# Equilibrium shapes and floatability of static and vertically vibrated heavy
liquid drops on the surface of a lighter fluid

Andrey Pototsky  Department of Mathematics, Swinburne University of
Technology, Hawthorn, Victoria, 3122, Australia<EMAIL_ADDRESS>Alexander Oron  Department of Mechanical Engineering, Technion-Israel
Institute of Technology, Haifa 3200003, Israel  Michael Bestehorn  Institute
of Physics, Brandenburg University of Technology, 03013 Cottbus-Senftenberg,
Germany

###### Abstract

A small drop of a heavier fluid may float on the surface of a lighter fluid
supported by surface tension forces. In equilibrium, the drop assumes a
radially symmetric shape with a circular triple-phase contact line. We show
theoretically and experimentally that such a floating liquid drop with a
sufficiently small volume has two distinct stable equilibrium shapes: one with
a larger and one with a smaller radius of the triple-phase contact line. Next,
we experimentally study the floatability of a less viscous water drop on the
surface of a more viscous and less dense oil, subjected to a low frequency
(Hz-order) vertical vibration. We find that in a certain range of amplitudes,
vibration helps heavy liquid drops to stay afloat. The physical mechanism of
the increased floatability is explained by the horizontal elongation of the
drop driven by subharmonic Faraday waves. The average length of the triple-
phase contact line increases as the drop elongates that leads to a larger
average lifting force produced by the surface tension.

##  I Introduction

Floatation of a small liquid droplet on the surface of a less dense carrier
fluid is a striking manifestation of the strength of tensile forces that exist
between immiscible liquid and gaseous phases. When in equilibrium, the drop
assumes a radially symmetric shape with a circular triple-phase contact line
that separates the drop into a sessile upper cap and a pendant lower part. The
quintessential question of determining the largest possible drop volume
capable of staying afloat was first formulated and studied over 40 year ago
(Hartland & Burri, 1976). From the theoretical point of view, a stationary
shape of a liquid drop can be found by solving the minimal surface problem for
liquid menisci, subject to a zero net force condition acting on the triple-
phase contact line (Princen, 1963; Princen & Mason, 1965). Due to its
nonlinear nature and complexity of the boundary conditions, this problem can
only be approached numerically for a particular combination of fluids. Prior
to the modern era of personal computers, the computation of stationary drop
shapes was a highly tedious task that involved manual manipulation of the
tabulated solutions of the minimal surface equations. This method was used by
(Hartland & Burri, 1976) to determine the maximum possible drop volume as a
function of the fluid densities and interfacial tensions. Extending the
original work (Hartland & Burri, 1976), subsequent studies focused on
experimental verification of the floatability of water drops on oil surfaces
(Phan et al., 2012; Phan, 2014), addressed the role of the line tension
(George et al., 2016; Bratukhin et al., 2001) and developed a simplified model
by assuming that the upper sessile cap of the drop is approximately flat
(Bratukhin & Makarov, 1994).

Here we use numerical continuation method and conduct a series of experiments
to reveal a previously unreported multistability of sufficiently small liquid
drops floating on the surface of a less dense bulk fluid. Depending on the
deposition method, a small quantity of a heavy fluid may form a floating drop
with two different equilibrium radially symmetric shapes: one with a smaller
and one with a larger radius of the triple phase contact line. It should be
emphasized that the existence of two different equilibrium shapes with a fixed
drop volume was first mentioned in (Hartland & Burri, 1976), but the shape
with the smaller value of the contact radius was discarded as being unstable.
We demonstrate here theoretically and experimentally that both shapes are, in
fact, stable for drop volumes below a certain critical value.

Our next goal is to experimentally study the dynamic floatability of a
vertically vibrated water drop on the surface of a more viscous and less dense
oil bath. Contrary to its anticipated destructive effect, external vibration
is known to suppress the Rayeigh-Taylor instability in stratified liquid films
and in liquid films on the underside of a solid plate (Wolf, 1969, 1970;
Lapuerta et al., 2001; Sterman-Cohen et al., 2017; Bestehorn & Pototsky, 2016;
Pototsky & Bestehorn, 2016). In a series of recent experiments, vertical
shaking was shown to create levitating layers of a heavy fluid up to 20 cm in
width floating on top of a lighter fluid (Apffel et al., 2020). A rather
remarkable phenomenon of the upside-down floatability was revealed, when a
solid body positioned at the lower interface of a levitating layer, acquired a
stable buoyancy position under the action of vertical vibration (Apffel et
al., 2020).

Earlier, we developed a long-wave hydrodynamic model to investigate the
saturation of the Rayleigh-Taylor instability in isolated vertically vibrated
two-dimensional liquid drops on the surface of a finite-thickness carrier
liquid film (Pototsky et al., 2019). In the absence of vibration, a small
quantity of a heavy fluid, deposited on the surface of a less dense carrier
film stretches into a liquid column. The tip of the extending column
eventually reaches the solid substrate and the carrier film ruptures. We have
shown that an external vertical vibration prevents film rupture at non-zero
Reynolds numbers, leading to a formation of a stable floating drop.

Motivated by these findings, we conduct here a series of experiments with
water drops on a vibrated oil surface and demonstrate that in a certain range
of the vibration amplitudes and frequencies, the floatability of the drop is
enhanced, allowing a larger quantity of water to stay afloat. In this regime,
the drop elongates horizontally driven by the subharmonic surface Faraday
waves that develop in its upper sessile cap (Pucci et al., 2011, 2013). The
time-averaged total length of the triple-phase contact line increases for the
elongated drop, leading to a larger lifting force generated by tensile forces.
This new dynamical regime exists in a narrow window of the vibration
amplitudes: above the onset of the Faraday waves in the drop and below the
onset of the Faraday waves on the more viscous oil surface. An even stronger
vibration destroys the balance of vertical forces and the drop sinks.

The paper is organized as follows. Section II presents the summary of the
analysis of equilibrium drop shapes in the static system in case when the line
tension is neglected. The set of the minimal surface equations for each of the
three interfaces is written in the form of a boundary-value problem with
integral constraints that take into account the vertical force balance and
satisfy the Neumann triangle condition at the triple-phase contact line. We
use the method of numerical continuation implemented with AUTO (Doedel et al.,
2007) to continue the analytically known solution that corresponds to a
spherical drop in case of zero gravity towards non-zero values of $g$. For any
fixed drop volume at sufficiently small gravity $g$, there exist two different
equilibrium shapes: one with a larger and one with a smaller radius of the
circular triple phase contact line. In Section III, we use the Helmzoltz free
energy to study the static stability of the drop and find that both
equilibrium shapes can be stable for sufficiently small drops. This conclusion
is experimentally verified using a small $<10$ $\mu$L water drop deposited
onto the surface of a commercial vegetable oil. The effect of the vertical
vibration is experimentally studied in Section IV.

