# Marangoni instability of a drop in a stably stratified liquid

Yanshen Li<EMAIL_ADDRESS>Physics of Fluids group, Max-Planck Center
Twente for Complex Fluid Dynamics, Department of Science and Technology, Mesa+
Institute, and J. M. Burgers Centre for Fluid Dynamics, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands  Christian Diddens  Physics of
Fluids group, Max-Planck Center Twente for Complex Fluid Dynamics, Department
of Science and Technology, Mesa+ Institute, and J. M. Burgers Centre for Fluid
Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The
Netherlands  Department of Mechanical Engineering, Eindhoven University of
Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands  Andrea
Prosperetti  Physics of Fluids group, Max-Planck Center Twente for Complex
Fluid Dynamics, Department of Science and Technology, Mesa+ Institute, and J.
M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500
AE Enschede, The Netherlands  Department of Mechanical Engineering, University
of Houston, Texas 77204-4006, USA  Detlef Lohse<EMAIL_ADDRESS>Physics of
Fluids group, Max-Planck Center Twente for Complex Fluid Dynamics, Department
of Science and Technology, Mesa+ Institute, and J. M. Burgers Centre for Fluid
Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The
Netherlands  Max Planck Institute for Dynamics and Self-Organization, Am
Faßberg 17, 37077 Göttingen, Germany

###### Abstract

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 as compared to
interfacial forces, especially on small scales. Consequently, the effect of a
stable stratification on the Marangoni instability has hitherto been ignored.
Here we report, for an immiscible drop immersed in a stably stratified
ethanol-water mixture, a new type of oscillatory solutal Marangoni instability
which 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 amplified in confined geometries, leading to an
earlier onset.

A concentration or temperature gradient applied to an interface can induce a
Marangoni instability of the motionless state, resulting in a steady
convection. Similarly, the steady state Marangoni convection can undergo
another instability, leading to an oscillatory motion rednikov1998two . Since
the first quantitative analysis in 1958 pearson1958convection , Marangoni
instabilities have been studied extensively due to their relevance for liquid
extraction sternling1959interfacial ; groothuis1960influence ;
rother1999effect ; berejnov2002spontaneous ; jain2011recent , coating
techniques pearson1958convection ; yarin1995surface ; demekhin2006suppressing
, metal processing gupta1992pore ; ratke2005theoretical ; zhang2006indirect
and crystal growth schwabe1978experiments ; schwabe1979some ;
chang1979thermocapillary ; chun1979experiments ; schwabe1982studies ;
preisser1983steady ; kamotani1984oscillatory , etc. Marangoni instabilities
are also the main mechanism to drive the self-propulsion of active drops
rednikov1994active ; rednikov1994drop ; herminghaus2014interfacial ;
yoshinaga2014spontaneous ; ryazantsev2017thermo ; maass2016swimming ;
morozov2019self , which have attracted lots of recent interest. Such drops are
an example of the rich physicochemical hydrodynamics of droplets far from
equilibrium lohse2020physicochemical  which are very relevant for food
processing degner2013influence ; degner2014factors  and modelling biological
systems maass2016swimming , etc.

Depending on the application, Marangoni instabilities have been investigated
in different configurations, such as a horizontal interface between two fluid
layers pearson1958convection ; sternling1959interfacial ;
reichenbach1981linear ; takashima1981surface ; levchenko1981instability ;
nepomnyashchii1983thermocapillary ; chu1988sustained ; chu1989transverse ;
hennenberg1992transverse ; rednikov1998two , the surface of a falling film on
a tilted plate nepomnyashchy1976wavy ; chang1994wave ; kliakhandler1997viscous
; miladinova2005effects ; demekhin2006suppressing , a vertical interface of a
liquid column schwabe1978experiments ; schwabe1979some ;
chang1979thermocapillary ; chun1979experiments ; schwabe1982studies ;
preisser1983steady ; kamotani1984oscillatory , and for drops submerged in a
solution rednikov1994active ; rednikov1994drop ; herminghaus2014interfacial ;
yoshinaga2014spontaneous ; ryazantsev2017thermo ; maass2016swimming ;
morozov2019self ; thanasukarn2004impact ; ghosh2008factors ;
degner2013influence ; degner2014factors ; dedovets2018five , etc. In many of
these situations, these systems are subjected to a stabilizing
temperature/concentration gradient takashima1981surface ;
levchenko1981instability ; demekhin2006suppressing ; schwabe1978experiments ;
schwabe1979some ; chang1979thermocapillary ; chun1979experiments ;
schwabe1982studies ; preisser1983steady ; kamotani1984oscillatory ;
chu1988sustained ; chu1989transverse , which induces a continuously stable
density stratification. However, except for a few cases for the horizontal
interface configuration welander1964convective ; wierschem2000internal ;
rednikov2000rayleigh , the effect of such a stable density stratification on
Marangoni convection has always been ignored, due to the generally accepted
view that on small scales bulk effects like buoyancy are negligible
nepomnyashchy2012interfacial . Here we report, for an immiscible drop immersed
in an ethanol-water mixture, that the stable stratification could actually
trigger an oscillatory instability once it is above a critical value.
Surprisingly, this critical value will decrease in a confined geometry,
implying that the effect of the stable stratification is actually amplified on
small scales. Our findings demonstrate that stable stratification can strongly
affect Marangoni convection and ask for further studies in related geometries.

