Finite-amplitude instability of thin free and wetting films: prediction of lifetimes

Abstract

The stability of thin fibs to long-wavelength, albeit finite-amplitude, initial perturbations is investigated analytically. In contrast to the linear stability analysis for the infinitesimal disturbances, the hydrodynamicand the intermolecular interaction nonlinearities cannot be ignored in such an event. The key idea of the proposed formalism is to determine the stability of a spatially nonhomogeneous stationary solution of the governing equations. It is in the limit of vanishingly small amplitudes that the general stationary solution reduces to the trivial solution (undisturbed, planar interface), and thus the results of the linear stability analysis are recovered. The overall objective of the study is to determine the conditions for the instability, the dominant wavelength of the disturbances, and the time of rupture of the thin films, all as functions of thin-film parameters and the amplitude of the initial disturbances. This set of information is important in the design of flotation, foam, and emulsion systems as well as in the design of thin-film heat-transfer equipment. The analytical results are derived both for a free film with soluble surfactants and for a wetting film (film in contact with a solid) with surfactants present in both the film and the bounding fluid phase. In the case of a wetting film, the effects of the difference in the densities of the film and the bounding fluid (the cause of the Rayleigh-Taylor instability) are shown to be important. For most cases of practical interest, the nonlinear theory predicts a significantly faster rate of thinning and a shorter dominant wavelength compared to the linear theory, because it predicts a greater destabilization due to the van der Waals interaction and a lesser stabilizing influence of the surface tension restoring force. It is deduced that the stabilizing influences of the Marangoni motion (surface elasticity) and the surface viscosity are over- and underestimated by the linear theory in the case of the free and the wetting films, respectively.