Hydrodynamics singularities associated with a static or moving three-phase contact line

Abstract

Stationary flows of viscous fluids with a static or moving contact line are considered. Contact line separates three phases: vapor, liquid and solid. No-slip condition on the solid-fluid surface and ordinary conditions with account constant surface tension for the fluid-vapor interface are supposed to be fulfilled. The flows presented in the report are induced by some physical mechanism concentrated in the very small region near the contact line. Such contact line is the origin of the flow and treated as a hydrodynamics singularity. As an example the flow in a two- dimensional viscous fluid drop which rests or steadily moves along a horizontal solid surface is considered. Motions of this type can be observed in experiments if the solid-fluid surface wettability is non-uniform. A sequence of solutions for the velocity field and the free surface shape with the successively increasing applicability region near the static or moving contact lines is obtained. At first stage the solution of the problem is found in the case when the distortion of the free surface of the drop during motion can be neglected. The problem is then reformulated using functions of a complex variable and expanded variables are introduced. In the new variables a more accurate solution of the same problem is found, with a much more narrow inapplicability region near the contact lines. Asymptotic behavior of the flow near the contact lines is discussed.