Anales AFA - XVI Meeting on Recent Advances of Physics of Fluids and its Applications 66-70

OSCILLATING BUBBLES RISING IN A DOUBLY CONFINED CELL

L. Pavlov*1, M. V. D’Angelo1, M. Cachile1, V. Roig2y P. Ern2

1Universidad de Buenos Aires, Facultad de Ingeniería, Grupo de Medios Porosos, Buenos Aires, Argentina,

CONICET, Buenos Aires, Argentina.

2Institut de Mécanique des Fluides de Toulouse, Université de Toulouse and CNRS, Toulouse, France.

Recibido: 30/10/21; Aceptado: 14/12/21

The motion of bubbles rising in conﬁned geometries has gained interest due to its applications in mixing and mass

transfer processes, ranging from bubble column reactors in the chemical industry to solar photobioreactors for algae

cultivation. In this work we performed an experimental investigation of the behavior of air bubbles freely rising at high

Reynolds numbers in a planar thin-gap cell of thickness h=2.8 mm ﬁlled with distilled water. The in-plane width of

the cell Wis varied from 2.4 cm to 21 cm. We focus on the inﬂuence of lateral conﬁnement on the motion of bubbles in

the regimes with regular path and shape oscillations of large amplitude, that occur for the size range 0.6 cm <d<1.2

cm. In addition, a rise regime that consists of a vertical rise path with regular shape oscillations, that does not appear in

the laterally unconﬁned case, is uncovered.

In the presence of lateral walls, the mean rise velocity of the bubble Vbbecomes lower than the velocity of a laterally

unconﬁned bubble of the same size beyond a critical bubble diameter dcV that decreases as the conﬁnement increases

(i.e. as Wdecreases). The inﬂuence of the lateral conﬁnement on the bubble mean shape can be determined from the

change in the mean aspect ratio χof the ellipse that best ﬁts the bubble contour at each instant. It is observed that

bubbles become closer to circular (χcloser to 1) as the conﬁnement increases. The departure from the values of χ

of the laterally unconﬁned case occur at a critical diameter dcχthat is lower for greater conﬁnement and also greater

than dcV for each conﬁnement, thus indicating that the effect of the lateral conﬁnement is seen earlier (i.e. on smaller

bubbles) on the velocity than on the aspect ratio.

Assuming that the wall effect is related to the strength of the downward ﬂow generated by the bubble, we introduce

the mean ﬂow velocity in the space let free for the liquid between the walls and the bubble, Uf, that can be estimated

by mass conservation as Uf=dVb/(W−d). We further introduce the relative velocity between the bubble and the

downward ﬂuid in its vicinity Urel =Vb+Uf=Vb/ξ, where ξ=1−d/Wis the conﬁnement ratio of the bubble. We

found that, for a given bubble size in the oscillatory regime, Urel is approximately constant for all the studied values

of W, and matches closely the value in the absence of lateral conﬁnement. This provides an estimation, at leading

order, of the bubble velocity that generalizes the expression proposed by Filella et al. (JFM, 2015) and accounts for the

additional drag experienced by the bubble due to the lateral walls. We then show that, for given dand ξ, the frequency

and amplitudes of the oscillatory motion can be predicted using the characteristic length and velocity scales dand Urel .

Keywords: Bubbles, bubble kinematics, bubble shape, conﬁned bubbles.

https://doi.org/10.31527/analesafa.2022.ﬂuidos.66 ISSN 1850-1168 (online)

I. INTRODUCTION

The motion of bubbles freely rising in a thin-gap cell at

high Reynolds numbers has interest in several fundamental

and practical problems, for example in applications invol-

ving conﬁned bubble reactors. This thin-space conﬁgura-

tion retains the speciﬁc properties associated with inertial

ﬂows, while operating limited volumes of liquid [1-5]. In

speciﬁc applications, bubbles rising in plane geometries are

also limited by side walls. This work then focuses on in-

vestigating the inﬂuence of an additional transverse conﬁ-

nement on the bubble behavior.

The problem of a 3D bubble rising in a viscous liquid has

been widely studied since the last century. The presence of

walls, as in the case of a Hele-Shaw geometry, will modify

the ﬂow ﬁeld around the bubble, and therefore the shape and

motion of the bubble during its rise [6-8]. Introducing addi-

tional lateral walls in the cell will modify the ﬂow around

* lpavlov@ﬁ.uba.ar

the bubble more drastically.

