The destabilization of a thin three-dimensional non-wetting film above a solid wall is examined for the special case in which surfactant is adsorbed onto the free surface of the film. Attention is restricted to the case of a Newtonian surface, with surfactant displaying rapid surface diffusion or exhibiting small Marangoni number, such that the dominant intrinsic interfacial stress is of a purely viscous origin. A surface-excess force approach is adopted for the purpose of incorporating into the analysis the attractive/repulsive dispersive forces acting between the solid wall and the film. Three coupled nonlinear partial differential equations are obtained that describe the ‘large-wavelength’ spatio-temporal evolution of the free film surface following a small initial disturbance. These equations are shown to reduce to results in the literature in the limit of zero interfacial viscosities. Employing linear stability analysis, an explicit dispersion equation is obtained relating the growth coefficient to interfacial viscosities. It is found, at least in the linear regime, that the sum of interfacial shear and dilatational viscosities – and not each separately – imparts a damping effect that in the most extreme case is four-fold relative to the case of no interfacial viscosities. Nonlinear stability analysis in the limiting case of a two-dimensional film indicates that interfacial viscosities may strongly hinder the onset of instability through large interfacial stresses that arise in the vicinity of trough and crest regions of the film.
Controlling fingering instabilities in Hele-Shaw flows in the presence of wetting film effects
