# Bubble rising near a vertical wall: experimental characterization of paths
and velocity.

C. Estepa-Cantero<EMAIL_ADDRESS>Área de Mecánica de Fluidos, Departamento de
Mecánica de Estructuras e Ingeniería Hidráulica, Universidad de Granada,
Campus Fuentenueva s/n, 18071, Granada, Spain.  Andalusian Institute for Earth
System Research, University of Granada. Avda. del Mediterráneo s/n, 18006,
Granada, Spain.  C. Martínez-Bazán<EMAIL_ADDRESS>Área de Mecánica de
Fluidos, Departamento de Mecánica de Estructuras e Ingeniería Hidráulica,
Universidad de Granada, Campus Fuentenueva s/n, 18071, Granada, Spain.
Andalusian Institute for Earth System Research, University of Granada. Avda.
del Mediterráneo s/n, 18006, Granada, Spain.  R. Bolaños-Jiménez
<EMAIL_ADDRESS>Área de Mecánica de Fluidos, Departamento de Ingeniería
Mecánica y Minera. Universidad de Jaén. Campus de las Lagunillas, 23071, Jaén,
Spain.  Andalusian Institute for Earth System Research, Universidad de Jaén.
Campus de las Lagunillas, 23071, Jaén, Spain.

###### Abstract

The trajectories of a single bubble rising in the vicinity of a vertical solid
wall are experimentally investigated. Distinct initial wall-bubble distances
are considered for three different bubble rising regimes, i.e. rectilinear,
planar zigzag, and spiral. The problem is defined by three control parameters,
namely the Galilei number, $Ga$, the Bond number, $Bo$, and the initial
dimensionless distance between the bubble centroid and the wall, $L$. We focus
on high-Bond numbers, varying $L$ from 1 to 4, and compare the results with
the corresponding unbounded case, $L\rightarrow\infty$. In all cases, the
bubble deviates from the expected unbounded trajectory and migrates away from
the wall as it rises due to the overpressure generated in the gap between the
bubble and the wall. This repulsion is more evident as the initial wall-bubble
distance decreases. Moreover, in the planar zigzagging regime, the wall is
found to impose a preferential zigzagging plane perpendicular to it when $L$
is small enough. Only slight wall effects are observed in the velocity or the
oscillation amplitude and frequency. The wall migration effect is more evident
for the planar zigzagging case and less relevant for the rectilinear one.
Finally, the influence of the vertical position of the wall is also
investigated. When the wall is not present upon release, the bubbles have the
expected behavior for the unbounded case and experience the migration only
instants before reaching the wall edge. This repulsion is, in general, more
substantial than in the initially-present-wall case.

††preprint: Manuscript in preparation

##  I Introduction

Bubble rising in liquids constitutes a relevant two-phase flow phenomenon that
has been the subject of vast research for decades. Its paramount importance
relies not only on its wide application in many different industrial
operations but also on its broad occurrence in numerous natural processes,
such as methane seep in marine fissures or gas bubbles in volcano eruptions
(Dean  _et al._ , 2018; Liu  _et al._ , 2015). Regarding technical
applications, bubble rising is relevant in areas as diverse as water
treatment, mineral flotation, carbon capture, drag reduction, ocean
microplastics scavenging, or medical treatment and diagnosis, among many other
techniques (Rodríguez-Rodríguez  _et al._ , 2015; Rajapakse  _et al._ , 2022;
Shaw and Deike, 2022; Wang, Wang, and Liu, 2022; Zhao, Wright, and Goertz,
2023; Lehmann, Häusl, and Gekle, 2023; Gao  _et al._ , 2023; Ri  _et al._ ,
2023).

The rising of an isolated bubble in an unbounded configuration has been
thoroughly investigated in the past (Clift, Grace, and Weber, 1978; Marusic
and Broomhall, 2021; Yang, Prosperetti, and Takagi, 2003) not only
theoretically and numerically (Blanco and Magnaudet, 1995; Magnaudet and
Eames, 2000; Mougin, 2002; Mougin and Magnaudet, 2006; Magnaudet and Mougin,
2007; Tripathi, Sahu, and Govindarajan, 2015; Cano-Lozano  _et al._ , 2015,
2016a; Zhang  _et al._ , 2021; Bonnefis, Fabre, and Magnaudet, 2023), but also
experimentally (Haberman and Morton, 1953; Bhaga and Weber, 1981; Duineveld,
1995; Maxworthy  _et al._ , 1996; Magnaudet and Eames, 2000; Ellingsen and
Risso, 2001; Vries, Biesheuvel, and Wijngaarden, 2003; Shew, Poncet, and
Pinton, 2006; Zenit and Magnaudet, 2008; Veldhuis, Biesheuvel, and Van
Wijngaarden, 2008; Legendre, Zenit, and Velez-Cordero, 2012; Aoyama  _et al._
, 2016; Rastello, Marié, and Lance, 2017; Agrawal  _et al._ , 2021; Kure  _et
al._ , 2021; She  _et al._ , 2021; Liu  _et al._ , 2022). However, most of
these studies are focused on bubbles at low-Bond and high-Reynolds numbers,
with the Bond number $Bo=\rho gD^{*2}/\sigma$ and the Reynolds number $Re=\rho
v^{*}D^{*}/\mu$, where $\rho$ and $\mu$ the liquid density and viscosity,
respectively, $\sigma$ the surface tension, $g$ gravity acceleration, $D^{*}$
the equivalent bubble diameter and $v^{*}$ the bubble terminal velocity. A
general outcome of these works is that the path that the bubble follows as it
rises in a quiescent liquid depends on Bond, $Bo$, and Galilei, $Ga$, numbers,
with $Ga=\rho g^{1/2}D^{*3/2}/\mu$, and therefore on the bubble shape (or the
major-to-minor bubble diameter ratio, $\chi$). At low $Bo$, the bubble is
spherical and, apart from a standing eddy on the rear, no vortical structures
are developed. Thus, it tends to follow a rectilinear path (Clift, Grace, and
Weber, 1978; Blanco and Magnaudet, 1995; Cano-Lozano  _et al._ , 2016a). As
$Bo$ increases, the bubble aspect ratio also increases, and the bubble wake
may become unstable, leading to path instability (Mougin and Magnaudet, 2002;
Magnaudet and Mougin, 2007), and providing two-dimensional (planar zigzagging
regime) or even three-dimensional (spiraling or flat-spiraling regimes) paths
(Yang, Prosperetti, and Takagi, 2003; Zenit and Magnaudet, 2008). The planar
zigzagging path is characterized by the shedding of two or four symmetric
counter-rotating trailing vortices(Cano-Lozano  _et al._ , 2016a; She  _et
al._ , 2021), while the wake of the spiraling bubble shows two intertwined
vortices wrapping around each other(Cano-Lozano  _et al._ , 2016a).

However, there are many real situations where the bubbles do not freely rise
but they unavoidably interact with solid boundaries. Under these conditions,
certainly, their rising dynamics are modified by the presence of the wall.
Many theoretical and numerical works have addressed this problem (Magnaudet,
2003; Magnaudet and Mougin, 2007; Sugiyama and Takemura, 2010; Sugioka and
Tsukada, 2015; Zhang  _et al._ , 2020; Hasan and Hasan, 2022) reporting a
clear migration away from the wall, caused by the constrain imposed in the
bubble movement. Indeed, Magnaudet (2003) performed an analytical study to
describe the deformation-induced lift force on the bubble. As for numerical
works, Sugiyama and Takemura (2010) corroborated that the migration of the
bubble away from the wall happened due to the deformation of the bubble. They
showed that the theoretical approaches available at that time did not provide
an accurate prediction of the amplitude of this migration for the smallest
bubble-wall distances and presented a new approach to predict the motion of
slightly deformed bubbles under these conditions. Zhang  _et al._ (2020)
performed unsteady, three-dimensional numerical simulations to describe the
dynamics of rectilinear and spiraling rising bubbles in the vicinity of a
vertical wall. They stated that the wall induced a destabilizing effect in the
wall-normal direction while it acted as a stabilizing factor in the spanwise
direction. They also observed an increase in the bubble-wall separation and in
the lateral oscillations which was stronger the closer the bubble was
initially to the wall. They found that the wall caused a lateral disturbance
whose magnitude increased as the wall-bubble initial distance decreased,
although these results have not been experimentally confirmed yet. Regarding
the spiraling motion, Zhang  _et al._ (2020) observed that the phase was
modified by the wall, but the amplitude and frequency of oscillation were both
unaffected by the initial distance. Moreover, Yan  _et al._ (2022) reported
numerically that a greater initial bubble-wall distance promoted less
deflection and that the path oscillations leading to an eventual spiral motion
began earlier as the distance to the wall decreased. In addition, they
observed that the bubble terminal velocity was slightly smaller than its
unbounded counterpart due to the wall additional viscous effect, which leads
to an increase in the bubble drag force. Hasan and Hasan (2022) studied the
migration dynamics of a deformable bubble in the presence of not one wall, but
two of them forming a corner. They reported that the bubble suffered a
migration perpendicular to the corner, i.e. forming 45o with each surface,
because its motion was constrained in two directions. They also concluded that
the wall was a destabilizing factor on the wall-normal direction and
attributed the strongest deformations of the bubble to the modification of
wake structure generated by the walls. Finally, Yan  _et al._ (2023) related
the lateral migration to changes in the flow distribution around the bubble by
means of simulations. They claimed that the wall presence favors bubble
oscillations along the wall-normal plane, being restricted across the parallel
direction. Moreover, they concluded that the wall promotes bubble vortex
shedding, and therefore the transition from a stable rising regime to an
unstable one.