##  II Stationary floating drop

We consider a sufficiently small drop of a heavy fluid $(1)$ capable of
floating at the interface between a lighter fluid $(2)$, and the ambient gas
$(g)$ as shown in Fig.1(a). In equilibrium, the drop is radially symmetric.
The shapes of the three interfaces that connect at the triple-phase contact
line are found from the balance between the Laplace and hydrostatic pressure.
A recent overview of basic phenomena related to floating droplets referred to
as liquid lenses at liquid-gas interfaces, including wetting, dewetting and
hydrodynamic instabilities can be found in (Nepomnyashchy, 2021). The upper
part of the drop, above the horizontal dashed line represents a sessile drop.
Using standard set of coordinates (Lohnstein, 1906; Boucher et al., 1975), the
upper tip of the drop is taken as the origin of the local coordinate system,
with the $z_{1}$-axis pointing downward. Introducing an angle $\phi_{1}$
between the tangent to the drop profile at point $(r_{1},z_{1})$ and the
horizontal, the pressure balance yields (Lohnstein, 1906; Boucher et al.,
1975)

$\displaystyle\sigma_{1g}\left(\frac{d\phi_{1}}{ds_{1}}+\frac{\sin(\phi_{1})}{r_{1}}\right)=\rho_{1}gz_{1}+\frac{2\sigma_{1g}}{R_{1}},$
(1)

where $s_{1}$ is the arc length of the drop profile from the origin and
$R_{1}$ is the radius of curvature at the upper tip. Note that the gas density
is neglected in Eqs. (1) and (2) below.

Similar, the lower part of the drop, below the horizontal dashed line
represents a pendant drop. The lower tip of the drop is taken as the origin of
the local coordinate system, with the $z_{2}$-axis pointing upward. At point
$\left(r_{2},z_{2}\right)$, $\phi_{2}$ is the angle between the tangent to the
drop profile and the horizontal, and $s_{2}$ is the arc length measured from
the lower tip. Similar to Eq.(1), the pressure balance can be written as
(Boucher et al., 1975)

$\displaystyle\sigma_{12}\left(\frac{d\phi_{2}}{ds_{2}}+\frac{\sin(\phi_{2})}{r_{2}}\right)=(\rho_{2}-\rho_{1})gz_{2}+\frac{2\sigma_{12}}{R_{2}},$
(2)

where $R_{2}$ is the radius of curvature at the lower tip. In addition to
Eq.(1) and (2), the coordinates $(r_{i},z_{i})$, $i=1,2$ satisfy

$\displaystyle\frac{dr_{i}}{ds_{i}}=\cos(\phi_{i}),~{}~{}\frac{dz_{i}}{ds_{i}}=\sin(\phi_{i}).$
(3)

Finally, the meniscus around the drop can be best described in terms of the
height $h(r)$, as shown in Fig.1(a). The requirement that pressure is constant
at any given vertical level, leads to (Landau & Lifshitz, 1987)

$\displaystyle\sigma_{2g}\left(\frac{h^{\prime\prime}}{(1+h^{\prime
2})^{3/2}}+\frac{h^{\prime}}{r\sqrt{1+h^{\prime
2}}}\right)=\rho_{2}gh-\rho_{2}gh_{0},$ (4)

where a prime stands for the derivative with respect to $r$, $h_{0}$ denotes
the height of the meniscus far away from the drop and $r$ is the radial
coordinate measured from the vertical symmetry axis of the system. The system
of Eqs. (1) - (4) must be solved as a boundary value problem, supplemented
with the following boundary and integral constraints.

Equations (1) and (3) with $i=1$ are solved on the interval $0\leq s_{1}\leq
S_{1}$, where $S_{1}$ is the arc length of the upper part of the drop measured
from the upper tip to the triple-phase contact line. Similar, Eq. (2) and Eq.
(3) with $i=2$ are solved on the interval $0\leq s_{2}\leq S_{2}$, where
$S_{2}$ is the arc length of the lower part of the drop measured from the
lower tip to the triple-phase contact line. At the tips of the drop, we set
the initial conditions

$\displaystyle r_{i}(0)=z_{i}(0)=\phi_{i}(0)=0,\,\,i=1,2.$ (5)

The requirement of continuity of the interface yields

$\displaystyle
r_{1}(S_{1})=r_{2}(S_{2})=R,~{}~{}{\rm{and~{}~{}}}z_{1}(S_{1})=z_{2}(S_{2}),$
(6)

where $R$ denotes the contact radius at the triple-phase contact line.
Equation (4) is solved on the interval $R\leq r\leq\infty$ with $h(R)=0$,
$h^{\prime}(\infty)=0$ and $h(\infty)=h_{0}$.

The Newmann triangle, formed by three tensile forces
${\bm{\sigma}}_{12}+{\bm{\sigma}}_{1g}+{\bm{\sigma}}_{2g}=0$, as shown in
Fig.1(a), reflects the requirement that the net force acting on a small
element of the contact line vanishes in equilibrium. Introducing the contact
angles $\Phi_{i}=\phi_{i}(S_{i}),~{}i=1,2$ between phase $(i)$ and the
horizontal at the triple-phase contact line, Neumann’s triangle is replaced by
two scalar equations

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sigma_{1g}\cos{(\Phi_{1})}+\sigma_{12}\cos{(\Phi_{2})}-\sigma_{2g}\frac{1}{\sqrt{1+\left[h^{\prime}(R)\right]^{2}}},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sigma_{1g}\sin{(\Phi_{1})}-\sigma_{12}\sin{(\Phi_{2})}+\sigma_{2g}\frac{h^{\prime}(R)}{\sqrt{1+\left[h^{\prime}(R)\right]^{2}}}.$
(7)

Figure 1:  (a) A liquid drop of a heavy fluid (1) floating at the interface
between a lighter fluid (2) and gas (g). Equilibrium Newmann’s triangle at the
triple-phase contact line is the requirement that the net force acting on a
small element of the contact line is zero, i.e.
${\bm{\sigma}}_{12}+{\bm{\sigma}}_{1g}+{\bm{\sigma}}_{2g}=0$. (b) Variation in
the total drop height $H$ on the gravity constant $g$ for a $50$ $\mu L$ water
drop floating at the oil-air interface. The fluid parameters are:
$\rho_{1}=1000$ kg/m3, $\rho_{2}=916$ kg/m3, $\sigma_{1g}=0.045$ N/m,
$\sigma_{2g}=0.022$ N/m, $\sigma_{12}=0.032$ N/m (Phan et al., 2012). The drop
profiles at points $(1,2,3)$ are shown in the right panel. Continuation starts
at point $P_{1}$ that corresponds to the analytical solution given by Eqs.
(12). The solution at the point $P_{2}$ corresponds to a circular drop fully
submerged in fluid $(2)$ and touching the interface $2-g$ at a point.

