Stokes flows in a sessile hemispherical drop due to evaporation and surface tension gradient

TL;DR

This paper provides analytical solutions for viscous flows in evaporating sessile hemispherical droplets, highlighting the influence of surface tension gradients and boundary conditions on flow regimes and the critical Marangoni number.

Contribution

It introduces a theoretical framework linking evaporation, surface tension gradients, and boundary conditions to flow behavior in sessile droplets, including a new perspective on the critical Marangoni number.

Findings

Rigid relationship between evaporation rate and surface tension gradient under no-slip boundary conditions.

Critical Marangoni number governs transition from capillary to Marangoni convection.

Flow sensitivity to boundary conditions influences droplet evaporation dynamics.

Abstract

Viscous hydrodynamic flow in a small, slowly evaporating, sessile hemispherical droplet with a pinned contact line is considered. Analytical solutions are obtained for the Deegan outward flow, which is responsible for the coffee ring effect, as well as the Marangoni flow excited by a surface tension gradient. It is assumed that the surface tension gradient may be caused by anisotropic cooling of droplet surface or other factors, such as nonuniform illumination of an optically active surfactant. Two main types of boundary conditions, no-slip and full-slip, are considered in describing the flow-substrate interaction. It is shown that under the no-slip condition, there is a rigid relationship between the evaporation rate and the surface tension gradient, which imposes strict requirements on the temperature regime inside the droplet. This result offers a new vision of the critical Marangoni number, which describes the threshold for the transition of an evaporating droplet from capillary flow to developed Marangoni convection. The results of this work may attract the attention of experimenters to the study of the sensitivity of viscous flow in an evaporating droplet to the liquid-substrate boundary conditions, especially if the system under consideration passes into the Marangoni regime, when the no-slip condition changes to a partial or full slip condition due to the increase in viscous shear stress near the substrate.