The migration of bubbles rising in the presence of a vertical wall has been
hardly explored experimentally due to the complexity involved in performing
these kinds of experiments in a controlled manner. Still, a few experimental
works (Takemura  _et al._ , 2002; Takemura and Magnaudet, 2003; Jeong and
Park, 2015; Lee and Park, 2017; Yan  _et al._ , 2022, 2023) can be found,
although at low-Bond regimes. In particular, Takemura  _et al._ (2002) and
Takemura and Magnaudet (2003) investigated the motion of a spherical bubble
near a wall for $Re<100$ and reported that the repulsion force acting on the
bubble happens due to the asymmetric distributions of the vortices with
respect to the bubble symmetry plane. Their work was corroborated numerically
by Sugioka and Tsukada (2015) with only slight differences and they clarified
that the drag force increased with the bubble-wall proximity. Moreover, Jeong
and Park (2015) tested a zigzagging bubble, finding that the bubble mostly
bounced off the wall for distances smaller than the bubble diameter. Under
very similar conditions, Lee and Park (2017) used Particle Image Velocimetry
to experimentally investigate the wake behind a zigzagging bubble in the
proximity of a wall.

The objective of the present work is to improve the knowledge on the rising of
bubbles near a wall by extending our study to regimes of high Bond numbers,
which have barely been explored experimentally. These millimeter-sized bubbles
are present in many natural and industrial processes, such as in aeration
tanks used in bioreactors and chemical reactors, or mineral purification by
flotation techniques, among many others. Furthermore, the experimental results
reported here might be useful to corroborate the numerical and theoretical
approaches.

This work is organized as follows. Section II describes the experimental
facility and the methodology followed. Section III reports the results
obtained for the three distinct regimes tested for different values of initial
wall-bubble distance. A comparison of the wall effect among different regimes
is included as well as the impact of the vertical position of the wall on
bubble trajectory. Finally, section IV is devoted to the conclusions extracted
from our analysis.

Figure 1: (a) Sketch of the experimental setup. (b) Sketch of the physical
problem including the physical and geometrical governing parameters. Note that
the reference-frame origin is set at the bubble centroid right after bubble
pinch-off. The trajectory is vertically defined by $z^{*}$ while $x^{*}$ and
$y^{*}$ are the horizontal axes in the direction perpendicular and parallel to
the wall, respectively. (c) Bubble images corresponding to the three regimes
studied here, taken from the high-speed recordings, showing the bubble contour
and their centroids (red line and asterisk, respectively). From left to right:
rectilinear, planar zigzag and spiral regimes.

##  II Experimental setup and methods

Figure 2: Images of an individual bubble at different instants, in the $xz$
(left panel) and $yz$ (right panel) planes, extracted from the high-speed
recordings for the unbounded case and the a) rectilinear, b) zigzagging and c)
spiral regimes. Note that small oscillations can be observed in the plane $yz$
in (b) because the views shown do not correspond with the zigzagging plane,
see section III.2.

The experimental setup is shown in Fig. 1(a). It consisted of an open tank 1.2
m high with a square cross-section 0.13 m wide. To assure optical access and
cleanliness, the four sides of the tank were constructed of glass and a
methacrylate cabin covered the facility entirely. Bubbles were generated
injecting a given flow rate of air through an injector placed at the center of
the tank base. Injectors of different diameters, ranging from 2 to 12 mm, were
used to generate bubbles of different sizes (see Table 1). A capillary tube
was inserted between the injector and the air feeding line to guarantee
constant flow rate conditions, and very small air flow rates were supplied to
ensure quasi-static bubble formation. Indeed, high-speed video images of the
bubble formation were analyzed to check this condition. Finally, a glass wall
was placed vertically inside the tank, parallel to the $yz$ plane, to study
its effect on the motion of the rising bubble. The horizontal distance from
the wall to the center of the injector was controlled with a rack and pinion
mechanism with micrometric precision.

Table 1: Main characteristics of the three cases studied in this work.
†$d^{*}$=12 mm is the diameter of a circular superhydrophobic surface used as
an air injector in this case. ‡ $St=$ 0.132$\pm$0.006 for wall-bounded cases
and $St=$0.160$\pm$0.007 for the free case. All tabulated data are stated as
average values $\pm$ standard deviation of all the experiments from both
bounded and unbounded cases and the three regimes, except $St$ for the spiral
regime, that exhibits different values for bounded and free case.

Case | Regime | Liquid |  $d^{*}$ (mm) |  $D^{*}$ (mm) | $Bo$ | $Ga$ |  $Mo$ ($\times 10^{6}$) | $St$
---|---|---|---|---|---|---|---|---
1 | Rectilinear | T11 | 2.50 | 3.31±0.07 | 5.0±0.2 | 59.7±1.8 | 9.87 | -
2 | Planar zigzag | T05 | 2.00 | 2.92±0.01 | 3.87±0.03 | 98.7±0.6 | 0.61 | 0.109±0.004
3 | Spiral | GW | 12.00† | 7.62±0.07 | 10.29±0.15 | 108±8 | 8.4±2.3 |  | 0.132±0.006‡
---
0.160±0.007‡

Figure 1(b) shows a schematic representation of the problem addressed in this
work. The reference-frame origin was set at the bubble centroid just after the
bubble pinch-off. Directions $x^{*}$ and $y^{*}$ define the horizontal
components of the bubble trajectory in the direction perpendicular and
parallel to the wall respectively, while $z^{*}$ defines the vertical
component. The variables involved in the free rise of a bubble are the
equivalent bubble diameter, $D^{*}$, the liquid physical properties,
$\rho,~{}\mu,~{}\sigma$, i.e. density, dynamic viscosity, and surface tension,
respectively, and the gravity acceleration, $g$. The air density and viscosity
are much smaller than those of the liquid, and thus they have been neglected.
Since the bubbles in this study are deformable, the bubble size was
characterized using the equivalent diameter $D^{*}$, defined as the diameter
of a sphere with a volume, $V^{*}$, identical to that of the bubble,
$D^{*}=(6V^{*}/\pi)^{1/3}$. Here asterisks denote dimensional parameters, in
contrast to their non-dimensional counterparts. Taking $D^{*}$, gravitational
velocity, $\sqrt{gD^{*}}$, and gravitational time, $\sqrt{D^{*}/g}$, as the
characteristic length, velocity, and time, respectively, to obtain the
dimensionless parameters for our analysis, the unbounded problem is fully
defined by Bond, $Bo=\rho gD^{*2}/\sigma$, and Galilei,
$Ga=\rho\sqrt{gD^{*3}}/\mu$, numbers (Cano-Lozano  _et al._ , 2016a). Morton
number, $Mo=g\mu^{4}/\rho\sigma^{3}$, related to Bond and Galilei numbers as
$Mo=Bo^{3}/Ga^{4}$, is also useful to describe this problem since it only
depends on the liquid properties. The values of $Bo$, $Ga$, $Mo$, as well as
the bubble diameters, corresponding to the experiments reported in this work,
are summarized in Table 1. In addition, in the bounded case, the initial
horizontal distance between the bubble centroid and the rigid wall,
$L=L^{*}/D^{*}$, represents a relevant parameter. Our experiments were
performed for four values of $L$, i.e. $L=1$, 2, 4 (wall-bounded cases) and
$L\rightarrow\infty$ (unbounded case). Reynolds, $Re=\rho
v^{*}D^{*}/\mu=Ga\,v$, and Weber, $We=\rho v^{*2}D^{*}/\sigma=Bo\,v^{2}$,
numbers will be outcomes of the problem for each pair of ($Ga,Bo$) values,
being $v^{*}$ the bubble terminal velocity and $v=v^{*}/\sqrt{gD^{*}}$ its
dimensionless counterpart. Note that the latter is indeed the Froude number,
$v=Fr$. Additionally, the unstable regimes exhibit oscillations with certain
amplitude $A^{*}$ and frequency $f$, characterized by the dimensionless
amplitude, $A=A^{*}/D^{*}$, and the Strouhal number, $St=fD^{*}/v^{*}$,
respectively.

