Synergistic effects of disjoining pressure and surface viscosity on film drainage process

Abstract

A mathematical model is established to investigate the gravity-driven draining process of a vertical thin liquid film containing insoluble surfactants when considering the synergistic effect of surface viscosity and disjoining pressure. Lubrication theory is used to derive a coupled equation set describing the evolution of the film thickness, surfactant concentration and surface velocity. The equation set is solved numerically by the FreeFem program based on the finite element method. The film is assumed to be supported by the wire frame at both the top and bottom, thus the mass of liquid and the mass of total surfactants are conserved in the simulation. The characteristics of film evolution under the constant and variable surface viscosity are examined. Simulation results show that the surface viscosity is a crucial factor affecting the film drainage process. When neglecting the effect of surface viscosity, the film surface exhibits the mobile mode, while the film surface presents the rigid mode in the case of the surface viscosity considered. Increasing the surface viscosity, the rate of film drainage is slowed down significantly, leading to a reduction of the film thinning and enhancement of film stability, which is consistent with the results obtained by Naire et al. The disjoining pressure is a key factor in the formation of black film. When the disjoining pressure is only involved in the model, the length of the black film region is relatively short, nevertheless, if the effect of surface viscosity is only considered, a stable black film does not form. Under the synergistic effect of the disjoining pressure and surface viscosity, a very long and thin but stable black film is found in the middle segment of the film. Additionally, the thickness of black film increases and the appearance time postpones with the increase of surface viscosity. Considering the influence of concentration-dependent surface viscosity, the drainage rate is greatly affected. In the early stage, due to the smaller overall surface viscosity, the surface velocity is relatively large. With increasing surface viscosity at the bottom of film, the strength of the film surface tends to be enhanced, and then the anti-perturbation ability of the film is promoted and the film thinning is retarded. There is no significant difference in the length nor the appearance time of black film except that the thickness of black film with concentration-dependent surface viscosity is lower than that with the constant viscosity, thus the flow stability is weaker than that with the constant viscosity. In addition, the presence of the disjoining pressure slows down the thinning of blackest portion of the film and the surfactant concentration at this position. In the numerical results of the variable surface viscosity given by Braun et al. it is observed that the concentration of surfactant could almost be swept to clean in the top part of film. That is possibly because the effect of the disjoining pressure is neglected by Braun et al. It should be pointed out that the surface elasticity plays an important role in the stability of film. Therefore, it is necessary to consider the effect of surface elasticity in the future investigation.