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\text{\,}\mathrm{m}\mathrm{m}$$, while the
smaller drop is levitating at a higher position
$h\approx$8.7\text{\,}\mathrm{m}\mathrm{m}$$. The snapshots are taken from one
experiment with two drops. To better show them, the upper/lower half of the
snapshots are shown with different scales. (b) Drop’s height $h$ as functions
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\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$) measured by PIV, in a mixture with
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y\approx$5\text{\,}\mathrm{m}^{-1}$$. The
resolution is not high enough to resolve the velocity close to the drop’s
surface. (d) The ethanol weight fraction $w_{\mathrm{e}}$ at the corresponding
height, with
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y\approx$5\text{\,}\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 $V_{\mathrm{M}}$. The ethanol concentration inside this layer is
enhanced & homogenized by Maragnoni 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 $\sigma(w_{\mathrm{e}})$ between
$5\text{\,}\mathrm{c}\mathrm{S}\mathrm{t}$ 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.

To determine the onset of the Marangoni instability, we experimentally explore
the parameter space spanned by the concentration gradient and the drop radius
$R$. Using the double-bucket method oster1965density , linearly stratified
liquid mixtures are prepared in a cubic glass container (Hellma, 704.001-OG,
Germany) with inner width of $L=$30\text{\,}\mathrm{m}\mathrm{m}$$ filled to
different depth, depending on the degree of stratification. The ethanol weight
fraction $w_{\mathrm{e}}$ at each height is measured by laser deflection
lin2013one ; li2019bouncing , from which the gradient of ethanol weight
fraction $\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y$ is calculated. The
concentration gradient $\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y$ is varied from
$\sim$3\text{\,}\mathrm{m}^{-1}$$ to $\sim$130\text{\,}\mathrm{m}^{-1}$$,
corresponding to density gradients ranging from
$-480\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{4}$ to
$-4200\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{4}$.
$5\text{\,}\mathrm{c}\mathrm{S}\mathrm{t}$ Silicone oil (Sigma-Aldrich,
Germany) is injected through a thin needle (with outer-diameter
$0.515\text{\,}\mathrm{m}\mathrm{m}$) to generate drops of different radii
$R$. The drops are released from the top of the stratified mixtures, and their
trajectories are recorded by a sideview camera. During the measurements, only
one single drop exists in the container at a time. The silicone oil has
density
$\rho^{\prime}=$913\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$$ and
viscosity $\mu^{\prime}=$4.6\text{\,}\mathrm{m}\mathrm{Pa}\cdot\mathrm{s}$$.

Two typical behaviors are observed after the initial sinking phase. See Fig.
1(a) for the successive snapshots of two silicone oil drops in a mixture with
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y\approx$5\text{\,}\mathrm{m}^{-1}$$:
While a smaller drop ($R=69\pm$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$)
stays at a fixed position around $h\approx$8.7\text{\,}\mathrm{m}\mathrm{m}$$,
a larger drop ($R=454\pm$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) bounces
continuously in the range
$$0\text{\,}\mathrm{m}\mathrm{m}$<h<$3\text{\,}\mathrm{m}\mathrm{m}$$. Here
$h=0$ marks the position where the density of the oil
($\rho^{\prime}=$913\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$$)
equals that of the mixture (at $w_{\mathrm{e}}\approx$49\text{\,}\%$$). The
drop’s position $h(t)$ as a function of time $t$ in the same stratified liquid
and the ethanol weight fraction $w_{\mathrm{e}}$ at the corresponding height
are respectively shown in Fig. 1(b) and (d). The smallest drop
($R_{1}\approx$44\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) is levitating at
$h\approx$9.1\text{\,}\mathrm{m}\mathrm{m}$$. As the drop size increases, it
levitates at a lower position, until above a critical radius $R_{\mathrm{cr}}$
it starts to bounce instead of levitating. If its size is further increased,
the drop bounces around a lower position (but still with $h>0$).

