Analytical solution of Stokes flow inside an evaporating sessile drop: Spherical and cylindrical cap shapes

TL;DR

This paper derives exact analytical solutions for the Stokes flow inside evaporating sessile drops with spherical and cylindrical caps, considering various contact angles, flux distributions, and wetting conditions, enhancing understanding of internal flow patterns during evaporation.

Contribution

It provides the first comprehensive analytical solutions for Stokes flow in evaporating sessile drops of both spherical and cylindrical shapes for arbitrary contact angles and flux conditions.

Findings

Flow patterns depend on contact angle and wetting conditions.

Viscous flows have streamlines farther from the substrate than inviscid flows.

Singular vapor flux at the contact line for certain evaporation conditions.

Abstract

Exact analytical solutions are derived for the Stokes flows within evaporating sessile drops of spherical and cylindrical cap shapes. The results are valid for arbitrary contact angle. Solutions are obtained for arbitrary evaporative flux distributions along the free surface as long as the flux is bounded at the contact line. The field equations, E^4(Psi)=0 and Del^4(Phi)=0, are solved for the spherical and cylindrical cap cases, respectively. Specific results and computations are presented for evaporation corresponding to uniform flux and to purely diffusive gas phase transport into an infinite ambient. Wetting and non-wetting contact angles are considered with the flow patterns in each case being illustrated. For the spherical cap with evaporation controlled by vapor phase diffusion, when the contact angle lies in the range 0<theta_c<pi, the mass flux of vapor becomes singular at the contact line. This condition required modification when solving for the liquid phase transport. Droplets in all of the above categories are considered for the following two cases: the contact lines are either pinned or free to move during evaporation. The present viscous flow behavior is compared to the inviscid flow behavior previously reported. It is seen that the streamlines for viscous flow lie farther from the substrate than the corresponding inviscid ones.