Solvability condition for the moving contact line.

We consider the motion of a contact line between a fluid, gas, and solid, as it occurs when a drop advances over a solid surface. This motion is controlled by a microscopic length scale near the contact line, such as a slip length or the precursor thickness. The capillary profile inside the drop is linked to the contact line through an intermediate region which is characterized by an interface slope which varies logarithmically. The intermediate solution contains a single adjustable constant, which can be computed either by matching to the capillary region or to the contact line. We describe a simple method to perform the matching and to compute the required constant. This extends and/or simplifies results known previously. We apply our results to the case of a spreading drop in the presence of an interface potential and derive the equation of motion by combining the inner and outer expansions.