The inertial regime of bubbles freely rising in a thin-gap

cell was investigated in detail by Roig et al. [9] and Filella

et al. [10]. They highlighted the existence of different types

of bubble motion and provided a characterization of the dif-

ferent paths observed for increasing bubble sizes. Filella et

al. [10] proposed a simple generic estimation for the mean

rise velocity of the bubble Vbvalid for a large range of bub-

bles’ sizes, Vb,∞≃0.7pgdeq, which can also be expressed

Vb,∞≃k(h/d)1/6pgd,(1)

where k=0.75 and Vbis denoted Vb,∞for consistency with

the remainder of the paper. In this expression, gis the gra-

vitational acceleration, and the diameters

deq =3d2h/21/3and d=p4A/π(2)

are, respectively, the three-dimensional equivalent diameter

of the bubble calculated with its volume (deq), and the pla-

nar equivalent diameter of the bubble determined from the

area Acovered by the bubble in the plane of the cell (d).

Equation (1) indicates that the mean vertical velocity of the

bubble Vbis not only proportional to the gravitational velo-

city √gd but also depends on the parameter h/dimposed

on the bubble by the small gap. This expression correctly

describes the rise velocity of bubbles in the regimes with

path and shape oscillations in cells with gap thicknesses of

1 mm [9] and 3 mm [10]. Nevertheless, it does not account

for the effect of the lateral conﬁnement on the rise velocity,

which is within the scope of the present investigation.

II. EXPERIMENTAL TOOLS

The experimental apparatus consisted of a thin-gap cell

of thickness h≃2.8 mm, height Hand width Wcorien-

ted vertically and ﬁlled with distilled water. The air bub-

bles were released individually from a syringe connected

to a nozzle located in the centre at the bottom of the cell.

The experiments were performed at ambient temperature of

(19 ±1)◦C, as described in Pavlov et al. [11]. PMMA pla-

tes were included inside the cell as internal lateral walls to

reduce the effective width from Wcto W. The experiments

were performed in three different cells, one made in PMMA

and two made of glass plates with heights Hbetween 30 and

50 cm and widths Wcbetween 9 and 21 cm. Internal walls

were included, when needed, in order to reduce the effective

width from Wcto W=9, 7, 4 and 2.4 cm. The motion of the

bubble was recorded using a high-speed camera with frame

rates ranging from 150 to 1000 frames per second, and a

spatial resolution between 100 and 200 µm/px, depending

on the experiments. A LED backlight illumination was used

to generate uniform lighting. The area, contour and centroid

of the bubble were obtained using typical image processing

techniques.

The behavior of the bubble can be further characterized

by two Reynolds numbers [12],

Re =Vbd

νand Reh=Reh

d2

.(3)

The ﬁrst, Re, characterizes the in-plane motion of the bub-

ble, and the second is associated with viscous diffusion in

the gap. The inertial regime, which is the focus of the pre-

sent work, corresponds to both Re ≫1 and Reh≫1.

III. RESULTS AND DISCUSSION

In a previous study [11], a description of the different rise

regimes is made for the case of laterally conﬁned bubbles ri-

sing in a thin-gap cell. In this paper we will focus on the re-

gimes with path and shape oscillations about a vertical path

and also in the regime that consists of a rectilinear path with

periodic shape oscillations. These regimes are presented in

Fig. 1. In these regimes, an inﬂuence of lateral conﬁnement

on the average rise velocity and on the shape of the bubble

is observed. The inﬂuence of the cell width Won the bubble

shape can be characterized by the aspect ratio χ=a/b, de-

ﬁned as the average over the stationary regime of the ratio

between the lengths of the major and minor axes (aand b,

respectively) of the ellipse that best ﬁts the bubble contour

at each time step (Fig. 2).

We are interested, in particular, in the inﬂuence of the

lateral conﬁnement on the different magnitudes that charac-

terize the movement of the bubble, whether they are magni-

tudes averaged over time (such as mean velocity or aspect

ratio) or magnitudes obtained from the analysis of periodic

signals (such as the frequency of oscillation).

FIG. 1: Different regimes observed in the experiments (from left

to right): ♢: path oscillations and moderate shape oscillations

around a vertical path, : path oscillations and large shape osci-

llations (usually with a horizontal drift in the path), ▽: path and

large shape oscillations around a vertical path, ▷: rectilinear path

with periodic shape oscillations.

FIG. 2: Comparison of the bubble contour with the ellipse used to

characterize its shape and the aspect ratio χ=a/b.

