Stability Analysis of Unsteady, Nonuniform Base States in Thin Film Equations

Abstract

We address the linear stability of unsteady and nonuniform base states within the class of mass conserving free boundary problems for degenerate and nondegenerate 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 traveling wave or exact 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 asymptotically self-similar form. 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 that the dispersion relation is linear in the long wave limit, which is in contrast to many other instability problems in thin film flows.