On application of lubrication approximations to nonunidirectional coating flows with clean and surfactant interfaces

Abstract

In this work, the characteristic properties of the lubrication approximation are studied and its weak ellipticity is established, in contradistinction to the commonly accepted parabolic character of the lubrication equations resulting from the underlying unidirectional flow assumption. The weak ellipticity property allows the lubrication analysis to capture flow topologies around stagnation points, contact lines, and flows over edges, all of which normally require elliptic operators to be accounted for. This is used to explain the empirically observed overperformance of the lubrication approximation from the perspective of characteristic analysis. While the analysis is developed in the context of the classical Landau–Levich problem of dip-coating, which is known to possess an interfacial stagnation point both in the clean and surfactant interface cases, the analysis is general since the Landau–Levich equation is common to many other lubrication problems. The analytical approach presented here when applied to the surfactant interface case, also allows one to establish a new physical result: a variation of the bulk surfactant concentration is the necessary condition for the film thickening phenomenon in the Landau–Levich problem to occur due to surfactant-induced Marangoni effects.