Potential flow inside an evaporating cylindrical line

An analytical solution to the problem of potential flow inside an evaporating line is obtained. The line is shaped as a half-cylinder lying on a substrate, and evaporates with either pinned or depinned contact lines. The solution is provided through the technique of separation of variables in the velocity potential and stream function formulations. Based on the flow field calculations, it is estimated that the coffee-stain phenomenon should be expected even for uniform evaporation flux throughout the cylindrical surface, provided that the contact lines remain anchored. A simple expression for the velocity potential is also suggested, which reproduces the local velocity vector with excellent accuracy. The vertically averaged velocity is calculated also for other contact line values, revealing for any value an outward liquid flow for pinned lines as opposed to inward flow for depinned lines.