The volumes $V_{i},~{}i=1,2$ of the sessile and pendant parts are given by

$\displaystyle V_{i}=\int_{0}^{S_{i}}\pi r_{i}^{2}\sin(\phi_{i})ds_{i}.$ (8)

The total buoyancy force $F_{b}$ experienced by the drop is given by the
weight of fluid $2$ displaced by the volume of pendant part $\rho_{2}gV_{2}$
and the weight of fluid $2$ in the cylindrical volume above the contact line
level ( the dashed line in Fig. 1(a)) with the radius $R$ and the meniscus
height $h_{0}$, i.e.

$\displaystyle F_{b}=\rho_{2}gV_{2}+\rho_{2}g\pi R^{2}h_{0}.$ (9)

Finally, the vertical force balance implies that the weight of the drop
$\rho_{1}g(V_{1}+V_{2})$ is balanced by the buoyancy force $F_{b}$ and the
vertical component of the surface tension force exerted by fluid $(2)$ onto
the drop. The latter is given by $\displaystyle
2\pi\sigma_{2g}Rh^{\prime}(R)\left[1+\left[h^{\prime}(R)\right]^{2}\right]^{-1/2}$
so that the balance of vertical forces can be written in the form

$\displaystyle\rho_{1}g(V_{1}+V_{2})=F_{b}+2\pi\sigma_{2g}R\frac{h^{\prime}(R)}{\sqrt{1+\left[h^{\prime}(R)\right]^{2}}}.$
(10)

We use numerical continuation method AUTO (Doedel et al., 2007) to solve the
above boundary value problem with an integral condition given by the
requirement that the total volume $V_{1}+V_{2}$ is fixed. Details of the
numerical continuation method are summarized in Appendix A.

As a starting point of the numerical continuation method, we use the
analytical solution in case of zero gravity, when the upper and the lower
parts of the drop are both spherical and the interface between fluid $2$ and
gas is non-deformed. By setting $h^{\prime}(R)=0$, we obtain from Eqs. (II)
the contact angles $\Phi_{i}$

$\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\cos^{-1}\left(\frac{\sigma_{1g}^{2}+\sigma_{2g}^{2}-\sigma_{12}^{2}}{2\sigma_{1g}\sigma_{2g}}\right),$
$\displaystyle\Phi_{2}$ $\displaystyle=$
$\displaystyle\cos^{-1}\left(\frac{\sigma_{12}^{2}+\sigma_{2g}^{2}-\sigma_{1g}^{2}}{2\sigma_{12}\sigma_{2g}}\right).$
(11)

It is easy to see that the solution of Eqs. (1) - (3) at zero gravity is

$\displaystyle\phi_{i}=\frac{s_{i}}{R_{i}},~{}~{}r_{i}=R_{i}\sin\left(\frac{s_{i}}{R_{i}}\right),~{}~{}z_{i}=R_{i}\left[1-\cos\left(\frac{s_{i}}{R_{i}}\right)\right],~{}~{}0\leq
s_{i}\leq R_{i}\Phi_{i},$ (12)

where the radii $R_{i}$ of the sessile and pendant parts are determined by the
contact angles $\Phi_{i}$ and the horizontal radius $R$ at the level of the
contact line, i.e. $R_{i}\sin(\Phi_{i})=R$. The volumes of the sessile and
pendant parts are found from Eq. (8) $V_{i}=\pi
R_{i}^{3}\left(\frac{2}{3}+\frac{1}{3}\cos^{3}(\Phi_{i})-\cos(\Phi_{i})\right)$.

First, we continue the solution Eqs. (12) in parameter $g$ with the fluid
parameters taken from (Phan et al., 2012). The total height of the drop
$H=z_{1}(S_{1})+z_{2}(S_{2})$ is displayed as a function of $g$ in Fig.1b. The
continuation starts at the point $P_{1}$ that corresponds to the analytical
solution Eqs. (12). As $g$ increases, the drop profile changes, as shown by
the three selected solutions $1,2,3$ in the right panel Fig. 1. No stationary
solutions exist past the saddle-node bifurcation point (solution $2$) in terms
of $g$. For any $g$ below the saddle-node point, there exist two stationary
drop profiles: one with a lower value of the radius $R$ (solution $1$) and one
with a larger value of $R$ (solution $3$). At vanishingly small $g$, the
second solution (point $P_{2}$) collapses to a spherical drop fully submerged
in fluid $2$ that only touches the interface $2-g$ at one point.

Figure 2:  (a) Two water drops with the identical volume $V=90$ $\mu$L at the
oil-air interface. (b) Contact line radius $R$ as a function of the volume $V$
at $g=9.8$ m/s2. Circles indicate the location of the solutions in (a).

After the value of $g=9.8$ m/s2 is reached, the solution is followed using
AUTO (Doedel et al., 2007) with the drop volume $V$ as a continuation
parameter. The radius $R$ of the contact line is shown as a function of $V$ in
Fig.2(b). Two different drops with the identical volume $V=90$ $\mu$L are
shown in Fig.2(a).

It is remarkable that multiple stationary profiles of floating drops have
never been studied in details. In the early study (Hartland & Burri, 1976),
the existence of multiple profiles has been mentioned, but the solutions with
the smaller value of $R$ were declared unstable. The reason for the
instability (Hartland & Burri, 1976) was the qualitative comparison of a
floating drop with a heavy rigid sphere capable of floating on the surface of
a fluid that does not wet the surface of the sphere (Hartland & Robinson,
1971). Indeed, in case of a rigid sphere, the stability condition is derived
by looking at the position of the center of mass $z_{c}(g)$ of the sphere as a
function of the gravity constant $g$. Thus, if $g$ is slightly increased
(decreased), the center of mass of a stable configuration must descend
(ascend) so that after $g$ is set back to its original value, the sphere will
bounce back up (down) to recover the original stationary position. It was
shown (Hartland & Robinson, 1971) that solutions with a larger radius of the
contact line are stable, while those with the smaller value of radius are
unstable. However, to the best of our knowledge, no such calculations are
available for a floating liquid drop. A comprehensive stability analysis that
takes into account static and dynamic perturbations is still missing.

##  III Static stability of a floating drop

We approach the stability of a floating drop from the point of view of the
Helmholtz free energy, by considering the so-called static axisymmetric
perturbations of the drop profile that satisfy the Laplace pressure balance.
Such static stability analysis neglects the motion of the fluids and,
therefore, can only be considered as a precursor to the true dynamic stability
with respect to non-axisymmetric perturbations. Note that a similar method has
been applied to study the static stability of pendant drops hanging from a
solid plate (Padday & Pitt, 1973).

We take the level of the carrier fluid $2$ far away from the drop, as a zero
level for the vertical axis directed upwards, as shown in Fig.3. The excess
total energy $E_{t}$ of the system is constructed not counting the infinite
energy of the semi-infinite fluid $2$ with a flat fluid-gas interface in the
absence of the drop.

Figure 3:  Excess potential energy $U_{e}$ of the fluid bath is given by the
potential energy of the shaded part filled with fluid $2$ with negative
density $-\rho_{2}$.

