Shape oscillations on bubbles rising in clean and in tap waterChristian Veldhuis, Arie Biesheuvel, and Leen van Wijngaarden Citation: Physics of Fluids (1994-present) 20, 040705 (2008); doi: 10.1063/1.2911042 View online: http://dx.doi.org/10.1063/1.2911042 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/20/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of rising motion on the damped shape oscillations of drops and bubbles Phys. Fluids 25, 112107 (2013); 10.1063/1.4829366 Bubble motion and size variation during thermal migration with phase change Phys. Fluids 25, 013302 (2013); 10.1063/1.4774329 Experimental studies on the shape and path of small air bubbles rising in clean water Phys. Fluids 14, L49 (2002); 10.1063/1.1485767 Steady and oscillatory thermocapillary convection generated by a bubble Phys. Fluids 12, 3133 (2000); 10.1063/1.1321263 Collective oscillations of fresh and salt water bubble plumes J. Acoust. Soc. Am. 107, 771 (2000); 10.1121/1.428253

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Shape oscillations on bubbles rising in clean and in tap waterChristian Veldhuis,1,2 Arie Biesheuvel,1,3 and Leen van Wijngaarden1,2

1J.M. Burgers Centre for Fluid Dynamics, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlands2Department of Applied Physics, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlands3Department of Mechanical Engineering, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlands

�Received 24 September 2007; accepted 24 February 2008; published online 30 April 2008�

This paper deals with air bubbles rising in purified water in the range of equivalent diameters wheresurface oscillations appear on the interface. The shape of the bubbles including these capillarydistortions is recorded by taking a large number of high speed pictures for each spiraling orzigzagging bubble trajectory. In analogy with surface harmonics, the oscillations are indicated as�2,0� axisymmetric and with wavelength equal to the distance from pole to pole and �2,2�nonaxisymmetric and with wavelength equal to one-half of the length of the equator. In the secondseries of experiments, the phenomena in the wakes of rising bubbles are made visible by usingSchlieren optics, which are applicable because a temperature gradient is applied to the water. Thefrequencies of vortex shedding correspond to the �2,0� mode of surface oscillation, whereas in otherworks reported in the literature, they correspond to twice the frequency of the spiraling orzigzagging bubble paths. By measurements and by analysis, it is shown here that the latter is due tocontamination of surfactants. © 2008 American Institute of Physics. �DOI: 10.1063/1.2911042�

I. INTRODUCTION

During the recent years, new knowledge has been ob-tained on the dynamics of bubbles rising in water. It is wellestablished that at a Reynolds number of about 600, the rec-tilinear path bifurcates into a zigzag or a helical path. Anumerical demonstration was done by Mougin andMagnaudet1 and experimental studies were done by Elling-sen and Risso2 de and Vries et al.3 More recently, muchattention has been given to details of the paths followed bythe bubbles in zigzag or helical motion, the wake behind thebubbles, and the forces on the bubbles �Mougin andMagnaudet,4 Shew et al.,5 and Veldhuis6�.

From these studies, the following picture emerges: Up toan equivalent diameter Deq of 2.8 mm, the shape of thebubble is an oblate ellipsoid with the short axis nearly, butnot quite, in the direction of the tangential velocity of thebubble along its path. The wake consists of two threads car-rying vorticities of opposite sign.

Figure 1�a� shows two mutually perpendicular images ofa bubble with an equivalent diameter, i.e., the diameter of asphere with the same volume, 2.2 mm, rising in pure water.This is taken from Veldhuis.6 Note that the figure does notshow two bubbles but two different projections of the samepath. The motion is made visible by applying a slight tem-perature gradient in the water, permitting visualization withthe help of Schlieren optics. This method7 was first used byde Vries et al.,3 to whom we refer for further details. Whenthe effective diameter reaches the value of 2.8 mm the oblateellipsoidal shape gets distorted by surface oscillations of cap-illary nature. Figure 1�b� pictures a bubble with surface os-cillations. A comparison with Fig. 1�a� shows that the wakeis also different. There is a shedding of vortices that is not

there in the absence of oscillations. In this paper, we inves-tigate the regime with surface oscillations. Previous work onthis has been done by Lindt,9 Brücker,10 and Lunde andPerkins.11

After a short description of the experimental device inSec. II, we report in Sec. III on the frequencies measured,both of the surface deformations and of the vortex sheddingin the wake of the bubbles.

