Flows with a moving contact line

The problem of a two-dimensional viscous fluid drop which steadily moves along a horizontal rigid surface is considered. Such motion arises if the rigid surface wettability is nonuniform. A sequence of solutions for the velocity field and the free surface shape with the successively increasing applicability region near the moving contact lines is obtained for small capillary numbers Ca. 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 moving contact lines. In the solution obtained the free surface approaches the receding contact line at an angle of 180° and the advancing line at a zero angle. The solution is applicable up to the receding contact line and here approaches the known asymptotics. Near the advancing contact line the solution is applicable until the angle between the free surface and the rigid substrate becomes of the order of Ca^1/3.