adfa, p. 1, 2011.

© Springer-Verlag Berlin Heidelberg 2011

Shape and Size of the Entrained Liquid Mass by a Taylor

Bubble Rising Through a Liquid-Liquid Interface

Abhimanyu Kar and Prasanta Kumar Dasa

Department of Mechanical Engineering, IIT Kharagpur

(aCorresponding author, e-mail: pkd@mech.iitkgp.ernet.in)

Abstract. Long gas bubbles in circular tubes, called Taylor bubbles, occur fre-

quently in industrial applications. If the tube contains two immiscible liquids

one on top of the other, a rising Taylor bubble after it passes though the inter-

face entrains the liquid below the interface into the liquid that is above the inter-

face. This article reports different shapes and sizes the entrained mass takes as it

rises beneath rising Taylor bubbles of different volumes.

1 Introduction

Entrainment of one fluid into another can be observed very commonly in daily life.

As an example, when the water jet from a tap strikes the surface of water in a bucket,

surrounding air is entrained into the bucket which is observed as bubbles coming to

the surface of water. The study of entrainment has been a vast domain and several

authors have tried to focus on different types on entrainment. Always a typical motion

of the interface can be identified as the cause of entrainment. There are examples [1-

3] of entrainment where one of the two fluids itself causes the motion that entrains.

Studies involving a rotating drum [4-5] entraining air on the surface of water or a

sphere falling onto the surface of water [6] have focused on entrainment of one fluid

into another caused by a solid body.

Long gas bubbles in pipes and closed channels are called Taylor bubbles in the

honour of Geoffrey Taylor who conducted one of the earliest studies [7] on this topic.

Several other studies have devoted on different aspects of Taylor bubbles, one of the

most well-known is by Viana et al.[8] where a universal empirical correlation has

been proposed to predict the velocity of Taylor bubbles for a wide range of fluid

properties.

Fig. 1. A 5 ml bubble rising through red dyed water and then through PDMS in a 25 mm pipe

entrains the water along its tail.

Fig. 2. Various elements of the two-fluid and Taylor bubble system related to the entrainment

phenomenon

The bubble, as it enters the lighter liquid, at first drags a column of heavier liquid

along its tail. From figure 1, it is clear that this column follows the intuitive Karman

vortex street behind a Taylor Bubble and often carries its wavy features. With time

the column stretches due to the rise of the bubble as well as erosion of mass due to

gravity. Finally a point comes when it breaks into two, often generating small droplets

in-between. The bottom portion quickly falls back to the bulk of the heavier liquid

while the top portion rises along the tail of the bubble. Figure 2 shows various ele-

ments related to entrainment: the original liquid-liquid interface, the entrained mass

following the Taylor bubble and the entrainment column.

Experiments conducted with two different tube diameters and several liquid pairs

show that all liquid pairs do not cause entrainment. But when there is entrainment, for

a tube of a particular diameter, the size of the entrained mass seems to vary with the

bubble volume. Entrainment caused by two liquid pairs, namely water/liquid paraffin

and water/PDMS, in a 25 mm Perspex tube are reported here.

2 Experimental Setup

Bubbles were generated in a Perspex tube having an inner diameter of 25 mm and

were released by a manually operated quick-closing ball valve. The bubble generating

mechanism shown in figure 3 shows a 150 mm pipe section below the ball valve with

a needle valve fixed below it. The section also has an air inlet pipe towards its top and

a liquid supply pipe towards its bottom which is fed by a reservoir placed higher than

the setup. To generate bubbles, the needle valve at the bottom is kept open and the air

inlet valve is also opened. They are kept open until the desired volume of drained

liquid has been collected. This creates an air cavity at atmospheric pressure whose

volume is presumed equal to that of the drained liquid and is used to specify the bub-

ble throughout this article. Closing these two valves and a sudden opening of the

quick-closing valve introduces a Taylor bubble into the pipe section which is used in

our experiments.

