013

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

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Author(s) : Marion Dziwnik , Maciek Korzec , Andreas Münch, Barbara Wagner

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 34-2013

MSC 2000

76M45 Asymptotic methods, singular perturbations
76E17 Interfacial stability and instability
35B40 Asymptotic behavior of solutions
35C20 Asymptotic expansions

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.

Keywords : multiple-scale methods, stability analysis, rim instability, free boundaries, dewetting films

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