About the dynamics and morphology of single
ellipsoidal bubbles in liquids

David Mikaelian, Aur?lie Larcy, Arnaud
Cockx, Christophe Wylock and Beno?t Haut*

* Corresponding author: Email:
bhaut@ulb.ac.be

Laboratory Transfers, Interfaces and Processes (TIPs), CP 165/67,

Universit? Libre de Bruxelles, Av. F. D.
Roosevelt 50, 1050 Brussels, Belgium

1.   
Introduction

Non-spherical
bubbles of a few millimeters rising in liquids with a non-linear trajectory are
commonly encountered in many industrial applications. Their dynamics and
morphology highly influence the efficiency of these industrial applications
because they control the mixing of the liquid phase and the mass transfers
between the bubbles and the liquid. Therefore, the dynamics and the morphology
of bubbles moving in various liquids have been extensively investigated, but
there are still open questions. In this paper three topics on the dynamics and
the morphology of single ellipsoidal bubbles rising in liquids are
investigated:

-      
the
potential alignment of the minor axis and the mass center velocity vector of an
ellipsoidal bubble, in the cases of a zigzag and a helical motion of the bubble;

-      
the
difference between the interface curvature radii at the front and at the rear
of an ellipsoidal bubble;

-      
the
identification and the quantification of a pulsation in the dynamics of the
vertical motion of the mass center of a bubble having a non-linear trajectory.

These three topics are investigated in this
work using raw images recorded with the experimental set-up presented in
Mikaelian et al. (2013) and a new data postprocessing method.

2.   
Experimental
set-up and experimental data set

The experimental set-up developed in
Mikaelian et al. (2013) is presented in Fig. 1. It is based on a shadowgraphy
technique. Bubbles of various sizes are generated in a column filled with a
water-glycerol mixture and their rises are recorded by a camera. Perspective
effects are avoided using two convergent lenses. When a bubble is between the
two lenses, the projection of the bubble onto the recording plane of the camera
appears black on the images recorded by the camera and it is called hereafter
the bubble projection. The high resolution and large field of view of the
set-up enable a simultaneous analysis of the dynamics and the morphology of the
generated bubbles.

Figure

1

: Sketch of the experimental
set-up used in Mikaelian et al. (2013). LS: light source, LG: light guide, D:
diffuser, I: iris, LR: light rays, L₁: first lens, L₂: second
lens, O: objective, Ca: camera, Co: computer, Mon: monitor, SP: syringe pump,
S: syringe, T: tube, N: needle, C: column, GT: graduated tube or bubble
collector.

The experimental set-up has been used for
a range of liquid properties and bubble equivalent diameters d_e
leading to the generation of ellipsoidal bubbles, as predicted by Grace et al.
(1976). For the shape of a bubble, the general case of an ellipsoid with a
fore-and-aft asymmetry is considered and sketched in Fig. 2, where a is defined
as the major axis length of the bubble, b₁ the semi-minor axis
length at the front of the bubble and b₂ the semi-minor axis length
at the rear of the bubble.

Figure

2

: Ellipsoidal shape of a
real bubble with a fore-and-aft asymmetry, where a is the major axis length of
the bubble, b₁ the semi-minor axis length at the front of the bubble,
b₂ the semi-minor axis length at the rear of the bubble and ●
the mass center.

In this paper, we select and postprocess
the experiments where bubbles rising with either a zigzag or a helical motion
and without interface wobbling were observed. These experiments are
characterized using the E?tv?s (Eo), the Morton (Mo), the Reynolds (Re) and the
Weber (We) numbers of the bubbles.

The selected experiments are
characterized by 0.8 < Eo < 8, 10^-11 < Mo < 10^-7,
140 < Re < 860, 2.6 < We < 6.3 and 2.5 mm < d_e <
6.8 mm.

3.   
Data
postprocessing

A
postprocessing method is developed for the raw images recorded during the
selected experiments above. This postprocessing method is used to determine:

-      
the
image acquired when the minor axis of a bubble is almost parallel to the
recording plane of the camera. This image is called the SMALP image and can be
used for the analysis of the morphology of the bubble because, on this image,
the length of the minor axis of the bubble projection is equal to the length of
the minor axis of the bubble (b₁ + b₂);

-      
the
type of the trajectory of a bubble;

-      
a
threshold λ for the binarization of the images, based on a well-defined
criterion (volume conservation);

-      
the
directions of the minor axis and the mass center velocity vector of a bubble
projection;

-      
the
interface curvature radii at the front and at the rear of a bubble, by fitting
the smoothed contour of the bubble projection on the SMALP image by two half
ellipses (one for the front and another for the rear of the bubble) with the
same center and the same major axis;

-      
a
possible pulsation in the vertical motion of the bubble mass center.

4.   
Results

For all the bubbles of a considered experiment,
the directions of the minor axis and the mass center velocity vector of each recorded
bubble projection can be evaluated for the successive positions of the bubbles
during their rises. These directions are compared either for all the recorded
images of the rise of a bubble randomly selected among all the bubbles of an
experiment or for all the SMALP images of all the bubbles of an experiment, and
their alignment is clearly shown. This observation is in agreement with the works
of Saffman (1956) and Ellingsen and Risso (2001). The way this alignment is
assessed here is different than in these works. Indeed, the directions of the
minor axis and of the velocity vector of a bubble are here directly determined
from experimental results and then compared to analyze their alignment, for the
successive positions of the bubble.

For all the bubbles of a considered
experiment, the interface curvature radii at the front and at the rear of the
bubbles are evaluated and the mean values (R_f and R_r) are
then deduced. For the values of Eo and Mo considered here, R_fis higher than R_r, meaning
that the interface at the front of the bubble is flatter than at the rear. This
observation is in agreement with the results of Ryskin and Leal (1984),
Duineveld (1995) and Zenit and Magnaudet (2008). In the case of a zigzag motion of a bubble
(3 < Eo <
8, 6 10^-10 < Mo < 10^-7), the following correlation for R_f/R_r is proposed as a function of Eo and Mo of
the bubble:

In the vertical motion of the mass center
of a bubble, a pulsation at twice the frequency of its horizontal motion is
identified in the case of a zigzag motion of the bubble. In the case of a
helical motion of a bubble, such a pulsation cannot be identified in the
vertical motion of the mass center of the bubble. These observations are in
agreement with the experimental results of Ellingsen and Risso (2001) and Shew
et al. (2006). The presence or not of a pulsation in the vertical motion of the
mass center of a bubble at twice the frequency of its horizontal motion can be
used to distinguish the zigzag and the helical motions of a bubble, when the
only data available are the projections of the bubble on a single vertical
plane.

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