Table 2: Main properties of the three liquids used in this work. The values corresponding to the silicon oils have been provided by the manufacturer. Liquid | Characteristics |  $\rho$ (kg/m3) |  $\mu$ (mPa$\cdot$s) |  $\sigma$ (mN/m)
---|---|---|---|---
T11 | Silicon oil | 935 | 9.35 | 20.1
T05 | Silicon oil | 913 | 4.57 | 19.7
GW | Glycerol-water | 1188.7±1.1 | 22.1±1.9 | 65.7±0.1

Three cases are considered in this work, each one exhibiting different rising
paths according to the numerical study by Cano-Lozano  _et al._ (2016a). In
particular, we selected the liquids and air injector diameters in order to
achieve conditions similar to those of bubbles 22, 19, and 26 in Table I of
Cano-Lozano  _et al._ (2016a). The three selected bubbles exhibit rectilinear,
planar zigzagging, and spiraling paths respectively, all of them being in the
proximity of the transition curves between regimes (Cano-Lozano, Bohorquez,
and Martínez-Bazán, 2013; Cano-Lozano  _et al._ , 2016b). The properties of
the liquids used in this work are shown in Table 2. Two different viscous
silicon oils, T11 and T05 (Dow Corning® XIAMETER™ PMX-200) were used for the
liquid phase, similarly to the reference work. However, in Case 3, the
capillary length associated with T11, the silicon oil considered for this case
in the numerical study, was much smaller than the air injector radius required
to generate the desired bubble size, preventing bubbles from being generated.
In order to overcome this issue, a glycerol-water (GW) mixture was employed
together with a superhydrophobic substrate (Rustoleum® NeverWet™) coating the
air injector. The necessary diameter of the superhydrophobic substrate was
obtained using the correlation given by Rubio-Rubio  _et al._ (2021). The GW
mixture composition was 74.16-74.89% in weight of glycerol, depending on the
case. Distilled water was used to make the GW mixtures. The effects of
surfactants on the bubble dynamics have been shown to be negligible when the
bubble is large enough (Duineveld, 1995; Ellingsen and Risso, 2001; Shew,
Poncet, and Pinton, 2006), like those considered in Case 3, as will be shown
later. The physical properties of the mixtures were measured in the laboratory
and compared to theoretical data (Dorsey, 1940; Glycerine Producers’
Association, 1963; Streit  _et al._ , 2005; Spann  _et al._ , 2021). The
temperature was continuously registered and the fluid properties were tested
for the whole range of temperature during the experiments using a Brookfield
DV3TLVCJ0 rheometer, a Krüss K20 force tensiometer, and a Mettler Toledo
Density2Go Densito densimeter. The average values of the GW physical
properties are shown in Table 2. The corresponding values of the resulting
dimensionless control parameters for the three experimental cases are
summarized in Table 1.

Before running any set of experiments, several steps were taken to ensure
optimal experimental conditions. The tank and the vertical wall were
thoroughly cleaned between different experimental runs using water and ethanol
to eliminate dust and surfactants, which may change the dynamics of the
problem (Duineveld, 1995). The liquids were replaced after some experimental
runs. To ensure the verticality of both the wall and the camera motion system,
a plumb bob was hung inside the tank using a nylon monofilament cord with a
calibrated diameter of 0.45 mm. The cord was recorded along the height of the
tank to eliminate possible minor horizontal displacements of the system in the
image processing. Moreover, before running each experiment, the bubble
terminal velocity was determined. Afterwards, a servo motor was used to move
the cameras similarly to previous works(Kure  _et al._ , 2021), being
controlled with a SMC Lecsa2-S4 driver and the software Melsoft MR
Configurator, which enabled us to program the cameras position, velocity, and
acceleration over time.

The ascent of the bubble was filmed using two synchronized high-speed cameras
(Fastcam SA1.1, Fastcam Mini Ax200) positioned perpendicularly to each other,
and mounted on a traverse rail that moved at a pace similar to the bubble
rising velocity. Thus, each camera recorded images from one side of the tank,
parallel and perpendicular to the glass wall, to allow for a three-dimensional
reconstruction of the bubble path. The cameras were also synchronized with the
servomotor that controlled the motion of the traverse they were attached to in
order to determine the bubble vertical position from the images. A laser beam
and a photodiode sensor were used to detect the bubble pinch-off and to
trigger the high-speed cameras recording and the traverse motion. A couple of
1.2-meter-long LED panels were used for backlighting, providing a highly
uniform background. The shutter speed was adjusted depending on the camera
model and the lighting conditions between 1/1000 and 1/30000 s. Every
individual experimental case was performed at least 10 times to ensure
repetitiveness. The time between each recorded bubble was long enough to avoid
possible effects of the previous bubble wake on the results.

Examples of the evolution of the bubbles along their paths are shown in Fig. 2
for the rectilinear, zigzagging and spiral regimes. The bubble contour and
centroid were detected in each image using an in-house image processing
routine based on the Matlab Image Segmentation Toolbox and an integral image
thresholding method (Otsu, 1979; Bradley and Roth, 2007), as shown in Fig.
1(c). The contours were used to compute the bubble size, shape, and equivalent
diameter $D^{*}$, while the centroid position was used to obtain bubbles
trajectories and velocities. The servomotor software allowed us to determine
the $z$ coordinate of the centroid with an accuracy of 1 $\mu m$. The overall
accuracy of the experiments depended on the temporal and spatial resolution of
the films, which ranged from 500 to 2000 frames per second, and between 17.79
and 36.31 $\mu$m per pixel respectively. The error of the centroid position
determined by image processing was evaluated using the estimation by Ho
(1983). The absolute errors corresponding to different calculated parameters
along the manuscript were estimated by applying propagation of uncertainty.

Figure 3: Bubbles silhouettes, extracted from the high-speed recordings at
different instants of time as they rise, for the rectilinear regime in the
$xz$ (left) and $yz$ (right) planes. (a) $L\rightarrow\infty$, (b) $L=4$, (c)
$L=2$, and (d) $L=1$. The wall, not shown for the sake of clarity, is placed
on the left-hand side of the $xz$ plane at (b) $x=-4$, (c) $x=-2$, and (d)
$x=-1$. The bubbles have been scaled down for clarity.

##  III Results

This section is devoted to presenting the results obtained from the
experiments. The section is organized as follows: the particular results
corresponding to the rectilinear, planar zigzag, and spiral regimes are
described in subsections III.1, III.2 and III.3, respectively. Subsection
III.4 presents a direct comparison between the three regimes. Finally, the
effect of the vertical distance from the wall to the injector is explored in
subsection III.5.

###  III.1 Rectilinear regime

Stable bubbles that rise with a rectilinear path were obtained using T11
silicon oil together with a 2.5 mm diameter injector (Case 1 in Table 1). The
resulting average Bond and Galilei numbers ($Bo$=5.0 and $Ga$=59.7) are in
good agreement with those of the targeted case (bubble 22 in Cano-Lozano  _et
al._ , 2016a, with $Bo$= 5.0 and $Ga$= 59.61).

Table 3: Experimental values of the terminal velocity and terminal Reynolds number obtained for the three cases described in the present work and the different wall distances. All tabulated data refer to the mean value ± standard deviation of all the experiments performed for the cases defined by each row and column. Case | $L=1$ | $L=2$ | $L=4$ | $L\rightarrow\infty$
---|---|---|---|---
1 |  $v$ | 1.02±0.04 | 1.07±.04 | 1.06±0.04 | 0.99±0.04
$Re$ | 59.0±0.7 | 62.0±0.6 | 61.6±0.6 | 60.8±0.3
2 |  $v$ | 1.06±0.04 | 1.1±0.04 | 1.11±0.04 | 1.15±0.01
$Re$ | 105.3±2.1 | 109.1±0.8 | 109.1±0.4 | 114.6±0.2
3 |  $v$ | 0.84±0.01 | 0.83±0.01 | 0.91±0.01 | 0.83±0.02
$Re$ | 90.4±0.6 | 90.0±0.6 | 87.5±0.6 | 99.6±2.7
Figure 4: Bubble paths in the rectilinear regime for $L\rightarrow\infty$
(black, $Bo=5.29,~{}Ga=62.2$), $L$=4 (green, $Bo=4.84,~{}Ga=58.2$), $L$=2
(blue, $Bo=4.79,~{}Ga=57.7$), and $L$=1 (red, $Bo=4.77,~{}Ga=57.6$). (a)
Three-dimensional view. (b) Side view ($xz$ plane), perpendicular to the wall
plane. Note that horizontal scales in (a) and (b) are magnified with respect
to the vertical one to make the migration movement more visible. (c)-(f) Top
views (xy plane) for different values of L. The vertical wall, not plotted for
the sake of clarity, is placed at plane $yz$ at $x=-4$ for $L=4$, $x=-2$ for
$L=2$, and $x=-1$ for $L=1$.