The excess surface energy is

$\displaystyle E_{s}=\sigma_{12}S_{12}+\sigma_{1g}S_{1g}+\sigma_{2g}S_{2g},$
(13)

where $S_{12}=\int_{0}^{S_{2}}2\pi r_{2}ds_{2}$ and
$S_{1g}=\int_{0}^{S_{1}}2\pi r_{1}ds_{1}$ are the surface areas of the fluid
$1$-fluid $2$ and fluid $1$-gas interfaces and $S_{2g}=\int_{R}^{\infty}2\pi
r(\sqrt{1+h^{\prime 2}}-1)dr$ is the excess surface area of the fluid $2$-gas
interface. The excess potential energy of the system is

$\displaystyle E_{p}=Mgz_{c}+U_{e},$ (14)

where $M=\rho_{1}(V_{1}+V_{2})$ is the total mass of the drop, $z_{c}$ is the
vertical coordinate of the centre of mass of the drop and $U_{e}$ is the
excess potential energy of the fluid bath. $U_{e}$ can be further represented
in the form

$\displaystyle U_{e}=-g\rho_{2}V_{e}Z_{e},$ (15)

where $V_{e}$ is the volume of the shaded part in Fig.3 and $Z_{e}$ is the
coordinate of the center of mass of the shaded part in Fig.3, filled with a
homogeneous density. The excess total energy $E_{t}$ is thus given by
$E_{t}=E_{p}+E_{s}$. Note that the values $z_{c},Z_{e},V_{e}$ and $E_{s}$
depend on $g$, thus

$z_{c}=z_{c}(g),~{}Z_{e}=Z_{e}(g),~{}V_{e}=V_{e}(g),~{}E_{s}=E_{s}(g).$ (16)

Now let us assume that the gravity constant has been slightly changed to
$g+\delta g$. This can be practically achieved by placing the system in a lift
that accelerates either upward or downward, or alternatively, as a result of a
localized in time displacing disturbance of the system. The drop will assume a
new stationary shape that corresponds to the new value of $g+\delta g$. Now
assume that the lift (or the disturbance) suddenly stops and the gravity
constant is instantaneously set back to the original value $g$. At this moment
of time, the drop and meniscus still have new shapes, so that the total energy
of the system is

$\displaystyle\tilde{E}_{t}=Mgz_{c}(g+\delta g)-g\rho_{2}V_{e}(g+\delta
g)Z_{e}(g+\delta g)+E_{s}(g+\delta g).$ (17)

The system is stable if $\tilde{E}_{t}>E_{t}$, i.e.

$\displaystyle\tilde{E}_{t}-E_{t}$ $\displaystyle=$ $\displaystyle
Mgz_{c}(g+\delta g)-Mgz_{c}(g)$ (18) $\displaystyle-$ $\displaystyle
g\rho_{2}V_{e}(g+\delta g)Z_{e}(g+\delta g)+g\rho_{2}V_{e}(g)Z_{e}(g)$
$\displaystyle+$ $\displaystyle E_{s}(g+\delta g)-E_{s}(g)>0.$

Expanding Eq. (18) into powers of $\delta g$, we obtain the leading term

$\displaystyle\delta
E_{t}=\left(Mg\frac{dz_{c}(g)}{dg}-g\rho_{2}\frac{d(V_{e}Z_{e})}{dg}+\frac{dE_{s}(g)}{dg}\right)\delta
g=0,$ (19)

which must vanish for arbitrary small $\delta g$, because the drop shape at
the original value of $g$ is in equilibrium.

Differentiating $\delta E_{t}/\delta g=0$ with respect to $g$, we obtain

$\displaystyle\frac{d}{dg}\left(\frac{\delta E_{t}}{\delta
g}\right)=M\frac{dz_{c}(g)}{dg}+Mg\frac{d^{2}z_{c}(g)}{dg^{2}}-\rho_{2}\frac{d(V_{e}Z_{e})}{dg}-g\rho_{2}\frac{d^{2}(V_{g}Z_{g})}{dg^{2}}+\frac{d^{2}E_{s}(g)}{dg^{2}}=0.~{}~{}$
(20)

The second term in the expansion of Eq. (18) in terms of $\delta g$ is

$\displaystyle\delta^{2}E_{t}=\left(Mg\frac{d^{2}z_{c}(g)}{dg^{2}}-g\rho_{2}\frac{d^{2}(V_{e}Z_{e})}{dg^{2}}+\frac{d^{2}E_{s}(g)}{dg^{2}}\right)\delta
g^{2}>0.$ (21)

Equation (20) can now be used to remove the second derivative terms in Eq.
(21) and to finally obtain the static stability condition

$\displaystyle\rho_{2}\frac{d(V_{e}Z_{e})}{dg}-M\frac{dz_{c}(g)}{dg}>0.$ (22)

Equation (22) is now recast in the form $df(g)/dg<0$, where

$\displaystyle f(g)=z_{c}(g)-\frac{\rho_{2}V_{e}(g)Z_{e}(g)}{\rho_{1}V},$ (23)

and $V$ is the fixed drop volume. Function $f(g)$ can be expressed in terms of
the variables $z_{i}$, $\phi_{i}$, $r_{i}$ and $h(r)$ from Eqs. (1) - (3), as
shown in Appendix B.

Figure 4:  Function $f(g)$ given by Eq. (23) determined for $V=50$, $20$ and
$10$ $\mu$L water drop floatation on oil with the parameters as in Fig.1(b).
Stable (unstable) solutions are marked by the solid (dashed) lines. The zero
gravity solution given by Eqs. (12) corresponds to the larger value of $f$ at
$g=0$. For each volume, the solutions on the upper branch have a larger
contact radius $R$. Drops smaller than $20$ $\mu$L have two stable floatation
shapes at $g=9.8$ m/s2: one with a larger and one with a smaller value of the
contact radius.

The function $f(g)$ from Eq. (23) is shown in Fig. 4 for three different
volumes $V=50$, $20$ and $10$ $\mu$L of a water drop on oil surface with fluid
parameters as in Fig.1(b). Statically stable shapes correspond to $df/dg<0$,
as indicated by the solid curves, whereas statically unstable states are
marked by the dashed parts of the curves. For each volume, the zero-gravity
solution given by Eqs. (12) has a larger value of $f(g=0)$. At any fixed value
of $g$ below the saddle-node bifurcation point, the solution with larger
(smaller) values of the contact radius $R$ belongs to the upper (lower) branch
of $f(g)$. At terrestrial gravity of $g=9.8$ m/s2, as indicated by the
vertical line in Fig. 4, larger drops with the volume $V>~{}20$ $\mu$L have
only one stable shape with a larger value of $R$ (upper branch). However,
smaller drops with the volume $V<~{}20$ $\mu$L have two stable shapes: one
with a larger (upper branch) and one with a smaller (lower branch) contact
radius $R$.

