On the fingering instability of a simultaneous thermocapillary and solutocapillary driven droplet

Abstract

We study the fingering instability in a droplet simultaneously induced to spread by a surfactant and temperature. The use of the lubrication approximation yields coupled evolution equations for the film thickness, surfactant concentration, and temperature. A direct numerical simulation is performed, and a stability analysis based on the disturbance energy is conducted. Four cases are considered for the substrate temperature field: a nonheated substrate, an isothermally heated substrate, a nonisothermally heated substrate, and a thick substrate. The results show that fluids always tend to “flee” from hotter areas and surfactant-enriched areas, and that the flow stability is greatly influenced by this effect. The uneven distribution of the velocity field caused by surface tension is the fundamental reason for the formation of fingerlike patterns. The contributions of the capillary effect, the solutocapillary effect, and the thermocapillary effect as driving forces are quantified in terms of their locations and relative strength during spreading. The solutocapillary and thermocapillary effects exert a destabilizing effect on the spreading. On a nonisothermally heated substrate, a stronger thermocapillary effect strengthens the unevenness of the surfactant, leading to the most unstable flow. Finally, a variable viscosity model is considered and the flow stability is examined. The results show that on a nonisothermally heated substrate, the unevenness of the surfactant and temperature distribution is strengthened due to better fluidity in hotter areas, leading to a more unstable flow. On an isothermally heated substrate, the overall liquidity increases the spreading velocity but does not affect the stability.