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

TL;DR

The study addresses Marangoni instabilities of cylindrical drops in a vertical Hele-Shaw cell within a stratified liquid, revealing two distinct instability mechanisms governed by advection–diffusion balance and confinement friction. A unifying scaling theory introduces the Marangoni number $Ma$ and Rayleigh number $Ra$, predicting a wrapping regime with $(Ma/Ra^{1/4})_{cr}\approx 170$ for large gradients and a local regime with $Ma_{cr}\approx 3490$ for small gradients; these are corroborated by phase-diagram data. The results show that confinement via the plate-induced friction, encapsulated in $k$, modifies boundary-layer thickness and velocity, giving rise to the two mechanisms and providing insight into confined Marangoni flows relevant to microfluidics and drop manipulation. Overall, the work advances understanding of Marangoni-driven transport in confined geometries and offers quantitative criteria for instability onset in practical applications.

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.

Figures (9)

• Figure 1: (a) Sketch of the experimental set-up (top view). The Hele-Shaw cell was formed by a quartz plate and the front side of a cubic glass container. The interference fringes formed by the reflections from the two inner surfaces of the Hele-Shaw cell -- surfaces ① and ② -- were recorded by a front view camera. The thickness of the Hele-Shaw cell is $d$. A 532nm laser (Laser 1) was used to generated the interference patterns. The two surfaces ① and ② were adjusted parallel before it was filled with linearly stratified ethanol-water mixture. The exact ethanol concentration at different hight of the mixture was measured by laser deflection using Laser 2. A 100cSt silicone oil drop was injected in the Hele-Shaw cell and its side view was recoded by camera 2. ($b$) The interference pattern after the two surfaces ① and ② have been adjusted parallel but before the stratified liquid was injected. ($c$) The interference fringes when the Hele-Shaw cell was filled with linearly stratified liquids. Fringes indicate isopycnals of the stratified liquid. ($d$) A sketch of the front view of the oil drop and the Hele-Shaw cell filled with stratified liquid. The height of the center of the drop is $y_0$. ($e$) A typical measured ethanol concentration $w_\mathrm{e}$ as a function of height $y$. ($f, g$) Typical front & side views of the oil drop. The contact angle of oil drop on surfaces ① and ② immersed in the stratified liquid is $\varphi$. The scale bar is 1mm. ($h$) Interfacial tension $\sigma$ of the 100cSt oil with the ethanol-water mixture at different ethanol concentrations.
• Figure 2: Snapshots of 100cSt silicone oil drops of different radii $R$ in linearly stratified ethanol–water mixtures with different ethanol concentration gradients $\mathrm{d}w_\mathrm{e}/\mathrm{d}y$. The snapshots start from some time $t_1$ (or $t_2$ or $t_3$) after the linear stratification has been generated. Point A is the intersection of a horizontal line passing through the drop's center and a dark fringe (labelled No. 1) right above this horizontal line. The isopycnals close to the drop all bend down because of the downwards Marangoni flow. (a) $R=0.26mm$, $\mathrm{d}w_\mathrm{e}/\mathrm{d}y=98.9m^{-1}$, $t_1\approx1600s$. The scale bar is 0.5mm. (b) $R=0.57mm$, $\mathrm{d}w_\mathrm{e}/\mathrm{d}y=107.9m^{-1}$, $t_2\approx1500s$. The oscillation period is $T=39.1s$. The scale bar is 1mm. ($c$) $R = 1.71mm$, $\mathrm{d}w_\mathrm{e}/\mathrm{d}y=44.5m^{-1}$, $t_3\approx1800\,\text{s}$. Zoomed views of the fringes close to the drop within an oscillation period $T=17s$ are show in the dashed boxes. The fringes at the equator (red box) oscillates but the fringes above (magenta box) are stable. (d) Temporal variation of the horizontal position $X$ of point A.
• Figure 3: The variation of horizontal position $X$ with time $t$ for drops of different radii in similar concentration gradients: (a) $\mathrm{d}w_\mathrm{e}/\mathrm{d}y\approx70m^{-1}$, (b) $\mathrm{d}w_\mathrm{e}/\mathrm{d}y\approx50m^{-1}$.
• Figure 4: Phase diagram of the 100cSt drops with a drop radius $R$ versus concentration gradient $\mathrm{d}w_\mathrm{e}/\mathrm{d}y$ parameter space. Black triangles stand for unstable situations and red circles for stable situations.
• Figure 5: (a) Streamlines and concentration field of a drop immersed in a stably stratified liquid in the ideal case, i.e., when $\mathrm{d}w_\mathrm{e}/\mathrm{d}y$ is very small so that the effect of gravity and advection (as compared to diffusion) are both negligible. Black lines represent streamlines and color strips represent isopycnals. (b) Interference pattern for a drop of $R=0.26mm$ at concentration gradient $\mathrm{d}w_\mathrm{e}/{\mathrm{d}y=98.9m^{-1}}$. The flow is stable. Also shown is the definition of coordinates and physical properties. The plane polar coordinate $(r, \theta)$ has its origin at the center of the drop. The cartesian coordinate $y$ is pointing upwards and gravity $g$ is pointing downwards. The thickness of the concentration boundary layer is $\delta_c$.