Figure 3 shows bubble silhouettes at different times in the planes
perpendicular and parallel to the wall, $xz$ and $yz$ planes respectively, for
three different initial wall distances, together with the unbounded case (Fig.
3a). Note that the bubbles do indeed go up vertically in both planes following
a straight path in the case without wall, $L\rightarrow\infty$ (Fig. 3a), in
agreement with Cano-Lozano  _et al._ (2016a). In particular, the bubble
quickly evolves until reaching a nearly fore-and-aft symmetric oblate
ellipsoidal shape (see images depicted in Figs. 1c and 2a). An average major-
to-minor bubble diameter aspect ratio $\chi=2.53\pm 0.06$ is obtained from all
the experiments performed in Case 1. This value agrees fairly well with that
obtained using the expression given by Legendre, Zenit, and Velez-Cordero
(2012), $\chi_{L}=[1-9We/64\,(1+0.2\,Mo^{1/10}\,We)^{-1}]^{-1}=2.19$, 13%
lower than our experimental value, with $We=Bo\,v^{2}=5.12$, being $Bo=5.22$
and $v=0.99$ (see Table 3) the average values corresponding to the unbounded
case. Unlike Yan  _et al._ (2022), who reported a decrease of $\chi$ in the
presence of a vertical wall, no remarkable effects of the wall on the overall
bubble aspect ratio have been found in our experiments. Within the rectilinear
regime, it is known that the flow around the bubble exhibits a standing eddy,
while it describes a stable straight upward trajectory (Blanco and Magnaudet,
1995; Cano-Lozano, Bohorquez, and Martínez-Bazán, 2013; Cano-Lozano  _et al._
, 2016a). Although this type of stable bubble has been theoretically and
numerically reported (Magnaudet and Mougin, 2007), very few experimental works
(Zenit and Magnaudet, 2008) can be found due to the complexity it implies to
generate them in a controlled manner, especially for large $Bo$. In this case,
when a vertical surface is placed (Figs. 3b-d), the bubble migrates
transversely away from the wall as it rises vertically in the plane parallel
to the wall, due to the overpressure generated in the liquid film between the
bubble and the wall, as can be noticed in both the $xz$ and the $yz$ planes.
This effect becomes more evident as the initial wall distance, $L$, decreases,
(see Fig. 3d for $L=1$). Consequently, the vertical wall induces a transversal
force to the bubble but does not modify the rising regime, keeping a stable
rising path regardless of the wall distance.

To analyze the migration effect in more detail, the dimensionless bubble
rising paths have been determined from the time evolution of the bubble
centroid (see Fig. 4). In particular, the three-dimensional paths
corresponding to the different initial wall distances, together with the case
without the wall, are shown in Fig. 4(a). In the absence of wall, the bubble
indeed follows a nearly vertical route. Considering that the wall is placed in
the $yz$ plane, notice that, when the wall is present, the bubble still rises
following a straight path, but it exhibits a transversal motion in the $xz$
plane, moving away from the wall, as shown in Figs. 4(c)-(f). The migration
effect is even more evident in Fig. 4(b), where the paths are plotted in the
plane perpendicular to the wall, $xz$. Note that the migration effect
increases as the wall is initially closer to the bubble, i.e., for a given
height $z$, the bubble is further away as $L$ decreases. In fact, while the
maximum transverse motion is almost negligible for $L=4$ ($x\simeq 0.11$), it
reaches $x\simeq 1.23$ for $L=1$, as observed in the top views (Figs. 4c-f).
Although for different values of $Bo$ and $Ga$, these results are in general
in good agreement with recent numerical simulations performed under similar
conditions (Zhang  _et al._ , 2020; Yan  _et al._ , 2022, 2023). It is
interesting to point out that the bubble keeps moving away from the wall when
it reaches the top of the tank for all the values of $L$, i.e. the migration
effect is still present at a height as large as $z\sim$ 200\. In particular,
contrary to the numerical results for similar cases (Zhang  _et al._ , 2020;
Yan  _et al._ , 2022), where the angle of the rising paths with the vertical
plane is shown to decrease as the bubble rises, a nearly constant angle is
found in our results. This fact indicates that probably a higher tank would
have been required to observe that the wall effect vanishes, being the bubble
path vertical again. Nevertheless, as far as we are aware, the vertical
distance explored in the present study ($z\gtrsim 200$) is larger than those
analyzed experimental and numerically in previous works for similar values of
the dimensionless parameters.

The lateral migration or repulsive force generated by the wall was already
reported theoretically (Leal, 1980; Magnaudet, 2003; Sugiyama and Takemura,
2010) for small-to-moderate Reynolds number, and explained as an effect of the
vorticity diffusion, as well as numerically (Yang, Prosperetti, and Takagi,
2003; Sugioka and Tsukada, 2015; Zhang  _et al._ , 2020; Yan  _et al._ ,
2022). These works showed that the flow velocity is restrained in the vicinity
of the wall, which breaks the wake symmetry, producing a perpendicular force
that causes the lateral motion of the bubble. The migration effect caused by
the wall has been also observed in experimental works (Takemura  _et al._ ,
2002; Takemura and Magnaudet, 2003; Jeong and Park, 2015; Lee and Park, 2017).
Nevertheless, the latter are mainly focused on nearly spherical bubbles,
corresponding to $Bo$ numbers smaller than those explored in the present work.
Specifically, the wall-normal movement agrees with the results obtained by
Takemura and Magnaudet (2003), who, for low $Bo$ and low-to-intermediate $Re$
spherical clean bubbles, showed the existence of a transverse force that
decreased with the separation distance between the bubble and the wall. They
proved that the wall-normal force could be repulsive or attractive to the wall
depending on $Re$. Particularly, for $Re<35$, the presence of the wall
generated an asymmetric vorticity distribution around the bubble surface,
resulting in a transverse force directed away from the wall. For higher $Re$,
an acceleration of the flow in the gap between the bubble and the wall was
predicted by the irrotational theory, giving rise to a force directed towards
the wall. In our results, where the bubbles are ellipsoidal, a repulsive lift
is observed, even though $Re>35$, indicating that the dynamics completely
change when the bubble deviates from the spherical shape. In this case, the
bubble’s interface forms a gap with the wall similar to that of a wedge and
the liquid moving through generates an overpressure that repels the bubble.
Regarding the numerical works, a repulsive lift force is obtained for similar
parameters, in good agreement with our results. Specifically, for $L=1$, Yan
_et al._ (2022) calculated a lateral deviation of $x\simeq$ 0.3 at $z$= 90 for
a similar $Ga$ but $Bo=2$, while $x\simeq$ 0.5 in our case. Taking into
account that $Bo$ is higher in the present work, larger bubble shape
deformations take place and the vortex structure becomes more asymmetrical,
thus enhancing the wall effect, the agreement is remarkable.

Figure 5: Temporal evolution of (a) the horizontal velocity, $v_{x}$, and (b)
the vertical one, $v_{z}$, for Case 1 (rectilinear) and different wall
distances. The inset in panel (b) shows the temporal evolution of the local
Reynolds number, $Re_{t}=v_{z}\,Ga$.

The rising bubble motion can be further analyzed by inspecting the bubble
velocity, obtained from the time derivative of the bubble paths shown in Fig.
4. In particular, Fig. 5(a) shows the time evolution of the horizontal
velocity perpendicular to the wall, $v_{x}$. As expected, when the bubble
rises freely ($L\rightarrow\infty$), the horizontal velocity is nearly
negligible. However, when the bubble ascents in the presence of the wall, a
positive value of $v_{x}$ can be clearly observed. For $L>1$ the horizontal
velocity is almost constant during the bubble migration process and increases
as $L$ decreases ($v_{x}\approx 0$ for $L\rightarrow\infty$ and $L=4$ and it
increases to $v_{x}\approx 0.002$ for $L=2$). Interestingly, for $L=1$ the
horizontal velocity initially increases due to the repulsive effect of the
wall until it is sufficiently far, when $v_{x}$ begins to decrease. However, a
positive horizontal velocity is still observed when the bubbles reach the top
of the tank, indicating that the effect of the wall persists at the final
stages of our experiments, even though the bubbles are already far from the
wall ($x\simeq 1.2$ at $t\simeq 200$ for $L=1$).

Figure 6: Bubble silhouettes in the zigzagging regime (Case 2), extracted from
the high-speed recordings at different instants of time in the $xz$ (left) and
$yz$ (right) planes. (a) $L\rightarrow\infty$, (b) $L=4$, (c) $L=2$, and (d)
$L=1$ The vertical wall is placed in plane $xz$ at $x=-4$ ($L=4$), $x=-2$
($L=2$), $x=-1$ ($L=1$). The bubbles have been scaled down for clarity. Note
that the main zigzagging plane does not coincides with plane $xz$ nor $yz$ in
cases (a)-(c), as shown in Fig.7.