Figure 5:  Two stable shapes of a water drop with approximately identical
volume $5\pm 0.5$ $\mu$L floating on the surface of a commercial vegetable
oil. The thick solid line is obtained by fitting the experimental drop profile
with a pendant drop solution of Eqs. (2), (3) with the fluid parameters as in
Fig.1 and the fitted value of the radius of curvature at the lower tip
$R_{2}\approx 1.13$ mm in both cases. The shape with a smaller contact radius
(a) is obtained by depositing a drop from a pipette directly onto the oil
surface. The shape with the larger contact radius (b) is obtained by
depositing the drop from the pipette onto the vertical wall of the container
and allowing the drop to slowly slide down the wall to touch the oil surface.

In order to experimentally verify the existence of two stable floatation
shapes, we use two different methods to deposit a small quantity of water
$<10$ $\mu$L onto a surface of a commercial vegetable oil. To obtain a shape
with a smaller contact radius, we use a similar technique as in (Phan et al.,
2012), namely, a pendant water drop with the maximal possible volume is
produced freely hanging in air from a pipette with the diameter of the nostril
of $2.5$ mm. After the drop is carefully brought into contact with the oil
surface, it separates from the pipette and starts to float on the oil surface,
as shown in Fig.5(a). In the second deposition method, we produce a drop with
a similar volume hanging freely from a pipette. The drop is carefully brought
into contact with a vertical wall of the polystyrene container filled with
oil. The drop slowly slides down the wall under the action of gravity, and
eventually comes into contact with the meniscus formed by oil. The spreading
coefficient of water on oil is generally larger than that of water on
polystyrene. Therefore, the drop is pulled by capillary forces away from the
container wall. This process creates the second stable shape with a larger
contact radius, as shown in Fig.5(b). In both cases, the estimated volume of
the water drop is approximately identical $V\approx 5\pm 0.5$ $\mu$L.

The drops are photographed and the contours of their pendant parts submerged
in oil are extracted using Matlab. Note that the contour of the upper cap of
the drop is difficult to extract, as it is obstructed by the meniscus formed
by oil with the container wall. The experimentally obtained profiles are
fitted by the analytical solution of the pendant drop Eqs. (2) and (3) with
fluid parameters as in Fig.1. We use the radius of curvature at the lower tip
$R_{2}$ as the fitting parameter and obtain the value of $R_{2}\approx 1.13$
mm for both shapes. The analytical solution is shown by the thick solid line
in Fig.5. The volumes of the submerged parts, determined using the fitted
analytical solution, are found to be $6$ $\mu$L in Fig.5(a) and $5$ $\mu$L in
Fig.5(b). The difference in volumes is due to the unaccounted volume of the
obstructed upper caps.

##  IV Vertical vibration helps liquid drops to stay afloat

Figure 6:  Phase diagram separating floating circular, floating elongated and
sinking water drops vibrated at $60$ Hz in an olive oil. Symbols indicate an
experimentally determined critical acceleration $a/g$, that corresponds to the
onset of the Faraday instability in a hanging water drop of the varying
volume. In the absence of vibration, smaller drops with $V\lesssim 0.1$ mL
hang at the oil-air interface, while larger drops $V\gtrsim 0.1$ mL sink to
the bottom of the container. For $0.1$ mL $\lesssim V\lesssim 0.15$ mL the
drop floats at the oil-air interface when vibrated with the amplitude $a/g>5$.
Insets: representative snapshots of a floating circular, floating elongated
and a sunk water drop. The vertical scale bar next to the floating circular
drop indicates the length of 5 mm. An ellipse (solid line) in the top view of
the floating elongated drop highlights an approximate location of the contact
line. The ratio between the major and the minor axes of the ellipse is $2:3$.

To investigate how external vertical vibration affects the floatability of the
drop, we conduct a series of experiments with drops of distilled water with
the volume $V\lesssim 0.15$ mL deposited on top of a vertically vibrated $1$
cm thick layer of commercial olive oil. The oil was placed in a circular
container with the diameter of 10 cm mounted on a $6.5$” $45$ W RMS audio
speaker (Sony, Japan) powered by a $30$W stereo amplifier (Yamaha TSS-15,
China) of a sinusoidal signal produced by a digital tone generator. The
container was vertically vibrated at frequencies in the range 40–80 Hz. The
vibration amplitudes were sufficiently large to excite subharmonic Faraday
waves on the surface of the water drop but not strong enough to excite Faraday
waves on the free surface of the oil layer.

The vertical coordinate of the container $z_{\rm{p}}(t)$ oscillates with the
amplitude $A$ and the period $T=1/f$ according to

$\displaystyle z_{\rm p}(t)=A\cos{(\phi+2\pi t/T)},$ (24)

where $f$ is the frequency in Hertz and $\phi$ is a fixed phase. In the co-
moving frame of reference, the time-dependent downward acceleration is

$\displaystyle
g+\ddot{z}_{\rm{p}}(t)=g-A\left(\frac{2\pi}{T}\right)^{2}\cos{(\phi+2\pi
t/T)},$ (25)

where $\dot{z}$ stands for the time derivative of $z$.

We use the ADXL 326 accelerometer (Analog Devices, USA) attached to the
container and a digital oscilloscope (Tektronix TDS 210, USA) to measure the
acceleration amplitude $\displaystyle\frac{a}{g}=\frac{A(2\pi f)^{2}}{g}$ in
units of $g$. The motion of the drop was recorded at up to 240 fps using a
high-speed camera and the obtained images were post-processed in Matlab for a
further analysis.

As a first step, we determine the maximal possible volume of a water drop
$V_{m}$ that hangs at the olive oil-air interface without vibration and find
that $V_{m}\approx 0.1$ mL. Note that with a commercial vegetable oil, a
stable hanging water drop of up to $0.17$ mL can be achieved (Phan et al.,
2012). Such a drop stretches down to reach a vertical size of $1$ cm and more
due to a larger value of the vegetable oil-air surface tension. In order to
avoid any direct contact between the drop and the bottom of the $1$ cm deep
container used in our experiments, we have chosen olive oil over vegetable
oil.

Previous studies have demonstrated that liquid lenses of a lighter fluid
floating on the surface of a heavier and more viscous fluid undergo a
spontaneous horizontal elongation under the action of vertical vibration
(Pucci et al., 2011, 2013, 2015; Pototsky & Bestehorn, 2018). The underlying
physical mechanism was traced down to the Faraday instability at the upper
drop surface. The onset of the Faraday instability corresponds to a period-
doubling bifurcation (Pototsky & Bestehorn, 2018): the lens starts to
oscillate at half of the driving frequency and elongates horizontally in a
randomly chosen direction. The resultant equilibrium shape of the elongated
lense, as seen from the top, is dictated by the balance between the radiation
pressure of the unidirectional Faraday waves and the Laplace pressure which
always tries to return the lens back into a circular shape (Pucci et al.,
2011, 2013).