We compare in Sec. IV these frequencies with the calcu-lated ones by Meiron12 and Lunde and Perkins.11 The latterauthors conducted similar experiments in tap water. Compar-ing our results with theirs, it appears that there is a strikingdifference regarding the wake frequencies. Therefore, wecarried out experiments in tap water also, which in Sec. Vare shown to agree with those by Lunde and Perkins.11

We suggest that the observed differences are due to thepresence of surfactants in tap water. This suggestion is cor-roborated by the prediction of a model presented in Sec. VI.

II. EXPERIMENTAL SETUP

For the experiments we used a Plexiglass tank with a0.50 m height and a cross section of 0.15�0.15 m2, whichwas filled with ultraclean water �electrical resistance of18 M� cm and less than 10 ppb organic particles�. The tem-perature was maintained at 21 °C, with a fluid density of998 kg /m3 and a kinematic viscosity � of 0.96�10−6 m2 /s.A bubble generator �see Ohl13 for more details� was con-nected to the bottom of the tank. After being generated, thebubbles rose through a long tube �with a length of 20 cm andan inner diameter of 5 cm� before entering the Plexiglasstank. There, they were allowed to rise over 60 cm before

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being recorded in the upper part of the water tank over adistance of about 7 cm.

In the second series of experiments, the water washeated from above, resulting in a small temperature gradient,1.0 K /cm. This made it possible to apply Schlieren optics forvisualizing the wakes of the bubbles. We used an improve-ment the details of which can be found from Veldhuis.6 Thetemperature in the recorded field of view increased from25 °C at the bottom to 32 °C at the top of the field of view.With the help of lenses and mirrors, two mutually perpen-dicular views of the rising bubbles were recorded with aPCO Imaging camera at 640 frames/s with 1024�1280 pixels, resulting in a resolution of 0.078 mm /pixel.Typical measurement times �from the bottom to the top ofthe field of view� were in the order of 0.2–0.3 s. By account-ing for at least six measurement points within one oscillationperiod of the bubble, we were able to measure shape oscil-lation frequencies up to about 100 Hz. Therefore, the powerspectra that will be presented later in this paper are limited tothat value. With a measurement time of 0.2–0.3 s, the mini-mum frequency that can be measured will be about 3–5 Hz.This is only slightly lower than the typical path oscillationfrequency of the bubbles of about 6 Hz. Therefore, the pathoscillation frequency was always taken from visual inspec-tion of the images and the power spectra start at about 10 Hz.

III. EXPERIMENTAL RESULTS WITH BUBBLESIN PERFECTLY CLEAN WATER

A. Without Schlieren optics

In the first series of experiments, we did not useSchlieren optics. All data were obtained from the images asdescribed in Sec. II. Here, we present data on six represen-tative bubbles with the parameters given in Table I. The �symbol means the ratio between the major and minor axes.The mean Reynolds number Re is defined with the equiva-lent diameter Deq and the mean vertical velocity UT, Re=DeqUT /�. Bubble paths, as shown in Fig. 1, are periodic.We denote the associated frequency with fpath.

Figure 2 shows the three dimensional �3D� bubble pathand Fig. 3 shows a top view of the six bubbles collected inTable I. The color values indicate the velocity of the bubbletangential to its path; they are plotted here in order to visu-alize the difference between velocity oscillations and pathoscillations of a bubble. The deviation from a straight pathstarts at an equivalent diameter of 1.8 mm.