Figure 5(b) shows the time evolution of the vertical component of the
velocity, $v_{z}(t)$, for the four values of $L$ considered. The vertical
velocity rapidly increases from $v_{z}$= 0 at $t=0$ until it reaches a nearly
constant value, i.e., the terminal velocity, $v$. Notice that for all values
of $L$, $v\simeq 1$ (see Table 3), indicating that the terminal velocity is
practically the gravitational velocity. The small differences observed in the
vertical velocity are mostly associated with little differences in the sizes
of the bubbles when they are generated rather than due to wall effects.
Nevertheless, although a similar evolution is observed, it can be inferred
that larger initial accelerations are promoted by the wall, and therefore the
terminal velocity is achieved sooner than in the free-rise case. Note also
that, after the initial acceleration, the velocity decreases before reaching
the terminal velocity. However, when the vertical velocity is expressed in
terms of the local Reynolds number, $Re_{t}(t)=v_{z}(t)\,Ga$ (inset in Fig.
5b), the Reynolds number associated to the terminal velocity, $Re=v\,Ga$,
barely varies with $L$, except for $L=1$, in which a slightly lower value of
$Re$ is reached (see Table 3). Although the presence of the wall is expected
to reduce $Re$, such an effect is only noticeable when the wall is
sufficiently close ($L=1$). The $Re$ reduction by the presence of the wall has
been shown to be evident for low $Ga$ numbers but becomes smaller as $Ga$
increases (Yan  _et al._ , 2022). Finally, we would like to remark that the
terminal Reynolds number given by Rastello, Marié, and Lance (2011), valid for
free clean bubbles rising within the stable regime,
$Re_{R}=2.05\,We^{2/3}\,Mo^{-1/5}=61.03$, with $Mo=9.87\times 10^{-6}$ and
$We=Bo\,v^{2}=5.12$, agrees very well with the experimental value obtained for
$L\rightarrow\infty$, $Re=60.8\pm 0.3$. This excellent agreement verifies that
bubbles in Case 1 are stable and their surface can be assumed to be clean.
Therefore, our results can be considered representative of stable bubbles
rising in a pure liquid.

###  III.2 Planar zigzag regime

Unstable bubbles that rise in a planar zigzag regime were generated using T05
silicon oil and an injector of diameter $d^{*}$ = 2 mm (Case 2 in Table 1).
The resulting average values of the Bond and Galilei numbers ($Bo$=3.87 and
$Ga$=98.7) reasonably agree with those of bubble 19 in Cano-Lozano  _et al._
(2016a), with $Bo$=4.0 and $Ga$=100.8.

Figure 7: (a)-(c) Three-dimensional rising path corresponding to the planar
zigzagging regime (Case 2) for (a) $L\rightarrow\infty$ ($Bo=3.86,~{}Ga=98.6$)
(b) $L=4$ ($Bo=3.85,~{}Ga=98.3$) (c) $L=2$ ($Bo=3.87,~{}Ga=98.7$) and (d)
$L=1$ ($Bo=3.89,~{}Ga=99.2$). Corresponding top views for (e)
$L\rightarrow\infty$ (f) $L=4$ (g) $L=2$ and (h) $L=1$, where the angle that
the main zigzagging plane forms with the wall, $\alpha$, is indicated. The
vertical wall, not plotted for the sake of clarity, is placed at plane $yz$ at
$x=-4$ for $L=4$, $x=-2$ for $L=2$, and $x=-1$ for $L=1$.

Figure 6 shows the silhouette of an individual bubble at different times as it
rises, in planes perpendicular ($xz$) and parallel ($yz$) to the wall for
$L\rightarrow\infty$, 4, 2, and 1. Note that in this case, the bubble does not
rise with a stable path, but oscillations in both planes $xz$ and $yz$ are
observed. In fact, oscillations are mainly contained in a vertical plane, as
shown in Fig. 7, described in detail below, indicating that this case
corresponds to a planar zigzagging regime. In this case, the ellipsoidal shape
of the bubble loses its fore-and-aft symmetry (see Fig. 6 and the central
image of Fig. 1c), in agreement with previous experimental (Brücker, 1999;
Zenit and Magnaudet, 2008, 2009) and numerical results (Mougin, 2002; Mougin
and Magnaudet, 2006; Cano-Lozano  _et al._ , 2016a) on freely rising bubbles.

An average aspect ratio $\chi$= 2.57 $\pm$ 0.09 is obtained, in reasonable
agreement with the experimental results by Zenit and Magnaudet (2008) for the
same oil. According to Legendre, Zenit, and Velez-Cordero (2012), considering
the values corresponding to the free rise ($We=Bo\,v^{2}=$ 5.13 and
$Mo=0.61\times 10^{-6}$),
$\chi_{L}=[1-9We/64\,(1+0.2\,Mo^{1/10}\,We)^{-1}]^{-1}=2.38$, being 7.4% lower
than our experimental value. Note that in this case, the average value of the
Bond number is similar to that of Case 1 (Table 1), although the viscosity of
the liquid used (T05 oil) is lower than that of the liquid used in the
rectilinear case (T11 oil). This increases the corresponding $Ga$ (and
consequently the Reynolds number) and the bubble paths become unstable. Bubble
shape and path instability are related to the wake structure behind the
bubble. Particularly, the three-dimensional wake consists of two counter-
rotating trailing vortices with a symmetry plane, that change their sign twice
during the zigzag period (Cano-Lozano  _et al._ , 2016a).

Figure 8: Rising trajectories in the main zigzagging plane, $y^{\prime}z$,
(left) and the orthogonal plane, $x^{\prime}z$, (right) corresponding to the
cases in Fig. 7 for (a) $L\rightarrow\infty$ ($\alpha=18.72^{\circ}$), (b)
$L=4$ ($\alpha=-12.54^{\circ}$), (c) $L=2$ ($\alpha=44.79^{\circ}$), and (d)
$L=1$ ($\alpha=85.99^{\circ}$). Central grey lines depict the average path.

The three-dimensional rising paths corresponding to individual experiments
have been reconstructed from the bubble centroid position detected in both
$xz$ and $yz$ planes, and are plotted in Figs. 7 (a)-(d). In the absence of
wall (Fig. 7a), it is shown that, after some transient oscillations, the
bubble begins to oscillate in a plane that, in this particular case, forms an
angle $\alpha=18.72^{\circ}$ with the $yz$ plane. Furthermore, when the wall
is present (Figs. 7b-d), the bubbles still follow a planar zigzagging
trajectory and no change is observed in the rising regime. However, as in the
rectilinear regime, the wall causes the bubble to move away from it as the
bubble rises, being this effect more pronounced as $L$ decreases, in agreement
with recent numerical studies (Zhang  _et al._ , 2020; Yan  _et al._ , 2023).
Indeed, when the bubble rises near a wall, Yan  _et al._ (2023) showed
numerically that the vortex closest to the wall gets smaller, giving rise to
an asymmetric vortex structure that causes the bubble to migrate away from the
wall. These characteristics can be better analyzed from the corresponding top
views shown in Figs. 7(e)-(h), where the bubble oscillates in planes that move
away from the wall in each cycle (especially evident in Fig. 7g).
Nevertheless, the angle $\alpha$ that the oscillation plane forms with the
wall, or similarly with plane $yz$, varies in each case. In fact, from our
experiments, it has been observed that when the wall is far from the bubble,
$\alpha$ is established randomly and it is not controllable. That is, when the
bubble rises freely or sufficiently far from the wall ($L=2$, $L=4$), each
individual experiment performed with the same wall distance exhibits a
distinct zigzagging plane orientation, being different from that in cases
plotted in Fig. 7. This result, which agrees with Cano-Lozano  _et al._
(2016a), indicates that the experimental facility does not introduce any
perturbation triggering the instability and imposing a preferential zigzagging
plane. However, when $L$ is sufficiently small the bubble trajectory tends to
oscillate in a plane perpendicular to the wall (see Fig. 7h). This phenomenon
has been reported in numerical studies Zhang  _et al._ (2020), but, as far as
we are aware, this is the first experimental proof of such a phenomenon.
Interestingly, the zigzagging regime is established earlier when the wall is
present since it imposes a lateral disturbance triggering the instability.
Then, the presence of the wall is observed to promote the onset of the
oscillations in a normal direction. This fact can be attributed to the
interaction of the bubble wake with the wall, as suggested in numerical works
in which the liquid flow field is analyzed (Yan  _et al._ , 2022). In
particular, the wall is shown to modify the flow field around the bubble, the
vortex structure, and the bubble shape, which becomes asymmetric along the
central axis. Specifically, the bubble presents different curvatures at the
near-wall and the free sides, what causes a certain vorticity accumulation on
the bubble surface that imposes the shedding of periodic vortices. The latter
are observed to be symmetrical in the plane parallel to the wall, while they
lose their symmetry in the wall-normal plane, imposing a wall-normal zigagging
plane together with the migration effect.

Figure 9: Time evolution of the horizontal coordinates of the bubble
trajectory, namely, $x(t)$ (red line) and $y$ (blue line), corresponding to
cases in Fig.7, for (a) $L\rightarrow\infty$ ($\alpha=18.72^{\circ}$), (b)
$L=4$ ($\alpha=-12.54^{\circ}$), (c) $L=2$ ($\alpha=44.79^{\circ}$), and (d)
$L=1$ ($\alpha=85.99^{\circ}$). Insets in each subfigure show the PSD of
$x(t)$, exhibiting a clear peak at $f\simeq 7.08$ Hz in the bounded cases.