Fig. 3. Setup for generating Taylor bubbles in a 25mm pipe.

The described bubble rising arrangement is fitted in a frame with an arrangement

of lighting (as shown in figure 4) and a computer controlled high speed camera is

used to capture images either from the same side as the lights (from lighting) or from

the directly opposite side (back lighting).

Fig. 4. Arrangement for lighting and photography of the rising Taylor bubbles

3 Post Processing Methods

The raw data captured in our experiments are image frames captured with a high

speed camera which need to be manipulated for quantification. This requires various

image processing methods.

The captured video or image sequence is subject to some basic processing such as

separation of colour channels i.e. separation of the image into the three monochrome

colour components which constitute the colour image. Colour in computer graphics is

described by 3 independent components according to different colour models. The

most common form is the RGB model which separates the colour into primary or red,

green and blue components.

Some standard noise reduction filters are also used on the image before further

processing. The most commonly used filter here is the median filter, though in some

cases Gaussian low pass filters have also been used.

3.1 Correlation Tracking

Correlation tracking (see figure 5) is used to determine the position of the bubble in a

particular frame. This algorithm requires the user to select an area of interest (AOI) in

the first frame around the object to be tracked. This is placed on each possible posi-

tion on each subsequent frame, the position which gives maximum statistical correla-

tion of pixel values for a particular frame is considered the position where the AOI

encompasses the object. In most cases breaking into colour channels (using the RGB

model) to get the red channel and sometimes segmentation with a manually set

threshold was done prior to tracking. Since the marker for the heavier liquid was red,

the use of the red channel minimized the effect of the surrounding fluid around the

bubble, in case it still fails, image segmentation is done to render the bubble with one

colour and the rest of the image with another.

Fig. 5. Snapshots of correlation tracking of a bubble being done by Image Pro Plus software.

the area marked by the box is the ROI which is being tracked.

3.2 Entrained mass calculations

Fig. 6. Snapshots from a MATLAB program to identify the volume of the entrained mass with

time. The mass shown in white is water that is above a manually set threshold and is considered

as entrained mass while the mass shown in light grey is waster that is below the threshold and

not counted as entrained mass. The dark blue label on the bottom left corner shows the volume

of the total entrained mass in mm3 while the single largest entrained mass is outlined with green

and its volume in mm3shown on it. The graph at the bottom represents the variation of en-

trained mass with time showing the total mass in thick yellow and the volume of the single

largest mass in thin black line. Snapshots are shown 20 frames apart.

The entrained mass has been recognised by its red colour. As a red dye was used for

the heavier liquid, a difference of red and green channels, with weights added manual-

ly gave an image sequence of only the red liquid. This was segmented with a manual-

ly set threshold to yield a black and white image sequence where the red coloured

heavier liquid is white and the rest all black. With this arrangement one can visualize

a two-dimensional entrained mass which is generally shaped irregularly without any

32mm

axis of symmetry. To reconstruct the volume from this two dimensional figure, the

following assumptions were made:

1. The portion of the heavier liquid which is above only a specified height is consid-

ered as “entrained mass”

2. Each horizontal portion of the liquid column is circular about its own midpoint.

Thus if a horizontal scanline finds the column of width d, and the difference be-

tween successive such scanlines is h, the volume of the section is

3. In case there are holes in the white portion, that is an error and all holes should be

filled with white

Taking successive scan lines of the width of each pixel and adding up gave the vol-

ume of the entrained mass. The scale factor in pixels per mm was determined by

measuring the outer diameter of the pipe which was 32 mm. Figure 6 shows the snap-

shots of a MATLAB programme that calculates the volume of the entrained mass for

a 15 ml bubble.

4 Results

4.1 Qualitative aspects

The shape and size of the water mass entrained by a Taylor bubble rising through

water and then liquid paraffin is mapped with increasing distance travelled by the

bubble in figure 7. While the mass entrained by the smaller bubble has a more pointed

tail, that entrained by a larger bubble has a more round bottom surface.