It is often stated �Ellingsen and Risso,2 Magnaudet andEames,14 Saffman,15 Shew et al.5� that for diameters as in �a�and �b� in Table I, the final state is a helix. Our experimentssuggest that a zigzag path can be a stable state of motion inthe presence of shape oscillations. To verify this, experi-ments in a device permitting a higher water level than wehad at our disposition should be carried out because the pos-sibility exists that bubbles in this range of diameters wouldeventually go over in a helix, as is the case in the experi-ments performed by Shew et al.,5 who used a 2 m high tank.These experiments will be very hard to do because of thelarge amount of purified water that is necessary.

The bubbles in Figs. 2�c� and 3�c� describe a perfecthelical path, whereas the spiral in �d� and �e� is flattened.Finally, the big bubble in �f� has a random, rocking motion.As mentioned in the Introduction, for each bubble at a largenumber of positions along their path, we made two mutuallyperpendicular projections. A sketch of such a pair of projec-tions is shown in Fig. 4.

The two planes on which the projections are made areindicated as ZY and ZX. From these images, the path and

1cm

t=0.205s

1cm

t=0.205s

(a) (b)

FIG. 1. �Color online� Bubbles rising in purified water with the bubble path indicated by the dot-dashed line and the bubble shape indicated by the closedcontours with an interval of 0.016 s; �a� without shape oscillations; �b� with shape oscillations. �The figures in the upper left-hand corner give the time elapsedfrom the appearance of a bubble in the image.�

TABLE I. Overview of experiments in clean water.

Subfigure Deq �mm� � fpath �Hz� Re Path

�a� 3.0 2.2 6.7 899 Zigzag

�b� 3.4 2.2 7.0 973 Zigzag

�c� 3.6 2.3 5.5 1018 Spiral

�d� 4.0 2.4 6.5 1096 Flattened spiral

�e� 4.5 2.7 6.5 1162 Tilted flattened spiral

�f� 5.2 2.8 5.2 1305 Chaotic

040705-2 Veldhuis, Biesheuvel, and van Wijngaarden Phys. Fluids 20, 040705 �2008�

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shape are obtained. Note that we do not obtain a fully 3Dreconstruction of the oscillating bubble. It is well known thatsuch a reconstruction can only be done for an axisymmetricbubble. Here, we only reconstruct the two two dimensional�2D� views of the bubble for further analysis of both the pathand general shape of the bubble.

For the bubbles with the properties given in Table I,these projections are shown in Fig. 5.

Images are recorded with 640 frames /s. Since thebubble shapes are plotted every ten frames, there is a timeinterval of 0.016 s between two consecutive bubble traces.The frequency of the path is about 6 Hz, so that roughly onewavelength is recorded. From the data as in Fig. 5, the nor-malized power spectrum �NPS� is computed; the NPS is de-fined as

NPS�f�t�� =FFT�f�t��FFT�f�t��

�FFT�f�t��FFT�f�t���,

in which FFT is the fast Fourier transform, the overbar indi-cating a complex conjugate and � � an average. The NPS iscomputed for a number of important quantities. To beginwith, the velocity tangential to the bubble path was com-puted. In each of cases �a�–�f�, the relevant spectrum isshown in the upper one of two plots in Fig. 6. Recall that werecorded with 640 frames /s with a measurement time of ap-proximately 0.2–0.3 s; therefore, the frequencies in thepower spectra can be fully trusted between 10 and 100 Hz.

The dominant modes in the oscillations appear to bethose with a wavelength equal to the distance over the ellip-

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FIG. 2. �Color online� Measured trajectories of bubbles rising in water. The color values represent the tangential velocity of the bubbles. The attached numbersindicate velocity in m/s. The coordinates along the axes are made nondimensional with the equivalent bubble diameter. See Table I for details of the bubbles.