To directly compare the zigzag paths for different values of $L$, taking into
account the value of $\alpha$ in each case, paths are plotted in the main
zigzagging plane $y^{\prime}z$, and its orthogonal view, $x^{\prime}z$ in Fig.
8, being $y^{\prime}=x\sin{\alpha}-y\cos{\alpha}$ and
$x^{\prime}=x\cos{\alpha}+y\sin{\alpha}$. In this way, it can be confirmed
that while the oscillations take place mainly in plane $y^{\prime}z$, they are
almost negligible in the perpendicular one, $x^{\prime}z$. This corroborates
that a planar zigzagging regime is indeed established. Additionally, the
migration effect caused by the wall is more evident as $L$ decreases, as
observed in the mean path depicted in each case. Regarding the zigzagging
amplitude, it slightly increases as the bubble rises in all cases. This
indicates that the path is still not completely developed at $z\simeq 200$,
differently from previous works on free bubble rise, where the regime is fully
established for shorter heights (Cano-Lozano  _et al._ , 2016a). Moreover,
smaller vertical distances have been used in previous numerical works
including a vertical wall for similar regimes (Zhang  _et al._ , 2020; Yan
_et al._ , 2022, 2023), in which fully developed rising paths are obtained for
shorter vertical distances. Concerning the effect of the wall on the
oscillating amplitude, it slightly increases when the wall is present, being
$A\simeq\pm 0.25$ for $L\rightarrow\infty$, while $A\simeq\pm$0.3, $\pm$0.32
and $\pm$0.34 for $L$=4, 2 and 1, respectively. The increase of the amplitude
in the bounded cases seems to be due to the destabilizing effect introduced by
the wall (Zhang  _et al._ , 2020), which also explains the slight increase of
$A$ as $L$ decreases. Nevertheless, the quantitative effect of $L$ on the
amplitude is hard to infer since the angle $\alpha$ of the zigzagging plane is
different in each case and may vary with time.

To further analyze the bubble movement in this regime, Fig. 9 shows the time
evolution of the path coordinates, $x(t)$ and $y(t)$. The migration effect of
the wall on the bubble trajectory is clear. While $y(t)$ oscillates around
$y=0$ in all cases, when the wall is present $x(t)$ oscillates around a value
that increases with time because the bubble migrates away from the wall.
Again, the migration effect is more evident as $L$ decreases. Furthermore,
note that for $L=1$ (Fig. 9d), $y(t)$ barely changes with time since the
oscillating plane is nearly perpendicular to the wall. In general, once the
zigzagging regime is achieved, a constant oscillation frequency is
established, whose value is not affected by the wall, as can be clearly
observed in the power spectral density graph shown as insets in each panel of
Fig. 9. In particular, a frequency $f\simeq$ 7.08 Hz is obtained for the
bounded cases depicted. Taking all the experiments carried out within this
regime, the mean oscillation frequency is $f=7.10\pm 0.10$ Hz, which provides
a Strouhal number $St=f\,D^{*}/v^{*}=0.109\pm 0.004$, as displayed in Table 1.
An excellent agreement is achieved with the numerical result $St=0.108$,
obtained for this particular bubble in Cano-Lozano  _et al._ (2016a) for the
unbounded case. The phase shift between $x(t)$ and $y(t)$ depends on the angle
$\alpha$, being in phase when $\alpha<0$ (Fig. 9b) and out of phase for
$\alpha>0$ (Figs. 9a,c).

Figure 10: (a)-(d) Time history of the velocity horizontal components,
$v_{x}$, $v_{y}$ for a) $L\rightarrow\infty$ ($\alpha=18.72^{\circ}$), (b)
$L=4$ ($\alpha=-12.54^{\circ}$), (c) $L=2$ ($\alpha=44.79^{\circ}$), and (d)
$L=1$ ($\alpha=85.99^{\circ}$). Velocities for $t\leq$100 are shown for the
sake of clarity. (e) Vertical velocity $v_{z}$ as a function of time for the
four wall distances. The inset shows the time evolution of the local Reynolds
number.

The bubble velocity components extracted from the trajectories are displayed
in Fig. 10. As expected, the horizontal components of the velocity begin to
oscillate after a transient time that decreases as $L$ is reduced, indicating
that the wall favors the onset of the instability (Figs. 10a-d). The velocity
components exhibit positive and negative values associated with the bubble
displacements. As in the trajectories, the oscillation amplitude of $v_{x}$
and $v_{y}$, as well as the phase difference, are related to the lateral
displacement and depend on the angle $\alpha$. The mean migration velocity of
the bubbles when the wall is present, of order $\mathcal{O}(10^{-3})$, is much
smaller than the oscillation amplitude, of order $\mathcal{O}(10^{-1})$, and
cannot be observed in the plots. With respect to the vertical velocity, Fig.
10(e) shows that, as in the rectilinear regime, the rise velocity initially
increases linearly with time until it reaches a state where it oscillates
around a mean value, slightly larger than the gravitational velocity in this
case. In fact, in this regime, given that the values of $Ga$ are larger than
the critical one corresponding to our experimental mean value of $Bo=$ 3.9,
the occurring path instability makes $v_{z}(t)$ oscillate around the mean
value, $v$, at a frequency twice that of the zigzagging motion, $f\simeq 14$
Hz, as anticipated numerically by Cano-Lozano  _et al._ (2016a) for a free-
rising bubble. The same temporal evolution can be observed for the local
Reynolds number depicted as an inset. The mean values of $v$, as well as those
of associated $Re$ are shown in Table 3. The mean terminal velocity (and thus
the Reynolds number) slightly decreases as $L$ decreases, due to the
additional drag induced by the presence of the wall. Based on the experiments
by Maxworthy  _et al._ (1996), Cano-Lozano  _et al._ (2015) proposed a
correlation to predict the Reynolds number for clean unstable bubbles,
providing $Re_{CL}=[We^{2}\,(We-2.14)/0.505\,Mo]^{1/4}=126.4$, with
$We=Bo\,v^{2}$=5.13, $Bo$=3.88 and $v$=1.15 the values corresponding to the
free case. Note that $Re_{CL}$ is only 10% larger than $Re=114.6$, the average
experimental value for $L\rightarrow\infty$ (Table 3), indicating that bubbles
in Case 2 can be considered to be clean.

###  III.3 Spiral regime

Figure 11: Bubble silhouettes in the spiral regime (Case 3), extracted from
the high-speed recordings at different instants of time in the $xz$ (left) and
$yz$ (right) planes. (a) $L\rightarrow\infty$, (b) $L=4$, (c) $L=2$, and (d)
$L=1$. The bubbles have been scaled down for clarity. The vertical wall, not
plotted for the sake of clarity, is placed at $x=-4$ for $L=4$, $x=-2$ for
$L=2$, and $x=-1$ for $L=1$.

Unstable bubbles that rise in a spiral path were generated using glycerol-
water mixtures (74.16% - 74.89% weight) and a 12 mm superhydrophobic circular
substrate (Case 3 in Table 1). The resulting average values of the Bond,
$Bo=10.29$, and Galilei, $Ga=108$, numbers are similar to those of the target
case (bubble 26 in Cano-Lozano  _et al._ , 2016a, with $Bo$=10 and
$Ga$=100.25). Experiments in this case were not as reproducible as those in
Cases 1 and 2, since the spiral regime takes longer to develop. Moreover, the
physical properties of GW mixtures are not as stable as silicon oils, and
therefore the values of the control parameters slightly varied from experiment
to experiment. In addition, the established rising paths depended on the
initial injection conditions, which were not exactly the same in all the
experiments, since the bubbles were generated using a superhydrophobic
substrate (see Rubio-Rubio  _et al._ (2021) for more details) instead of a
conventional injector as in Cases 1 and 2. Furthermore, bubble shape
oscillations were observed along the rise of these bubbles. However, the
deformation of the bubble never led to a collision with the wall even for the
smallest values of $L$.

Figure 12: (a)-(c) Three-dimensional rising path corresponding to the spiral
regime (Case 3) for (a) $L\rightarrow\infty$ ($Bo=9.94,~{}Ga=121.4$) (b) $L=4$
($Bo=10.37,~{}Ga=103.0$) (c) $L=2$ ($Bo=10.38,~{}Ga=103.1$) and (d) $L=1$
($Bo=10.43,~{}Ga=103.5$). Corresponding top views for (e) $L\rightarrow\infty$
(f) $L=4$ (g) $L=2$ and (h) $L=1$. The vertical wall, not plotted for the sake
of clarity, is placed at $x=-4$ for $L=4$, $x=-2$ for $L=2$, and $x=-1$ for
$L=1$.