Here we observe that similar to floating lenses, a heavy water drop that
floats on the surface of a lighter and more viscous oil also undergoes a
horizontal elongation, when Faraday waves are excited on the upper drop
surface. Remarkably, the floatability of the elongated drop increases,
allowing drops with volumes greater than the maximal static volume
$V_{m}\approx 0.1$ mL to stay afloat.

Our experimental protocol consists of a careful deposition of a water drop of
various volumes with a pipette on the surface of the vibrated oil bath, and
different acceleration amplitudes $a$. Below the Faraday instability
threshold, the drop assumes a circular shape when viewed from the top as seen
in the upper right inset of Fig. 6. A side view of such a floating circular
drop is also shown in the inset of Fig. 6 (floating circular). Above the
Faraday threshold the drop elongates as displayed in the inset in Fig. 6
(floating elongated). The shape of the contact line in the elongated state is
approximately elliptic with a relatively small eccentricity (ratio of the
major and minor axes) of $2:3$. Contrary to an anticipated destructive effect
of shaking, the drop continues to float, when vibrated at up to $a=6.5g$. For
$a>6.5g$, the Faraday instability in the oil layer sets in.

Following this protocol we record the Faraday stability threshold for
different drop volumes in Fig. 6 vibrated at $f=60$ Hz (a representative
value). The vertical and horizontal error bars indicate the uncertainty in
measurements of the acceleration amplitude and the drop volume, respectively.

We found that drops with the volume $0.1$ mL < $V~{}<~{}0.15$ mL can float at
the oil-air interface when vibrated at $a>5g$. These drops will otherwise sink
to the bottom of the container when the vibration amplitude is reduced below
$a<5g$. Notably, the action of vibration delays the emergence of the critical
event of drop sinkage similar to delaying of film rupture (Sterman-Cohen et
al., 2017) or delaying of liquid bridge breaking (Benilov, 2016).

###  IV.1 The origin of the excess lifting force

In order to understand the physical mechanism responsible for the excess
lifting force, required to keep larger drops afloat we consider the balance of
the vertical forces acting on the drop averaged over at least two forcing
periods $2T$:

$\displaystyle Mg=\langle F_{b}\rangle+\langle F_{t}\rangle,$ (26)

where $M$ is the total mass of the drop, $F_{b}$ is the buoyancy force,
$F_{t}$ is the lifting force due to surface tension acting at the triple-phase
oil-water-air contact line and
$\langle\dots\rangle=(2T)^{-1}\int_{0}^{2T}(\dots)\,dt$ denotes averaging over
time. Note that averaging over $2T$ is necessary due to a subharmonic nature
of the Faraday waves.

Similar to Eq. (10) in the static case, the instantaneous value of $F_{t}$ for
the elongated drop can be represented in the form

$\displaystyle F_{t}=\sigma_{2g}\int_{C(t)}\sin(\alpha(l,t))\,dl,$ (27)

where $\alpha(l,t)$ is the local instantaneous contact angle between the oil-
gas interface and the horizontal, and the integration contour $C(t)$
represents the instantaneous shape of the contact line of the elongated drop,
parametrised by the arc length $l$. In the static case,
$\sin{\left(\alpha(l,t)\right)}=h^{\prime}(R)/\sqrt{1+\left[h^{\prime}(R)\right]^{2}}$
is constant and $C$ is a circle with the radius $R$ so that Eq.(10) is
recovered.

Figure 7: (a) Series of $50$ contours of the upper part of a $0.12$ mL
floating drop vibrated at $60$ Hz extracted from a 200 fps slow motion video.
The vertical axes in (a,b) represent time in the units of the vibration period
$T=1/60$ s. The horizontal axis in (b) is the vertical coordinate of the
container $z_{\rm{p}}(t)$ vibrating at $60$ Hz. (c) Circles correspond to the
scaled volume factor $s(t)$ associated with the upper part of the drop above
the oil-air interface estimated from (a). The dashed curve in (c) is the
fitting line $\sim S_{0}\cos(\psi+2\pi t/T)$ to the experimental data. (d) The
function $s(t)\ddot{z}_{\rm{p}}(t)$ together with its time-average value
represented by the thick horizontal line. Panels 1 - 6 display snapshots of
the drop showing also the blobs emerging above the oil-gas interface. The six
snapshots correspond to the first six contours in (a) from the time interval
between $0$ and $1.5T$.

The excess lifting force can be explained if $\langle F_{t}\rangle$ is larger
than its static value, given by $2\pi
R\sigma_{2g}h^{\prime}(R)/\sqrt{1+\left[h^{\prime}(R)\right]^{2}}$. However,
it is important to emphasize that the average value of the integral in Eq.
(27) cannot be simply estimated, due to the time-dependent contact angle
$\alpha(t,l)$, as can be appreciated from the side view of the floating
elongated drop in the inset in Fig.6.

As we explain below, the estimation of the average value of the buoyancy force
$\langle F_{b}\rangle$ is experimentally easier accessible than the estimation
of $\langle F_{t}\rangle$. We therefore proceed to show that $\langle
F_{b}\rangle$ measured for the vibrated drop is, in fact, smaller than the
static value of a fully submerged drop, i.e. $\langle
F_{b}\rangle<(\rho_{1}-\rho_{2})gV_{0}$, where $V_{0}$ represents the total
volume of the drop. This confirms that the excess lifting force is due to the
increased value of $\langle F_{t}\rangle$.

We record a side-view slow-motion video of a $0.12$ mL drop. To increase
contrast with yellow oil, we add a small amount of red food coloring fluid to
the water. The drop is stretched horizontally and the stretching direction is
perpendicular to the view line of the camera, as shown in the inset in Fig.6.

Individual frames are extracted from the slow-motion video and post-processed
in Matlab to detect the contour of the drop. Figure 7(a) displays $50$
contours of the upper part of the drop exposed above the oil-air interface.
The time interval between any two neighboring contours is $1/200$ s. For
comparison, we show in Fig.7(b) the vertical coordinate of the vibrating
container $z_{c}(t)$ in the laboratory frame. It is clear that the drop
oscillates sub-harmonically at half of the driving frequency $f/2=30$ Hz. The
left and the right edges of the drop periodically bulge out to develop a small
blob every $1/30$ s in anti-phase to each other.

