Leveling of a film with stratified viscosity and insoluble surfactant



The evolution of a film with insoluble surfactant on a wavy horizontal wall differs from flow without surfactant (the way it usually is studied) in that the film passes through different stages. The first stage is as if the surfactant were absent. Once the surface tension gradient—induced by the nonuniform surfactant concentration adsorbed at the free surface—starts resisting the flow effectively, the evolution enters a transitional stage. A final stage is reached once the free surface becomes rigid due to the surface-tension gradient (high elasticity limit) or becomes virtually leveled before the surface-tension gradient is released (low elasticity limit). The velocity profile through the film changes with time, sol fluid is depleted or accumulated at different strata in the film as the flow evolves. The velocity profile and resulting deformations throughout the film can be influenced significantly by the viscosity distribution or stratification, which occurs, for example, when multiple layers of different viscosity are coated simultaneously. A model and applications for the leveling of such a film are presented. The evolution is described in general terms for a film of uniform viscosity and for a film of two discrete layers of different viscosity. Then the three limiting cases are established. For two of these limits, the effect on the exponential decay rate of the flow and the deformation of the different strata or layers is examined when the viscosity is changed in an infinitesimally thin layer or stratum, and in a layer of finite thickness in films of two and three discrete layers.