The smaller drops are able to levitate above the density matched position
$h=0$ against gravity because of a stable Marangoni flow around it, as shown
in Fig. 1(c). The flow field is obtained by PIV measurements for a drop
levitating in the gradient
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y\approx$5\text{\,}\mathrm{m}^{-1}$$. The
interfacial tension of the drop $\sigma$ decreases with increasing ethanol
concentration of the mixture $w_{\mathrm{e}}$, as shown in Fig. 1(f). This
interfacial tension gradient at the drop’s surface pulls liquid downwards,
generating a viscous force acting against gravity, which levitates the drop.
When the drop becomes large enough, however, the equilibrium becomes
oscillatory, and the drop starts to bounce between two different levels. Thus,
the transition from a levitating drop to a bouncing one signals the onset of
the instability.

While exploring the parameter space, we use an easily distinguishable
criterion to determine whether a drop is bouncing: If the drop’s bouncing
amplitude $h_{\mathrm{A}}$ is larger than its radius $R$, then the drop is
considered to be bouncing (see Supplemental Material for more details). The
results are shown in Fig. 2. Surprisingly, while for weak gradients (like for
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y\approx$10\text{\,}\mathrm{m}^{-1}$$)
there is a critical radius $R_{\mathrm{cr}}$
($\approx$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) above which the
Marangoni flow becomes unstable, the Marangoni flow is always unstable for
stronger gradients
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y>$23\text{\,}\mathrm{m}^{-1}$$ in all
performed experiments. Note that for larger drops
($R>$0.1\text{\,}\mathrm{m}\mathrm{m}$$), we could not explore the full
parameter space for
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y<$3\text{\,}\mathrm{m}^{-1}$$ since it
would require an unrealistically large container.

Figure 2: Phase diagram of the levitating & bouncing drops in the parameter
space of drop radius $R$ vs. concentration gradient
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y$. Black triangles stand for bouncing
drops, red circles for levitating ones. Measurement errors in the $x$
direction are comparable with the size of the symbols.

To get a better understanding of the onset of this Marangoni instability, the
key is to understand the coupling between the Marangoni flow and the
concentration field: The Marangoni flow is induced by the ethanol (solute)
concentration gradient around the drop, which is subjected to change by
advection (caused by the Marangoni flow itself) and diffusion, see the sketch
in Fig. 1(e). The Marangoni flow tends to homogenize the ethanol concentration
around the drop, thus weakening the Marangoni flow force and thus itself. At
the same time, diffusion acts to restore the ethanol gradient in the vicinity
of the drop to its undisturbed value, i.e., the value it takes in the far
field. This competition between advection and diffusion around the drop
determines whether the Marangoni flow is stable or not. Furthermore, once it
becomes unstable, a temporarily strong Marangoni flow homogenizes the
concentration field around the drop, consequently weakening itself. Later the
Marangoni flow restarts once diffusion has restored the concentration field
around the drop, so that the flow is oscillatory and leads to the continuous
bouncing of the drop.

The liquid layer whose concentration is affected by the Marangoni advection is
effectively the Marangoni flow boundary layer with thickness $\delta$ (see
Fig. 1(e)). The time scale for advection to change the concentration in this
layer is the time needed for the Marangoni flow to bring down the ethanol-rich
liquid from the top: $\tau_{\mathrm{a}}\sim R/V_{\mathrm{M}}$, where
$V_{\mathrm{M}}$ is the Marangoni flow velocity at the equator of the drop.
For the drop in the concentration gradient it holds young1959motion  (see
Supplementary Material) $V_{\mathrm{M}}\sim-\mathrm{d}\sigma/\mathrm{d}y\cdot
R/(\mu+\mu^{\prime})$. The time scale for diffusion to restore the
concentration across this layer is $\tau_{\mathrm{d}}\sim\delta^{2}/D$, where
$D$ is the diffusivity of ethanol in water. The flow will become unstable when
advection is faster than diffusion, $\tau_{\mathrm{a}}<\tau_{\mathrm{d}}$.
Substituting the two time scales into this relation, we obtain
${V_{\mathrm{M}}R}/{D}>{R^{2}}/{\delta^{2}}$. The left hand side has the form
of a Péclet number, which is the ratio between advection and diffusion, and
which in problems of this type is referred to as the Marangoni number