Figure 11 shows the bubble silhouettes at different times in both transverse
($xz$) and parallel ($yz$) planes for three different initial wall distances
and for the unbounded case. Like in the previous cases, the bubble silhouettes
have been scaled down so they don’t look so big in the figures. Note that
oscillations in both planes take place and, as observed in Fig. 12, the
bubbles rise describing an elliptical spiral or a zigzag trajectory, depending
on the wall distance. In this case, as seen in Fig. 1(c), the bubble
deformation is stronger than in Case 2 (zigzag), presenting large deviations
from ellipsoids. This large deformation is attributed to the reduced effect of
surface tension since the Bond and the Weber numbers are higher than in the
previous cases (see Table 3). In particular, besides fore-and-aft asymmetry,
right-left asymmetry along the major diameter can also be observed, presenting
a more pointed rim at the exterior part of the path, in agreement with the
numerical shapes obtained by Cano-Lozano  _et al._ (2016a) for the same bubble
and by Zhang  _et al._ (2020) and Yan  _et al._ (2023) for cases under similar
conditions. The latter works show that the wake structure of this regime
consists of a double spiral composed of double-threaded vortex pairs.
Specifically, an average aspect ratio $\chi$= 2.97 $\pm$ 0.17 is obtained, in
reasonable agreement with the experimental results by Zenit and Magnaudet
(2008) using a silicon oil of the same Morton number. According to the
expression given by Legendre, Zenit, and Velez-Cordero (2012),
$\chi_{L}=[1-9We/64\,(1+0.2\,Mo^{1/10}\,We)^{-1}]^{-1}=3.27$, with
$Mo=5.0\times 10^{-6}$ and $We=Bo\,v^{2}=$6.96 the mean value provided with
$v=0.83$ (Table 3) and $Bo=10.11$ (values for $Mo$, $Bo$ and $v$ correspond to
the unbounded cases, $L\rightarrow\infty$). Notice that a good agreement is
achieved, being $\chi_{L}$ 10% larger than our experimental value.

The rising paths are shown in Fig. 12 where the freely ascending bubble first
exhibits a flattened spiraling trajectory that eventually evolves into a
nearly circular helix (Fig.12 a,e). The dimensionless vertical distances
reported in this case are smaller than in Cases 1 and 2 because the bubble
diameters are larger (see Table 1). As in the rectilinear and the zigzagging
cases, an overall migration effect is also observed when the wall is present
(Fig.12 b-d). This effect, similar to the other regimes, is promoted by the
asymmetric vortex structures generated in the wake of the bubbles when the
wall is present. However, unlike in the other cases, a change in the rising
regime is observed here. Although a direct comparison with the free case must
be done with caution since $Ga$ is lower in the wall-bounded cases, it is
clearly seen that the spiral regime is no longer established when the wall is
present (Fig.12 f-h). Instead, a flattened spiral, or even almost a zigzag
pattern, is established. Since the wall is already present when the bubble is
released, it imposes a boundary condition that is different from that given in
the free-rising case, preventing the bubble from following a complete spiral
path from the very beginning. Such spiraling or zigzag trajectory is expected
to be established at sufficiently long distances from the injection point when
the bubble is already far from the wall. For example, for $L=$ 4, the bubble
describes an elliptic spiraling path whose major axis is tilted with respect
to the wall while the $L=2$ case is prone to start following a zigzagging
regime in a plane nearly parallel to the wall and develops a flattened spiral
path as it evolves. However, for $L=1$ the bubble describes a flattened
spiraling path whose major and minor axes rotate as the bubble rises. This
behavior differs from recent numerical works (Zhang  _et al._ , 2020), where
only the migration effect is observed when the wall is present, but the bubble
keeps its helical trajectory.

Figure 13: (a) Rising paths corresponding to the spiraling regime for the
different values of $L$ in the (a) wall-normal plane ($xz$) and (b) wall-
parallel plane ($yz$). In (b), the centers of the trajectories were displaced
$\sim$ 0.5 units each in the positive $y$ direction for the sake of clarity.
Grey lines denote the mean rising path.

Figure 14: Time evolution of horizontal coordinates of the bubble centroid,
namely, $x(t)$ (red line), $y$ (blue line) corresponding to cases in Fig.12

, for a) $L\rightarrow\infty$, (b) $L=4$, (c) $L=2$, and (d) $L=1$. Insets in
each subfigure show the PSD of $x(t)$.

The migration effect caused by the wall can be clearly seen in Fig. 13(a),
where the projection of the bubble rising path on the plane perpendicular to
the wall is shown. As in Cases 1 and 2, the average bubble trajectory is
nearly vertical in the absence of the wall, and it moves further away from the
wall as $L$ decreases. However, no wall effect is observed in the projection
of the mean trajectory of the bubble in the plane parallel to the wall, $yz$,
which, after a transient period in which it deviates from the injection point,
follows an almost vertical trend (Fig. 13b). Unlike Case 2, paths exhibit
oscillations in both $xz$ and $yz$ planes since they are three-dimensional in
this regime. Note that, for $L=1$, the amplitude of the oscillations increases
until $z\approx 55$ in the $xz$ plane and begins to decrease for $z>55$, while
it keeps increasing in the $yz$ plane, indicating that the major axis of the
elliptic spiraling path is rotating. Continuing with the oscillation
amplitude, shown in Fig. 14 as a function of time, while the amplitudes are
comparable in both planes for $L\rightarrow\infty$ because a nearly helical
path is established, $L=4$ and $L=2$ cases show much larger amplitudes in one
of the planes since they follow flattened spiral paths as mentioned above. The
case in which $L=1$ is more complex, where a flattened spiral path is
developing as the bubble rises and migrates away from the wall. A more
detailed analysis of the effect of the wall on the trajectory amplitude cannot
be performed because the rising regimes are not exactly the same for each $L$.
Considering the oscillation frequency (insets in Figs. 14 a-d), a constant
value ($f=4.15$ Hz) slightly lower than that of the free case ($f=5.00$ Hz)
has been found independently of $L$. The higher oscillating frequency in the
unbounded case, whose Strouhal number is $St=0.160$, can be attributed to the
fact that the corresponding Galilei number is slightly higher than those of
the wall-bounded cases, where $St=0.132$. Nevertheless, these values are in
good agreement with those reported by Cano-Lozano  _et al._ (2016a) for bubble
26, who found $St$ to vary from 0.136 to 0.174 as the bubble evolves from a
planar zigzag to a spiral path.

Figure 15: Time history of the velocity horizontal components, $v_{x}(t)$ and
$v_{y}(t)$ corresponding to cases in Fig. 12. (a) $L\rightarrow\infty$, (b)
$L=4$, (c) $L=2$, and (d) $L=1$. (e) Vertical velocity $v_{z}(t)$ as a
function of time for all the different wall distances. The inset represents
the time evolution of the local Reynolds number.

The velocity components extracted from the trajectories are presented in Fig.
15, where the horizontal components (Fig. 15 a-d) can be observed to oscillate
with time at the same frequency as the corresponding trajectories (Fig. 14
a-d). Similarly to Case 2, both components $v_{x}(t)$ and $v_{y}(t)$ present
positive and negative values as the bubble alternatively approaches and moves
away from the wall. The oscillation amplitude of $v_{x}$ and $v_{y}$, as well
as the phase difference, are related to the orbital excursions described
above. Notice that the amplitude of the velocity oscillations decreases when
the wall is present since different regimes are established, i.e. a helical
regime for $L\rightarrow\infty$ and tends to develop a flattened spiral one as
$L=$ decreases. Regarding the vertical velocity, in this case, it increases,
reaching a quasi-terminal value before dropping and beginning to oscillate
around a mean value smaller than the gravitational velocity with an amplitude
that decreases with the presence of a wall (Fig. 15e). The same temporal
evolution can be observed for the local Reynolds number depicted as an inset.
The drop in the velocity (Reynolds) may be attributed to the increase in drag
when the spiral regime is established, which is retarded by the presence of
the wall. Note that $Re_{t}$ decreases when the wall is present, a feature
already observed in the rectilinear and zigzagging regimes, especially for
$L=1$. The mean values of the terminal velocity $v$, as well as those of $Re$
are shown in Table 3. The Reynolds number predicted by the correlation by
Cano-Lozano  _et al._ (2015) gives
$Re_{CL}=[We^{2}(We-2.14)/0.505Mo]^{1/4}=98.12$, which compares very well with
the average experimental value for $L\rightarrow\infty$, $Re=99.6$ (Table 3).
This excellent agreement indicates that contamination, although probably
present due to the use of water, did not alter the experimental results
obtained here.

###  III.4 Comparison of regimes

In this section, the effect of the bubble-rising regime on the migration
process is analyzed. To that aim, the average rising paths in the vertical
$xz$ plane, normal to the wall, for the different regimes are directly
compared for a given value of $L$. In particular, the rectilinear path in the
$xz$ plane is displayed in Fig. 16 together with the corresponding averaged
paths of zigzagging and spiraling cases for $L=4$ (Fig. 16a), $L=2$ (Fig. 16b)
and $L=1$ (Fig. 16c). For both unstable regimes, the depicted paths correspond
to those shown in solid gray lines in Figs. 8 and 13. It can be observed that,
for the three values of $L$, the zigzag regime (Case 2) shows the largest
average lateral displacement away from the wall, while the rectilinear regime
(Case 1) is the least affected. The same result is presented for all the
analyzed values of $L$, being more evident as $L$ decreases. Thus, the zigzag
regime, besides presenting the largest amplitude, as shown above, also
promotes the strongest bubble lateral migration away from the wall. This
result is related to the characteristic wake structure that the bubble
presents when rising within this regime, which has been described above.
Notice that, when comparing the three regimes, both $Bo$ and $Ga$ vary. On the
one hand, cases 1 and 2 present a similar $Bo$, while $Ga$ is larger for Case
2. On the other hand, $Ga$ is comparable in cases 2 and 3, being $Bo$ larger
for Case 3. These results represent relevant findings since they can be used
to optimize many applications involving bubble-wall interaction phenomena.

