Role of natural convection in the dissolution of sessile droplets

TL;DR

This study investigates how natural convection influences the dissolution of small sessile alcohol droplets in water, revealing universal scaling laws for mass transfer and flow dynamics in the convective regime.

Contribution

It identifies universal scaling relations for Sherwood and Reynolds numbers in convective dissolution, linking buoyancy-driven flow to droplet size and solubility.

Findings

Universal Sherwood-Rayleigh scaling relation ($Sh \,\sim\, Ra^{1/4}$)

Convective dissolution time scales as $\tau_c \sim R^{5/4}$

Plume Reynolds number scales as $Re_p \sim Sc^{-1} Ra^{5/8}$

Abstract

The dissolution process of small (initial (equivalent) radius $R_0 < 1$ mm) long-chain alcohol (of various types) sessile droplets in water is studied, disentangling diffusive and convective contributions. The latter can arise for high solubilities of the alcohol, as the density of the alcohol-water mixture is then considerably less as that of pure water, giving rise to buoyancy driven convection. The convective flow around the droplets is measured, using micro-particle image velocimetry ($\mu$PIV) and the schlieren technique. When nondimensionalizing the system, we find a universal $Sh\sim Ra^{1/4}$ scaling relation for all alcohols (of different solubilities) and all droplets in the convective regime. Here Sh is the Sherwood number (dimensionless mass flux) and Ra the Rayleigh number (dimensionless density difference between clean and alcohol-saturated water). This scaling implies the scaling relation $\tau_c \sim R^{5/4}$ of the convective dissolution time $\tau_c$, which is found to agree with experimental data. We show that in the convective regime the plume Reynolds number (the dimensionless velocity) of the detaching alcohol-saturated plume follows $Re_p \sim Sc^{-1} Ra^{5/8}$, which is confirmed by the $\mu$PIV data. Here, Sc is the Schmidt number. The convective regime exists when $Ra > Ra_t$, where $Ra_t = 12$ is the transition Ra-number as extracted from the data. For $Ra < Ra_t$ and smaller, convective transport is progressively overtaken by diffusion and the above scaling relations break down.