Before proceeding, we estimate the average dynamic pressure $P_{d}$ acting on
the water-oil interface at the submerged part of the drop. Due to continuity
of the velocity field, the difference of the dynamic pressures in oil and
water is given by $P_{d}=|\rho_{1}-\rho_{2}|u^{2}$, where $\rho_{1}$ and
$\rho_{2}$ are the densities of water and oil, respectively. The typical
velocity inside the water drop can be extracted from Fig.7(a). The maximal
height of the blobs periodically developing above the oil-air interface are
$\sim 1\dots 3$ mm. Since the blobs develop over the interval of time
$1/f=1/60$ s, the average speed associated with this motion is $u\sim 10^{-1}$
m/s. This yields the average dynamic pressure at the oil-water interface of
$P_{d}\sim|\rho_{1}-\rho_{2}|\times 10^{-2}$ N/m2. The dynamic pressure must
be compared with the hydrostatic pressure $P_{s}=|\rho_{1}-\rho_{2}|gh$, where
$h\approx 10^{-2}$ m is the vertical size of the drop. Thus, we estimate that
the dynamic pressure $P_{d}\approx 10^{-1}P_{s}$ can be neglected and the
total buoyancy force is given by

$\displaystyle\langle
F_{b}\rangle=(\rho_{1}-\rho_{2})\langle(g+\ddot{z}_{\rm{p}}(t))V_{s}(t)\rangle,$
(28)

where $V_{s}(t)$ is the time-dependent volume of the oil displaced by the
drop.

We assume $V_{s}(t)\approx V_{0}-\Delta V(t)$, where $V_{0}$ is the total
volume of the drop and $\Delta V(t)\geq 0$ is the instantaneous volume of the
part of the drop above the oil-air interface. It is important to note that
this assumption neglects the existence of the air pockets (dark regions in the
insets in Fig.6). Finally, we obtain

$\displaystyle\langle
F_{b}\rangle=(\rho_{1}-\rho_{2})(gV_{0}-\langle\ddot{z}_{\rm{p}}(t)\Delta
V(t)\rangle-g\langle\Delta V(t)\rangle),$ (29)

where it was used that $\langle\ddot{z}_{\rm{p}}(t)\rangle=0$. Equation (29)
shows that the average buoyancy force $\langle F_{b}\rangle$ is larger than
the static buoyancy $(\rho_{1}-\rho_{2})gV_{0}$ only if
$\langle\ddot{z}_{\rm{p}}(t)\Delta V(t)\rangle+g\langle\Delta V(t)\rangle<0$.

Note that the average of a product of two sinusoidal functions $\langle
f(t)g(t+\phi)\rangle$ with a zero average value $\langle f(t)\rangle=\langle
g(t)\rangle=0$ can be positive or negative depending on the phase shift
between them. For example, $\langle a\cos(\omega t)\cos(\phi+\omega
t)\rangle=a\cos(\phi)/2$.

Our analysis is continued to show that $\langle\ddot{z}_{\rm{p}}(t)\Delta
V(t)\rangle>0$, which together with $\langle\Delta V(t)\rangle>0$ implies that
the average buoyancy force is smaller than the static buoyancy $\langle
F_{b}\rangle<(\rho_{1}-\rho_{2})gV_{0}$. To show that
$\langle\ddot{z}_{\rm{p}}(t)\Delta V(t)\rangle>0$, we assume that the true
volume $\Delta V(t)$ of the upper part of the drop is proportional to the area
$S(t)$ under the two-dimensional drop contour, shown in Fig. 7(a). The scaled
normalized area $s(t)=S(t)/\langle S(t)\rangle-1$ is calculated for each of
the 50 profiles in Fig. 7(a) and presented versus time in Fig. 7(c). The data
is fitted by $S_{0}\cos(\psi+2\pi t/T)$ shown by the dashed line in Fig. 7(c).
The function $\ddot{z}_{\rm{p}}(t)s(t)$ is also shown in Fig. 7(c). along with
the time-average value represented by the thick horizontal line. Since
$\langle\ddot{z}_{\rm{p}}(t)s(t)\rangle>0$, we finally conclude that
$\langle\ddot{z}_{\rm{p}}(t)\Delta V(t)\rangle>0$.

Consequently, the floatation of the drop is necessarily enabled by the
increased surface tension force $\langle F_{t}\rangle$, which supports the
drop at the triple-phase contact line. The average $\langle F_{t}\rangle$
becomes larger than the corresponding force in the static case, due to the
increased total length of the contact line in a horizontally elongated drop,
as observed in our experiments.

##  V Conclusion

To conclude, we have studied theoretically and experimentally the stability
and dynamics of a liquid drop of a heavier fluid, floating on the surface of a
lighter and more viscous fluid. In equilibrium, small drops may stay afloat
assuming two different radially symmetric shapes. One stable shape has a
smaller and one has a larger value of the triple-phase contact-line radius. We
have experimentally demonstrated the possibility to create both stable shapes
using two different deposition methods of the water drop onto the oil surface.
As the volume of the drop is gradually increased beyond a certain critical
volume, the shape with a smaller contact radius loses its stability and the
drop sinks. The second stable shape with a larger contact radius remains
stable until another critical volume is reached, beyond which no static
floating drops exist.

Remarkably, the floatability of the drop can be slightly increased if the drop
is vibrated vertically with the frequency in Hz-order. We have performed a
series of experiments with water drops on an olive oil surface and found that
drops with the volume between $0.1$ mL and $0.15$ mL remain afloat when
vibrated at $60$ Hz with the acceleration amplitude between $5g$ and $6g$. In
the absence of vibration, water drops larger than $0.1$ mL detach from the oil
surface and sink. The origin of the excess lifting force is rooted in the
horizontal elongation of the drop, driven by the unidirectional Faraday waves
that develop on the upper drop surface. The average length of the contact line
in an elongated state appears to be larger than in a static radially symmetric
shape. As a result, the average lifting force exerted by tensile forces
increases, allowing for heavier drops to stay afloat when vibrated.

Finally, the results presented here, apart of their academic interest, may
serve as a basis for a research in emulsification of dispersed systems
containing several immissible liquids of different densities and utilized in
pharmaceutical applications for stabilization of emulsions in their suspended
state by shaking.

## Acknowledgments

## Appendix A

In order to use the numerical continuation method (Doedel et al., 2007), we
write the minimal surface equations Eqs. (1)- (3) as a system of nine first-
order autonomous equations

$\displaystyle\frac{d\phi_{1}}{ds_{1}}$ $\displaystyle=$
$\displaystyle-\frac{\sin(\phi_{1})}{r_{1}}+\frac{\rho_{1}g}{\sigma_{1g}}z_{1}+\frac{2}{R_{1}},$
$\displaystyle\frac{dr_{1}}{ds_{1}}$ $\displaystyle=$
$\displaystyle\cos(\phi_{1}),$ $\displaystyle\frac{dz_{1}}{ds_{1}}$
$\displaystyle=$ $\displaystyle\sin(\phi_{1}),$
$\displaystyle\frac{d\phi_{2}}{ds_{2}}$ $\displaystyle=$
$\displaystyle-\frac{\sin(\phi_{2})}{r_{2}}+\frac{(\rho_{2}-\rho_{1})g}{\sigma_{12}}z_{2}+\frac{2}{R_{2}},$
$\displaystyle\frac{dr_{2}}{ds_{2}}$ $\displaystyle=$
$\displaystyle\cos(\phi_{2}),$ $\displaystyle\frac{dz_{2}}{ds_{2}}$
$\displaystyle=$ $\displaystyle\sin(\phi_{2}),$ $\displaystyle\frac{dh}{dr}$
$\displaystyle=$ $\displaystyle h^{\prime},$
$\displaystyle\frac{dh^{\prime}}{dr}$ $\displaystyle=$
$\displaystyle(1+h^{\prime
2})^{3/2}\left(\frac{\rho_{2}g}{\sigma_{2g}}h-\frac{\rho_{2}g}{\sigma_{2g}}h_{0}-\frac{h^{\prime}}{\tilde{r}\sqrt{1+h^{\prime
2}}}\right),$ $\displaystyle\frac{d\tilde{r}}{dr}$ $\displaystyle=$
$\displaystyle 1,$ (30)

