Asymptotic Law of the Falling Liquid Film Form in Partial Wetting

The paper considers a problem of falling the partially wetting liquid across solid surface taking into account the disjoining pressure for liquid particles in its thin layers close to the three phase contact line. Within the frame of dropping liquid hydrodynamics it is easy to take into consideration the surface tension. On the other hand, with available surface tension, the attempts to fulfill Yung condition for a contact angle encounter truly insurmountable difficulties. Generally speaking, this problem has no solution within the limits of standard hydrodynamic theory of viscous liquid if the surface tension occurs and, at the same time, it is necessary to fulfill Yung condition for partial wetting. These problems can be solved if, instead of Yung condition, which is specified on the three phase contact line, the task is reformulated taking into account an additional chemical potential (or, in other words, the disjoining pressure) of the liquid particles in thin liquid layers close to the three-phase contact line. This approach allows us to formulate the appropriate problem of mathematical physics in the consistent closed-form. The unique dependence of the disjoining pressure on the liquid film thickness and the decline angle of the free surface leads to degeneration of hydrodynamics equations on the three phase contact line, on the one hand, and to fulfilling the Yung condition, on the other one. The paper demonstrates application of the developed theory in the context of calculating liquid surface form at different movement velocities. The asymptotic relationships obtained from the results of numerical experiments allow us to describe the variation law of the apparent 'dynamic' contact angle versus rate of the falling liquid from the wetted surface in the way suitable for calculations