$Ma=\frac{V_{\mathrm{M}}R}{D}=-\frac{\mathrm{d}\sigma}{\mathrm{d}w_{\mathrm{e}}}\frac{\mathrm{d}w_{\mathrm{e}}}{\mathrm{d}y}R^{2}\cdot\frac{1}{(\mu+\mu^{\prime})D},$
(1)

where we have used above expression for $V_{\mathrm{M}}$ with an equal sign
and where $\mathrm{d}\sigma/\mathrm{d}w_{\mathrm{e}}$ is a material property
(see Fig. 1(f)) and $\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y$ the undisturbed
ethanol gradient of the mixture. The instability criterion thus is

$Ma>{R^{2}}/{\delta^{2}}.$ (2)

The liquid within the boundary layer is lighter than its surroundings as it is
entrained from the top, and it is held in place by the Marangoni induced
viscous stress against buoyancy:

$\mu\frac{V_{\mathrm{M}}}{\delta^{2}}\sim g\Delta\rho,$ (3)

where $\Delta\rho=\rho^{*}-\rho$ is the density difference between the liquid
inside and outside of the kinematic boundary layer, see Fig. 1(e). The lighter
liquid is brought down by the Marangoni flow along the drop’s surface, so
$\Delta\rho\sim-R\cdot\mathrm{d}\rho/\mathrm{d}y$. Cancelling $\delta$ from
Eqs.(2)&(3), we obtain the instability criterion

$Ma/{Ra}^{1/2}>c,$ (4)

where

$Ra=-\frac{\mathrm{d}\rho}{\mathrm{d}y}\cdot\frac{gR^{4}}{\mu D}$ (5)

is the Rayleigh number for characteristic length $R$ and $c$ is a constant to
be determined.

To calculate the Marangoni and Rayleigh numbers, ethanol weight fractions at
the positions where the drops levitate are used to obtain the viscosity $\mu$,
diffusivity $D$ and the interfacial tension $\sigma$ (see Supplemental
Material for the concentration dependence of $\mu$ and $D$). In the following,
for bouncing drops, we use values corresponding to their lowest position.

The phase diagram shown in Fig. 2 is replotted with $Ma/Ra^{1/2}$ vs. $Ra$ in
Fig. 3(a). It clearly shows that there is indeed a critical value
$(Ma/{Ra}^{1/2})_{\mathrm{cr}}$ above which the drop will always bounce, and
the instability threshold $c$ in Eq.(4) is measured to be $c=275\pm 10$ in the
range $6\times 10^{-3}\lesssim Ra\lesssim 3$. We cannot carry out experiments
for $Ra<6\times 10^{-3}$ because drops with
$R<$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ are too small to observe. For
experiments in the range $Ra>3$, the finite size of the container comes into
play. However, we speculate that $c\approx 275$ still holds for $Ra>3$ as long
as the container is large enough. The existing data on bouncing are consistent
with this value.

Figure 3: (a) Phase diagram replotted in dimensionless numbers: $Ma/Ra^{1/2}$
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 with the size of the symbols. (b) Phase diagram replotted with
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y$ vs. $w_{\mathrm{e}}$, where
$w_{\mathrm{e}}$ 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_{\mathrm{e}}>$98\text{\,}\mathrm{w}\mathrm{t}\%$$
($w_{\mathrm{e}}<$50\text{\,}\mathrm{w}\mathrm{t}\%$$) corresponds to
$Ra<6\times 10^{-3}$ ($Ra>3$). Measurement errors are comparable with the size
of the symbols.

We now express our stability criterion Eq.(4) in dimensional quantities by
substituting the definition of $Ma$ and $Ra$ to obtain:

$\left(\frac{\mathrm{d}w_{\mathrm{e}}}{\mathrm{d}y}\right)_{\mathrm{cr}}=c^{2}\left(\mu+\mu^{\prime}\right)^{2}\cdot\frac{gD}{\mu}\frac{\mathrm{d}\rho}{\mathrm{d}\sigma}\frac{\mathrm{d}w_{\mathrm{e}}}{\mathrm{d}\sigma}.$
(6)

Eq.(6) actually predicts a critical concentration gradient above which the
equilibrium is unstable. Note that remarkably the drop radius $R$ does not
enter into this equation. All the fluid properties $\mu$, $D$,
$\mathrm{d}\rho/\mathrm{d}\sigma$ and
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}\sigma$ depend on $w_{\mathrm{e}}$ – the
ethanol weight fraction at the levitation height. Thus the critical gradient
$(\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y)_{\mathrm{cr}}$ as a function of
$w_{\mathrm{e}}$ is shown in Fig. 3(b) as the blue curve. The data shown in
Fig. 2 are also replotted in Fig. 3(b). As can be seen, the blue curve as
predicted by Eq.(6) nicely separates the levitating drops and the bouncing
ones. The dashed blue line in the range
$w_{\mathrm{e}}>$98\text{\,}\mathrm{w}\mathrm{t}\%$$
($w_{\mathrm{e}}<$50\text{\,}\mathrm{w}\mathrm{t}\%$$) corresponds to
$Ra<6\times 10^{-3}$ ($Ra>3$), i.e., the region in which we could not perform
experiments.