An example of the studied magnitudes (instantaneous ve-

locity and deformation) is shown in Fig. 3for a bubble with

path and shape oscillations and in Fig. 4for the case with

only shape oscillations.

The presence of the walls modiﬁes the ﬂow around the

bubble. For sufﬁciently strong lateral conﬁnement, the spa-

ce available for the descent of the liquid around a rising bub-

ble is reduced. We can deﬁne the parameter ξ=1−d/W,

that compares the space let free for the downward motion

of the liquid with the total width of the cell. To satisfy mass

conservation, the mean downward velocity of the ﬂuid su-

rrounding the bubble Ufcan be estimated in a ﬁrst approxi-

mation from the mean bubble rise velocity Vband the mean

bubble width ⟨Wb⟩as:

Uf=Vb⟨Wb⟩

W−⟨Wb⟩.(4)

If we further approximate ⟨Wb⟩by d(in practice, the ratio

⟨Wb⟩/dtakes values between 0.9 and 1.1), Eq. (4) can be

Pavlov et al. / Anales AFA - XVI Meeting on Recent Advances of Physics of Fluids and its Applications 66-70 67

-1 0 1

x (cm)

y (cm)

FIG. 3: Example of a bubble rising in a cell with W =2.4cm.

Left: path of the center of mass of the bubble with d =1.09 cm and

its contours at different times. The contours are shown in colors,

which serve to identify the values of the different signals at those

particular instants in time. In addition, the characteristic ellipse in

each case and its minor axis are shown as a gray dotted line. Top

right: instantaneous velocity of the center of mass in the vertical

(Vy, black solid line) or horizontal (Vx, black dashed line) direction

as a function of time. The gray dotted lines correspond to ﬁts by a

sine function. Bottom right: aspect ratio χas a function of time.

The mean value of the signal is shown as a horizontal line.

-2 0 2

x (cm)

y (cm)

0.6 0.8 1 1.2 1.4

t (s)

1.8

1.85

1.9

1.95

2.05

2.1

FIG. 4: Left: contours of a bubble with d =1.20 cm which shows

only shape oscillations, at different times. Right: aspect ratio χas

a function of time for this bubble.

written as:

Uf=Vb

W−d.(5)

A mean relative velocity between the bubble and the liquid

can then be introduced as

Urel =Vb+Uf=Vb

W−d=Vb

ξ.(6)

Fig. 5shows both Vband Urel as a function of dfor the

different lateral conﬁnements. As can be seen, while the

mean rise velocity Vbis lower for stronger lateral conﬁne-

ments, the values of Urel are located in a single curve for

all the studied values of W. This indicates that a bubble of a

given size in these regimes always rises with the same value

of Urel , regardless of the lateral conﬁnement. Therefore, ξ

FIG. 5: Bubble mean rise velocity (top) and mean relative velo-

city between the bubble and the surrounding ﬂuid (bottom), as a

function of d. The color of the data points is associated with the

value of the cell width W (empty black: W =21 cm, ﬁlled black:

W=15 cm, orange: W =9cm, blue: W =7cm, green: W =4cm,

red: W =2.4cm), and each symbol (being it empty or ﬁlled) re-

presents a rise regime (see Fig. 1).

is a parameter that quantiﬁes, in the studied regimes, how

much slower a bubble of diameter drises in a cell of width

Wwith respect to the velocity that the same bubble would

have in the absence of lateral conﬁnement. In the limit of no

lateral conﬁnement, ξ=⇒1, and thus Urel ≃Vb,∞, which

can be used to generalize equation (1) in order to account

for the lateral conﬁnement [13].

FIG. 6: χas a function of d. The symbol convention is the same as

in Fig. 5.

For a given bubble size, a stronger conﬁnement also pro-

duces less elongated shapes, i.e.,χcloser to unity (Fig. 6).

The effect of conﬁnement, that is, the departure from the un-

Pavlov et al. / Anales AFA - XVI Meeting on Recent Advances of Physics of Fluids and its Applications 66-70 68

conﬁned case (which can be taken as the W=21 cm case

[11]) occurs for smaller bubble sizes in the case of the mean

rise velocity as compared with the mean aspect ratio of the

bubble. This can be particularly noticed in the W=7 cm

case, for which the departure from the mean rise velocity of

unconﬁned occurs already at d≈0.7 cm, while for bubbles

of that size the value of χis the same in that conﬁnement as

compared with the unconﬁned case. More generally, if one

deﬁnes dcV (W)(resp. dcχ(W)) as the diameter for which the

effect of the conﬁnement begins to be noticed in Vb(resp.

