Thermal Marangoni Convection in a Hemispherical Evaporating Liquid Droplet: Analytical Solution

Abstract

Quasi-stationary hydrodynamic flows have been studied in a small slowly evaporating hemispherical droplet with a pinned contact line on a flat horizontal substrate. Analytical solutions of the Stokes equations have been obtained and visualized for a capillary flow, which corresponds to the so-called “coffee-ring effect,” as well as for the Marangoni flow induced by a surface tension gradient associated with the cooling of the droplet surface due to the evaporation. The flow model corresponds to low Reynolds and Peclet numbers. The boundary conditions corresponding to the presence or absence of liquid slip at the substrate, which facilitate or hinder the Marangoni flow, have been analyzed. A hypothesis has been proposed that a critical Marangoni number corresponds to the system parameters that provide the separation of the flow from the substrate and the transition to the slip mode, in which the liquid velocity immediately at the substrate surface is nonzero. The obtained result can be used to study the flow structure and evaluate the flow rate in an evaporating microdroplet.