Marangoni instabilities of cylindrical drops in a vertical Hele-Shaw cell immersed in stratified liquids

TL;DR

This study investigates the onset of Marangoni instabilities in cylindrical drops within vertical Hele-Shaw cells immersed in stratified liquids, revealing two distinct regimes based on drop size and developing a unifying scaling theory.

Contribution

The paper provides the first detailed experimental and theoretical analysis of Marangoni instability regimes in cylindrical drops in confined geometries, including a unifying scaling model.

Findings

Two instability regimes depending on drop radius.

Instability occurs when advection exceeds diffusion.

Scaling theory agrees with experimental observations.

Abstract

The Marangoni instability of cylindrical drops in vertical Hele-Shaw cells immersed in stably stratified liquids has been studied previously, yet the underlying mechanism has not been explored thoroughly. Here we study the onset of the Marangoni instability of such a system by experimentally explore the parameter space of the drop radius and concentration gradient. The concentration field is directly observed with laser interferometry. The flow is found to become unstable when advection is too strong for diffusion to maintain a stable concentration field. However, two different instability regimes are found depending on the drop radius. When the drop is small, the friction force caused by the two plates of the Hele-Shaw cell is small so that it does not change much the velocity field. Marangoni advection in such a regime can be very strong so that the entire periphery of the drop can become unstable. When the drop is large, the friction becomes so large that the Marangoni velocity plateaus and the boundary layer thickness is also reduced. The modified velocity and concentration fields lead to another instability regime, where only liquid close to the equator of the drop becomes unstable. A unifying scaling theory that includes both instability regimes is developed, which agrees well with the experimental results. Our findings may shed new light on the understandings of Marangoni flows in confined geometries.