χ) for a given W, then dcV (W)<dcχ(W).

For sufﬁciently conﬁned bubbles, there is a small size

range for which the bubble rises with shape oscillations but

with a vertical center of mass trajectory. This regime is not

observed in the unconﬁned case because the trajectory of

the bubble eventually becomes unstable and begins to show

oscillations, which may or may not be regular. The lateral

conﬁnement thus provides a stabilizing effect to the move-

ment of the bubble in this scenario.

For bubbles with both path and shape oscillations the fun-

damental frequency fis that corresponding to the path osci-

llations. This frequency, fis related with the frequency fχ,

corresponding to shape oscillations, being f=fχ/2 (Fig.

3). This indicates that the bubble aspect ratio has two os-

cillations at every period of path oscillation, which in turn

reﬂects the symmetry of the path. In contrast, in the case of

bubbles that only have shape oscillations the only frequency

will be fχ. We present, then, in Fig. 7the frequency values

corresponding to the shape oscillations, for all the bubbles

studied in this work.

FIG. 7: Frequency fχas a function of d. The symbol convention is

the same as in Fig. 5.

For each W, the values of fχfor bubbles in the regime of

only shape oscillations are close to those of the biggest bub-

bles showing path oscillations. This means that, even if the

characteristics of the rise regime are completely different,

the frequency of the shape oscillations does not change bet-

ween the two regimes.

Moreover, the frequency of oscillation does not depend

on the cell width (except for the most conﬁned cases, ξ<

0.6, for which a “bouncing” effect appears [11]). The same

can be said for the amplitude of oscillation of the vertical

component of the velocity (Fig. 8). This means that the cha-

racteristic velocity scale that controls those parameters is

the relative velocity between the bubble and the ﬂuid and

not the bubble mean rise velocity [11].

FIG. 8: Amplitude of oscillation of the vertical component of the

velocity, Vy, as a function of d. The symbol convention is the same

as in Fig. 5.

It is interesting to point out some similarities between the

laterally conﬁned cases and the problem of a bubble rising

in a thin-gap cell that is laterally unbounded, in the presen-

ce of a moderate counterﬂow. In that problem, it was found

[14] that the bubble velocity slows down in the presence

of a counterﬂow of mean velocity Uc f , its ﬁnal rise velo-

city being Vb−Uc f (where Uc f was always smaller than Vb).

This means that the mean relative velocity between the bub-

ble and the ﬂuid remains unchanged with respect to the case

of the bubble rising freely (in the absence of counterﬂow),

as it is the case in the laterally conﬁned cases. Moreover,

the velocity amplitudes and frequency of oscillation also

remain the same regardless of the counterﬂow, as it is the

case with the lateral conﬁnement. However, some differen-

ces arise, particularly with respect to the bubble shape. The

presence of a counterﬂow barely modiﬁes the mean aspect

ratio χof the bubble, while it was seen (Fig. 6) that the la-

teral conﬁnement produces a decrease in the value of χfor

a bubble of a given size. A similar behavior can be seen in

the amplitude of the orientation of the bubble (not shown

in this work), which is in fact a magnitude that is strongly

related to χ[11].

IV. CONCLUSIONS

We investigated the oscillatory motion of inertial bubbles

rising in a thin-gap cell with an additional lateral conﬁne-

ment. The inﬂuence of the cell width Won the bubble rise

velocity, shape and oscillations was characterized. A mean

relative velocity between the bubble and the ﬂuid was in-

troduced and shown to be independent of W. The analogy

with the results for a counterﬂow experiment without lateral

conﬁnement was discussed. Finally, for strong enough late-

ral conﬁnements we also showed the existence of a regime

for which the bubble exhibits only shape oscillations.

ACKNOWLEDGMENTS

The authors are grateful to G. Ehses, G. Albert and L.

Mouneix for building the IMFT experimental set-up and

to S. Cazin for his help with the optical technique. This

collaboration was possible thanks to the ﬁnancial support

from CONICET and Universidad de Buenos Aires, Argenti-

na and from INPT, Toulouse (SMI fundings 2016 and 2018)

and became part of the IRP CNRS-CONICET IVMF in

Pavlov et al. / Anales AFA - XVI Meeting on Recent Advances of Physics of Fluids and its Applications 66-70 69

2019.

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