Stokes flow in a drop evaporating from a liquid subphase

TL;DR

This paper models the flow fields in a two-liquid system during drop evaporation, deriving self-similar solutions near the contact line and analyzing how flow contributions depend on contact angles and viscosity ratios.

Contribution

It provides a novel analytical framework for understanding coupled flow dynamics in evaporating drops on liquid subphases, focusing on the near-contact-line behavior.

Findings

Flow driven by evaporation dominates for contact angles > 90°.

Moving contact-line flow dominates for angles < 90°.

Eigenmode flow is asymptotically negligible.

Abstract

The evaporation of a drop from a liquid subphase is investigated. The two liquids are immiscible, and the contact angles between them are given by the Neumann construction. The evaporation of the drop gives rise to flows in both liquids, which are coupled by the continuity of velocity and shear-stress conditions. We derive self-similar solutions to the velocity fields in both liquids close to the three-phase contact line, where the drop geometry can be approximated by a wedge. We focus on the case where Marangoni stresses are negligible, for which the flow field consists of three contributions: flow driven by the evaporative flux from the drop surface, flow induced by the receding motion of the contact line, and an eigenmode flow that satisfies the homogeneous boundary conditions. The eigenmode flow is asymptotically subdominant for all contact angles. The moving contact-line flow dominates when the angle between the liquid drop and the horizontal surface of the liquid subphase is smaller than $90^\circ$, while the evaporative-flux driven flow dominates for larger angles. A parametric study is performed to show how the velocity fields in the two liquids depend on the contact angles between the liquids and their viscosity ratio.