Figure 16: Rising path in $xz$ plane for the rectilinear regime (black),
together with the average path for zigzag (red) and spiral (blue) regimes
corresponding to (a) $L=4$, (b) $L=2$, and (c) $L=1$.

###  III.5 Effect of the vertical position of the wall leading edge

Figure 17: Bubble paths in the lateral plane for (a) rectilinear, (b)
zigzagging, (c) spiraling regimes. Solid and dashed lines correspond to a long
wall and a short wall, respectively. Here, (b) and (c) show averaged paths.

Additional experiments were carried out in order to investigate the effect of
the vertical distance from the air injector to the wall edge. To that aim,
unlike in the initial experiments, a short wall was placed into the tank. In
this case, the wall edge did not reach the bottom of the tank and the bubbles
initially began to rise freely before reaching the wall. The vertical distance
from the wall edge to the injector, $z_{w}$, was fixed so that the bubbles
reached the terminal velocity in each regime before they encountered the wall.
In particular, $z_{w}\approx 60$ for the rectilinear regime, $z_{w}\approx
160$ for the zigzagging regime, and $z_{w}\approx 85$ for the spiraling one.

Figure 18: 3D (top row) and xy (bottom row) bubble paths for zigzagging regime
using a short wall and (a), (d) $L$=4 (green, $Bo=3.27,~{}Ga=86.4$), (b), (e)
$L$=2 (blue, $Bo=3.28,~{}Ga=87.3$), and (c), (f) $L$=1 (red,
$Bo=3.24,~{}Ga=87.0$). The angle of the zigzag plane is indicated at the
bottom row. The wall, not displayed for the sake of clarity, is placed along
plane $yz$ at $y$=-4, -2 and -1 for $L=$4, 2 and 1, respectively.

In order to compare the different cases together, Fig. 17 shows the vertical
distance from the bubble centroid to the wall, $z-z_{w}$ as a function of $x$
coordinate. In this way, $z-z_{w}$=0 indicates the wall edge, while in the
experiments with a long wall, $z_{w}$= 0. The results for the rectilinear
regime are presented in Fig. 17(a). Notice that, when the short wall is used
(dashed lines), the migration effect starts before reaching the wall, at
$z<z_{w}$, and the bubble displacement is much larger than that observed with
a long wall ($z_{w}$= 0). This effect is more pronounced as $L$ decreases. The
same comparison is made for the zigzag and the spiral regimes in Figs. 17(b)
and (c), respectively. In these cases, averaged paths are plotted for the sake
of clarity. The same qualitative picture as the one described for the
rectilinear regime can be observed, but notice that the migration occurs at
even lower vertical positions. In the zigzag case, depending on the phase of
the zigzagging movement when the bubble reaches the wall edge, the induced
migration effect can take place even sooner. The early migration is more
notable in the spiral regime due to the three-dimensional nature of the paths,
being the bubbles bigger and more deformed than in the other regimes. It is
worth mentioning that both unstable cases with the short wall are not quite
reproducible since the final migration effect depends on the exact relative
position between the bubble and the wall, which is different for each
experiment.

Since Figs. 17(b) and (c) show the average bubble paths of the unstable
regimes, it is interesting to explore the complete rising trajectories within
these unstable regimes when the short wall is used. In this regard, Figs.
18(a)-(c) show the three-dimensional rising path corresponding to the planar
zigzagging regime. Before reaching the wall, the bubble freely rises in a
zigzag regime. Notice that, just before finding the wall edge, the lateral
displacement from the wall takes place while keeping the zigzag motion. As in
its long-wall counterpart (see Fig.7), this effect is particularly clear for
the smallest value of $L$ (Fig. 18c). Moreover, as shown in the corresponding
top views displayed in Figs. 18(d)-(f), the established zigzagging plane was
completely random, even for $L$=1, differently from the long wall experiments,
in which the wall sets a boundary condition imposing a perpendicular
zigzagging plane when the bubble is close enough. Except in the local region
close to the wall edge, the wall has been found not to modify the oscillation
frequency or the oscillating amplitude.

Figure 19: 3D (top row) and xy (bottom row) bubble paths for spiral regime
using a short wall and (a), (d) $L$=4 (green, $Bo=12.34,~{}Ga=128.1$), (b),
(e) $L$=2 (blue, $Bo=12.33,~{}Ga=128.5$), and (c), (f) $L$=1 (red,
$Bo=12.13,~{}Ga=124.7$). The wall, not displayed for the sake of clarity, is
placed along plane $yz$ at $y$=-4, -2 and -1 for $L=$4, 2 and 1, respectively.

With regard to the spiraling regime, Fig.19 shows the three-dimensional and
top views of the bubble path when a short wall is placed into the tank. The
bubble clearly migrates from the wall just before reaching its edge. Compared
to its long-wall counterpart (Fig.12) notice that, when the bubble is
released, it freely rises without any constraint, differently from the long-
wall case. Thus, paths are more spiral-like than in the case with the long
wall. This fact can be clearly observed in the top views (Figs.19d-f), where
the path horizontal projections show a more elliptical shape than in the cases
with the long wall, which tend to evolve into flattened spirals. While in the
long-wall case the wall imposes a restriction for the spiral to develop, in
the current case the final regime is completely established before reaching
the wall.

##  IV Conclusions

In this work, the rising dynamics of bubbles with high $Bo$ in the presence of
a rigid solid wall are experimentally analyzed. Three different rising
regimes, namely rectilinear, planar zigzagging, and spiraling are explored.
The initial horizontal distance between the wall and the bubble centroid is
also varied. Some discrepancies have been found with previous works studying
spherical bubbles, which justifies our focus on the high $Bo$ regime, typical
of ellipsoidal bubbles, whose deformation leads to complex asymmetric vortex
structures in the flow, enhancing the wall effect.

The main observed effect for all cases and bubble-wall initial distances is a
transverse migration of the bubble away from the wall that increases with
bubble-wall initial proximity. In the rectilinear and zigzagging cases, the
bubble keeps a similar behavior to the unbounded cases, i.e. they maintain the
expected trajectories, that are only altered by this transverse motion. When
comparing the impact of the wall on the average paths of the different
regimes, it can be observed that the largest migration effect takes place for
the zigzagging regime, being the lowest for the stable case. Moreover, in the
spiraling regime, besides the bubble migration from the wall, a change in the
rising path is promoted when the wall is initially close to the bubble. In
particular, the limitation imposed by the wall prevents the bubble from
following a complete spiral path from the very beginning, promoting a more
zigzag-like trajectory that eventually leads to a flattened spiraling motion.

Unstable regimes seem to be established earlier due to the wall presence,
which means that the wall triggers instability. A slight increase in amplitude
due to this destabilizing effect was also observed in the unstable regimes, as
found numerically by Zhang  _et al._ (2020). However, when the wall is
sufficiently close to the bubble, the zigzagging plane is established normal
to its surface. This fact indicates that the wall can impose an initial
condition on the bubble trajectory development as soon as the bubble is
generated.

Regarding the vertical bubble velocity, the presence of the wall promotes a
larger initial acceleration which shortens the time necessary for the bubble
to reach its terminal velocity. Besides, this velocity decreases with the
proximity of the wall due to an increase in the drag force. Nevertheless, this
velocity drop is only observable in our experiments when the bubble and wall
are very close or when the spiraling regime is established. Additionally,
oscillations in the vertical velocity have been obtained in the zigzagging
regime, as reported numerically by Cano-Lozano  _et al._ (2016a). Moreover,
the experimental Reynolds number associated to the terminal velocity and
corresponding to the unbounded case are in good agreement with the ones
obtained from correlations valid for clean bubbles. This indicates that
contamination, if present, is not relevant in our experiments.

In addition, it has been observed that, when the wall edge is placed far above
the bubble injection point, the bubble displacement starts before it reaches
the wall, and deviates longer distances from the surface. Focusing on the
planar zigzagging case, since the oscillating plane is already established
before the bubble reaches the wall position, this plane is not altered, only
the migration effect is added to the already developed trajectory. Indeed, the
zigzagging plane is established arbitrarily, as happens in the long-wall
experiments when the initial wall distance to the bubble is sufficiently
large. For the spiraling case, a later wall interaction allows for more
spiral-like paths as the wall spatial limitation near to the injector is
eliminated.

###### Acknowledgements.

This work was supported by the coordinated project, PID2020-115961RB-C31,
PID2020-115961RB-C32 and PID2020-115961RA-C33, financed by
MCIN/AEI/10.13039/501100011033. C. Estepa-Cantero would like to thank the
Spanish Ministry of Universities for the financial support provided by the
Fellowship FPU20/02197.

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