Comparison shows that the entrained mass has a tendency to be confined in the wake

region below the Taylor bubble (figure 8 shows PIV generated streamline[9] for a

Taylor bubble in a single liquid). Thus the vortex formation behind a bluff body (in

this case a Taylor bubble) can be explained as a primary cause of entrainment and the

stability of the entrained mass can be attributed as a fight between these forces and

gravity is observed to increase with the size of the bubble as shown in figure 6, where

a comparison of the mass that entrains and the mass that falls back clearly shows low-

er tendency to fall back with increasing volume.

Fig. 7. Shape of the entrained mass at different distances from the original position of the inter-

face for different bubble volumes

Fig. 8. A typical vortex at the tail of a Taylor bubble: PIV results by RLG de Souza[9]

3 ml 3.8 ml 5 ml 7.5 ml 10 ml 12.7 ml 15.1 ml

Bubble volume

4.2 Entrained mass

Figure 9 shows the volume of the entrained mass with time. It is observed that it de-

creases due to detrainment just after it crosses the interface. The steep rise in en-

trained volume shown in the first portion of this graph shows the rise of the bubble

and the steep drop at the end shows the bubble rising above the frame of the imaging

area. While small Taylor bubbles (such as 3 ml) are observed to entrain more, medi-

um sized bubbles (10 ml) are found to entrain the least while it slightly increases for

larger bubbles upto 15.1 ml.

Fig. 9. The entrained mass of liquid erodes with time due to constant detrainment. The smallest

bubble is observed to have large entrained volume in the beginning while it erodes to the same

levels later. Other bubble volumes do not show any regular variation.

4.3 Entrainment column

After the bubble crosses the interface a column of heavier liquid still connects it with

the bulk of that liquid. This is termed as the entrainment column. A point comes when

the this column breaks apart leaving a mass dangling with the bubbles, known as the

entrained mass and a column which swiftly falls back. Intermediate droplet may be

left for certain liquids such as for water-PDMS shown in figure 1.

The height of the entrainment column from the original interface position before de-

tachment is shown in figure 10. This clearly shows a drop with increasing bubble

volume, however, adding the drop in interface in figure 11 shows that the maximum

column height is almost constant.

Fig. 10. Height of the liquid column from the original interface position before detachment

Fig. 11. The length of liquid column before it detaches the interface, shown above and below

the original position of the interface in cream and violet respectively\

3 ml 3.8 ml 5 ml 7.5 ml 10 ml 12.7 ml 15.1 ml

Bubble volume

As the column breaks into the entrained mass and a retracting column the total length

is almost constant, while some loss is sometimes seen due to the intermediate droplet

formations. Figure 12 shows the comparison of the volume of the entrainment column

from the initial interface level and the volume of the two liquid masses after detach-

ment. As described in section 4.2 the smaller bubbles tend to entrain larger masses. It

is to be noted that the volume of water could be calculated only from the initial level

of the interface for the ease of calculations.

Fig. 12. The comparison of the volume of rising bubble column with the volume of the en-

trained mass and the volume of the bubble column that is falling back.

5 Conclusion

This article investigates several aspects of the liquid mass entrained by rising Taylor

bubbles of different volumes and the results are summarised as follows:

1. A Taylor bubble rising through the interface of two liquids, drags some heavier

liquid up into the lighter liquid along its tail.

2. The entrained mass follows roughly the wake below the bubble.

3. A column of heavier liquid spans from the interface to the entrained mass, which

eventually breaks and the column falls back.

4. The volume of liquid entrained varies slightly with the bubble volume first increas-

ing and then decreasing.

5. The maximum length of the column before it breaks is almost constant with in-

creasing bubble volume.

Further exploration into other liquid pairs and other tube diameters are possible and

may yield newer insights.

6 References

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3. Bubble entrainment by the impact of drops on liquid surfaces, Oguz Hasan N and Prosper-

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