Stokes flow inside an evaporating liquid line for any contact angle

Evaporation of droplets or liquid films lying on a substrate induces internal viscous flow, which affects the transport of suspended particles and, thus, the final deposit profile in numerous applications. In this work, the problem of Stokes flow inside a two-dimensional droplet, representing the cross section of an evaporating liquid line lying on a flat surface, is considered. The stream function formulation is adopted, leading to the biharmonic equation in bipolar coordinates. A solution in closed form is obtained for any contact angle in (0,pi) and is, thus, valid for both hydrophilic and hydrophobic substrates. The solution can be used with any type of evaporation mechanism, including diffusion, convection, or kinetically controlled modes. Both pinned and depinned contact lines are considered. For the boundary conditions to be compatible at the contact lines, the Navier slip boundary condition is applied on the substrate. Numerical results are presented for kinetically and diffusion controlled evaporation. For pinned contact lines, the flow inside the evaporating liquid line is directed towards the edges, thus, promoting the coffee stain phenomenon. In the case of depinned contact lines and contact angle less than pi/2 , the flow is directed towards the center of the droplet, whereas, for strongly hydrophobic substrates it is directed outwards.