The above results are all obtained in a large enough container. We will now
discuss the effect of a geometrical confinement, i.e., the dependence of our
findings on the container size $L$. Let $\mathcal{L}$ denote the maximum
extent of the flow field induced by the drop. Then $\mathcal{L}>L$ means that
the flow is confined. In the case of no confinement, i.e., $\mathcal{L}<L$,
the liquid in the far field is not disturbed by the Marangoni flow, so that
the density in the far field is maintained at $\rho^{*}$ (see Fig. 1(e)).
However, when the flow is confined, i.e., $\mathcal{L}>L$, the liquid close to
the side wall is affected by the Marangoni flow. In such a situation, because
the liquid is pulled down in the center by the drop, the liquid close to the
wall will be pushed up due to mass conservation. This effectively increases
the density $\rho^{*}$. Consequently, the density difference
$\Delta\rho=\rho^{*}-\rho$ is increased, which means that the effect of
buoyancy is amplified. According to Eqs. (3)&(4), the instability threshold
$c$ will thus decrease. Either decreasing the container size $L$ or increasing
$\mathcal{L}$ both leads to a stronger confinement effect. Since for stable
stratifications $\mathcal{L}\sim\left(-{\mathrm{d}\rho}/{\mathrm{d}y}\cdot{\mu
D}/{g}\right)^{-1/4}$ phillips1970flows ; wunsch1970oceanic , one can thus
also increase the confinement effect by using very weak stratifications. We
have performed experiments for weaker gradients
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y<$3\text{\,}\mathrm{m}^{-1}$$ and also in
a larger container to confirm the effect of the confinement. Indeed, for
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y\approx$2\text{\,}\mathrm{m}^{-1}$$, a
cubic container with $L=$50\text{\,}\mathrm{m}\mathrm{m}$$ is already not
large enough, and the instability threshold is reduced to $c\approx 172$. A
smaller container ($L=$30\text{\,}\mathrm{m}\mathrm{m}$$) further decreases
the threshold to $c\approx 157$. For
$\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y\approx$1\text{\,}\mathrm{m}^{-1}$$, the
geometry is more confined, and the threshold is further decreased to $c\approx
122$ in the larger container and even to $c\approx 117$ in the smaller one.
The fact that a weaker stratification leads to a more amplified effect of
buoyancy demonstrates that the stable stratification is very relevant for the
Marangoni instability, in particular on small scales where the confinement is
more relevant.

In conclusion, we have discovered a new type of oscillatory Marangoni
instability for an immiscible drop immersed in a stably stratified ethanol-
water mixture. The commonly ignored stable density stratification induced by
the concentration gradient is vital in triggering this instability. Its onset
is indicated by the transition from a levitating drop to a bouncing one. By
experimentally exploring the parameter space spanned by the concentration
gradient $\mathrm{d}w_{\mathrm{e}}/\mathrm{d}y$ and the drop radius $R$, the
instability is found to be determined by the balance between the advection and
diffusion through the kinetic boundary layer set by the Marangoni flow. This
yields a critical concentration gradient as the instability criterion.
Remarkably, the critical gradient is decreased in a confined geometry, i.e.,
the effect of the stable stratification is amplified on small confined scales.
Our findings indicate that the stable stratification induced by the
corresponding concentration gradient is very relevant, especially in confined
geometries, and should be further explored in other geometries.

Our results for solutal Marangoni flows can also be extended to thermal
Marangoni flows. We found that a stabilizing temperature gradient as low as
$3\text{\,}\mathrm{K}\mathrm{/}\mathrm{m}\mathrm{m}$ can trigger a similar
oscillatory instability on a bubble immersed in water. Such low temperature
gradient is smaller than what is occurring in various applications
schwabe1978experiments ; ratke2006destabilisation ; dedovets2018five , where a
stabilizing temperature gradient can easily go beyond
$10\text{\,}\mathrm{K}\mathrm{/}\mathrm{m}\mathrm{m}$.

We thank Chao Sun and Vatsal Sanjay for valuable discussions. We acknowledge
support from the Netherlands Center for Multiscale Catalytic Energy Conversion
(MCEC), an NWO Gravitation programme funded by the Ministry of Education,
Culture and Science of the government of Netherlands, and the ERC-Advanced
Grant Diffusive Droplet Dynamics (DDD) with Project No. 740479.

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