The new variable $\tilde{r}=r$ is introduced to allow writing the meniscus
equation as an autonomous system of first-order differential equations, where
we have taken a special care of Eq. (4) by truncating the semi-infinite
interval $R\leq r<\infty$ to $R\leq r<$ $R_{\infty}$, with a fixed upper bound
of $R_{\infty}=10$ cm. Equations (1)-(3) are solved on the interval $0\leq
s_{1}\leq S_{1}$ and Eqs. (4)-(6) are solved on the interval $0\leq s_{2}\leq
S_{2}$. Note that the first and the fourth equations in Eqs. (A) are singular
at $s_{i}=0$ due to the presence of the term
$\displaystyle\frac{\sin(\phi_{i})}{r_{i}}$. In order to avoid singularity we
introduce the following set of fourteen regularized boundary conditions

$\displaystyle\phi_{1}(0)$ $\displaystyle=$ $\displaystyle 10^{-5},$
$\displaystyle r_{1}(0)$ $\displaystyle=$ $\displaystyle 10^{-5}R_{1},$
$\displaystyle z_{1}(0)$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
r_{1}(S_{1})$ $\displaystyle=$ $\displaystyle R,$ $\displaystyle\phi_{2}(0)$
$\displaystyle=$ $\displaystyle 10^{-5},$ $\displaystyle r_{2}(0)$
$\displaystyle=$ $\displaystyle 10^{-5}R_{2},$ $\displaystyle z_{2}(0)$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle r_{2}(S_{2})$
$\displaystyle=$ $\displaystyle R,$ $\displaystyle h^{\prime}(R_{\infty})$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle h(R)$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\tilde{r}(R)$ $\displaystyle=$ $\displaystyle
R,$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sigma_{1g}\cos(\phi_{1}(S_{1}))+\sigma_{12}\cos(\phi_{2}(S_{2}))-\sigma_{2g}\frac{1}{\sqrt{1+\left[h^{\prime}(R)\right]^{2}}},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sigma_{1g}\sin(\phi_{1}(S_{1}))-\sigma_{12}\sin(\phi_{2}(S_{2}))+\sigma_{2g}\frac{h^{\prime}(R)}{\sqrt{1+\left[h^{\prime}(R)\right]^{2}}},$
$\displaystyle\rho_{1}g(V_{1}+V_{2})$ $\displaystyle=$
$\displaystyle\rho_{2}gV_{2}+\rho_{2}g\pi R^{2}h_{0}+\sigma_{2g}2\pi
R\frac{h^{\prime}(R)}{\sqrt{1+\left[h^{\prime}(R)\right]^{2}}},$ (31)

where the last condition represents the balance of the vertical forces acting
at the triple-phase contact line. The condition $r_{i}(0)=10^{-5}R_{i}$
together with $\phi_{1}(0)=10^{-5}$ ensure that
$\sin(\phi_{i})/r_{i}=R_{i}^{-1}$ at $s_{i}=0$.

Finally, we add two integral conditions associated with the volumes of the
pendant part $V_{2}$ and the total drop volume $V_{1}+V_{2}$

$\displaystyle V_{2}$ $\displaystyle=$ $\displaystyle\int_{0}^{S_{2}}\pi
r_{2}^{2}\sin(\phi_{2})\,ds_{2},$ $\displaystyle V_{1}+V_{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{S_{1}}\pi
r_{1}^{2}\sin(\phi_{1})\,ds_{1}+\int_{0}^{S_{2}}\pi
r_{2}^{2}\sin(\phi_{2})\,ds_{2}.$

To be able to continue a solution of the set of nine equations with fourteen
boundary and two integral conditions one requires $14+2-9+1=8$ active
continuation parameters. These are chosen to be
$(S_{1},S_{2},R_{1},R_{2},R,h_{0},V_{2})$ with one additional principal
continuation parameter, such as, for example, the gravity constant $g$ or the
total drop volume $V_{1}+V_{2}$. The AUTO files are available on demand.

## Appendix B

This section summarizes the computation of the stability function $f(g)$, from
Eq. (23) in terms of the variables $z_{i}$, $\phi_{i}$, $r_{i}$ and $h(r)$
based on Eqs. (1) - (3). Taking the level of the fluid bath far away from the
drop as a zero level, the vertical coordinate of the centre of mass of the
drop $z_{c}$ can be expressed as

$\displaystyle
z_{c}=\frac{\pi}{V}\left(\int_{0}^{S_{1}}r_{1}^{2}(z_{1}(S_{1})-h_{0}-z_{1})\sin(\phi_{1})ds_{1}+\int_{0}^{S_{2}}r_{2}^{2}(z_{2}-h_{0}-z_{2}(S_{2}))\sin(\phi_{2})ds_{2}\right),$

where $V=V_{1}+V_{2}$ is the total volume of the drop and all other variables
have been introduced in Section II. After some simplification we obtain

$\displaystyle z_{c}=\frac{V_{1}z_{1}(S_{1})-\int_{0}^{S_{1}}\pi
r_{1}^{2}z_{1}\sin(\phi_{1})ds_{1}-V_{2}z_{2}(S_{2})+\int_{0}^{S_{2}}\pi
r_{2}^{2}z_{2}\sin(\phi_{2})ds_{2}}{V_{1}+V_{2}}-h_{0}.$ (34)

The excess potential energy $U_{e}$ from Eq. (15) of the fluid bath is given
by the potential energy of the shaded part in Fig.3, filled with fluid $2$
with the negative density $-\rho_{2}$. $U_{e}$ can be split into the energy of
the pendant part of the drop and the energy of the remaining upper part of the
shaded region in Fig.3

$\displaystyle U_{e}=-g\rho_{2}\left[\int_{0}^{S_{2}}\pi
r_{2}^{2}(z_{2}-h_{0}-z_{2}(S_{2}))\sin(\phi_{2})\,ds_{2}+\int_{R}^{\infty}\pi
r^{2}(h(r)-h_{0})h^{\prime}(r)\,dr\right],$

where $R$ is the contact radius. Finally, the stability function $f(g)$ is

$\displaystyle f(g)=z_{c}+\frac{U_{e}}{gV\rho_{1}}.$ (36)

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