Marangoni instabilities of drops of different viscosities in stratified liquids

TL;DR

This study investigates the mechanisms triggering Marangoni flow instabilities in oil drops of varying viscosities within stratified liquids, revealing a new viscosity-driven instability at high viscosities and developing a unifying scaling theory.

Contribution

The paper introduces a new instability mechanism for high-viscosity drops and unifies low and high viscosity behaviors through a scaling theory based on two length scales.

Findings

High-viscosity drops exhibit a viscosity-driven instability.

A critical drop radius determines stability for high-viscosity drops.

The unifying theory accurately predicts experimental results across viscosities.

Abstract

For an immiscible oil drop immersed in a stably stratified ethanol-water mixture, a downwards solutal Marangoni flow is generated on the surface of the drop, owing to the concentration gradient, and the resulting propulsion competes against the downwards gravitational acceleration of the heavy drop. In prior work of Li et al. (Phys. Rev. Lett., vol. 126, issue 12, 2021, 124502), we found that for drops of low viscosity, an oscillatory instability of the Marangoni flow is triggered once the Marangoni advection is too strong for diffusion to restore the stratified concentration field around the drop. Here we experimentally explore the parameter space of the concentration gradient and drop radius for large oil viscosities and find a different and new mechanism for triggering the oscillatory instability in which diffusion is no longer the limiting factor. For such drops of higher viscosities, the instability is triggered when the gravitational effect is too strong so that the viscous stress cannot maintain a stable Marangoni flow. This leads to a critical drop radius above which the equilibrium is always unstable. Subsequently, a unifying scaling theory that includes both the mechanisms for low and for high viscosities of the oil drops is developed. The transition between the two mechanisms is found to be controlled by two length scales: The drop radius $R$ and the boundary layer thickness $\delta$ of the Marangoni flow around the drop. The instability is dominated by diffusion for $\delta<R$ and by viscosity for $R<\delta$. The experimental results for various drops of different viscosities can well be described with this unifying scaling theory. Our theoretical description thus provides a unifying view of physicochemical hydrodynamic problems in which the Marangoni stress is competing with a stable stratification.