Marangoni Instability of a Drop in a Stably Stratified Liquid

Marangoni instabilities can emerge when a liquid interface is subjected to a concentration or temperature gradient. It is generally believed that for these instabilities bulk effects like buoyancy are negligible compared to interfacial forces, especially on small scales. Consequently, the effect of a stable stratification on the Marangoni instability has hitherto been ignored. Here, however, we show that they can matter. We report, for an immiscible drop immersed in a stably stratified ethanol-water mixture, a new type of oscillatory solutal Marangoni instability that is triggered once the stratification has reached a critical value. We experimentally explore the parameter space spanned by the stratification strength and the drop size and theoretically explain the observed crossover from levitating to bouncing by balancing the advection and diffusion around the drop. Finally, the effect of the stable stratification on the Marangoni instability is surprisingly strongly amplified in confined geometries, leading to an earlier onset.

Figure 1

Bouncing and levitating drops in a linearly and stably stratified mixture of ethanol (lighter) and water (heavier). (a) Snapshots of two 5 cSt silicone oil drops at the given time after they were released in the mixture. The larger drop bounces at $h<3\mathrm{mm}$, while the smaller drop is levitating at a higher position $h\approx 8.7\mathrm{mm}$. The snapshots are taken from one experiment with two drops. To better show them, the upper and lower halves of the snapshots are shown with different scales. (b) Drop height $h$ as function of time $t$ for different drop radii $R$ after the initial sinking period. $h=0$ is the position where the density of the drop equals that of the mixture. The filled circles represent the relative size of the drops. (c) Flow field around a levitating drop ($R=31\pm 1\mu \mathrm{m}$), measured by particle image velocimetry, in a mixture with $d{w}_e/dy\approx 5{\mathrm{m}}^{-1}$. The resolution is not high enough to resolve the velocity close to the drop’s surface. (d) The ethanol weight fraction $w_e$ at the corresponding height, with $d{w}_e/dy\approx 5{\mathrm{m}}^{-1}$. (e) A sketch of the levitating drop (with radius $R$ and density ${\rho }^{\prime }$) and the ethanol concentration around it. Deeper red means higher ethanol concentration. The shaded ring inside the dashed circle represents the kinematic boundary layer with thickness $\delta$, set by the Marangoni velocity $V_M$. The ethanol concentration inside this layer is enhanced and homogenized by Marangoni advection bringing down the ethanol-rich liquid. The Marangoni flow is represented by the solid arrows. Dashed arrows represent diffusion across this layer. $\rho$ is the representative density inside this layer, and ${\rho }^{*}$ is the undisturbed density in the far field. $\mu$ and ${\mu }^{\prime }$ are the viscosities of the mixture and the drop, respectively. (f) Interfacial tension $\sigma (w_e)$ between 5 cSt silicone oil and the ethanol-water mixture. Each point is an average of six measurements, and the error bar is the standard deviation. The solid line is a polynomial fit to the data points.

Figure 2

Phase diagram of the levitating and bouncing drops in the parameter space of drop radius $R$ vs concentration gradient $d{w}_e/dy$. Black triangles stand for bouncing drops, red circles for levitating ones. Measurement errors in the $x$ direction are comparable to the size of the symbols.

Figure 3

(a) Phase diagram replotted in dimensionless numbers: $Ma/R{a}^{1/2}$ vs $Ra$. Black triangles stand for bouncing drops, red circles for levitating ones. The blue line is the instability threshold $(Ma/{Ra}^{1/2}{)}_{\mathrm{cr}}=275$, above which the flow is oscillatory and all drops bounce. The blue solid line (in the range $6\times {10}^{-3}<Ra<3$) is confirmed by experiments. Measurement errors in the $y$ direction are comparable to the size of the symbols. (b) Phase diagram replotted with $d{w}_e/dy$ vs $w_e$, where $w_e$ is the ethanol weight fraction at the levitation height. The blue curve is calculated from Eq. (6) with $c=275$. The dashed blue line in the range $w_e>98\mathrm{wt}\%$ ($w_e<50\mathrm{wt}\%$) corresponds to $Ra<6\times {10}^{-3}$ ($Ra>3$). Measurement errors are comparable to the size of the symbols.