040705-3 Shape oscillations on bubbles rising Phys. Fluids 20, 040705 �2008�

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soid from pole to pole and another with wavelength equal toone-half of the length of the equator. In analogy with a de-scription in terms of surface harmonics,

Pnm�cos ��exp i�m� − �nmt� , �1�

we indicate the former here as n=2, m=0, or simply �2,0�,oscillation and the latter as �2,2� oscillation. Later in thispaper, by means of comparison with literature, we will showthat indeed the dominant frequencies represent the �2,0� and�2,2� modes. In Eq. �1�, � and � give the angular position ofa point on the ellipsoid in the lateral and azimuthal direc-tions, respectively, whereas Pn

m�cos �� denotes a Legendrepolynomial with argument cos �. This does not mean that we

FIG. 4. Two mutually perpendicular projections in an XYZ laboratory frame.The orientation of the side views is arbitrary.

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FIG. 3. �Color online� Top views of bubbles rising in water.The color values represent the tangential velocity of the bubbles. The attached numbers indicatevelocity in m/s. * indicates the starting point of the trajectory. See Table I for details of the bubbles.

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identify these modes with spherical harmonics. Of course,analytically, they could be described as an expansion of suchfunctions, which we do not attempt. The �n ,m� are only usedto indicate the wavelength and axisymmetry or the absenceof these. We made these two modes visible in the spectra bylooking, following Lunde and Perkins,11 at the two longeraxes d1 and d2, as indicated in Fig. 4. First, we take the �2,0�mode and consider the plane �=� /2. Since this mode isaxisymmetric, d1 and d2 are the same at any time, for ex-ample,

d1 = d2 = a + cos �2,0t . �2�

So, the quotient d1 /d2 is constant but the product d1d2 hasthe term 2 cos �2,0t. Hence, the frequency �2,0 will show upin the spectrum of d1d2. The �2,2� mode is nonaxisymmetric.For �=0 and �=� /2, if at a given time

d1 = a + cos �2,2t , �3�

then, at the same time, at the position �=� /2, �=� /2�where the second of the two mutually perpendicular imagesis�

d2 = a − cos �2,2t .

Now, the product d1d2 is in the first order in a constant andits quotient has to this order the term 2 cos �2,2t. Hence, wesee that we can determine the general shape of the bubble,being axisymmetric or nonaxisymmetric, from the recon-struction of the two 2D perpendicular projections of thebubble, without full 3D reconstruction. For the experimentsof Table I, the power spectra of d1d2 and of d1 /d2 are shownin Figs. 6�a�–6�f� �the lower one in each pair of plots�.

The analysis of these spectra reveals an interesting be-havior. Figures 6�a� and 6�b� are for a zigzag and 6�c� is fora helix. In all three cases, we see that the tangential velocityhas oscillations of the same frequency as d1d2, indicating acoupling with the �2,0� mode of oscillation. This can be un-derstood by considering the dynamics of a bubble. The �2,0�mode changes the aspect ratio � of the ellipsoid and therebythe added mass. The periodic change of the added mass leadsto a change in velocity, since the impulse, which is the prod-uct of velocity and added mass, is approximately conserved.The relation between velocity changes and changes of addedmass was previously noted by de Vries et al.16 The nonaxi-

a) b) c)

1cm 1cm 1cm

d) e) f)

1cm 1cm 1cm

FIG. 5. Stereoscopic images of bubbles with their paths and shapes. Images are recorded with 640 frames /s. The bubble shapes are plotted every ten frames,giving a time interval of 0.016 s between the bubble shapes. See Table I for further details.

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symmetric �2,2� mode affects the added mass much less, tosecond order in terms of in Eqs. �2� and �3�. Therefore, themode mode �2,2� cannot be found in the velocity spectra butis very prominent in the zigzag motion. Again, this can beexplained by considering the dynamics.

A bubble moving along a zigzag path is schematicallyshown in Fig. 7. All the time forces are exerted on the bubblebut there are extra forces each time when the path turns tothe opposite direction. The force is mainly caused by pres-sure differences, as indicated in the figure. The pressure in-side the bubble is constant and these pressure differences aresupported by changes in curvature leading to shapes that areno longer axisymmetric. The adaptation from one shape tothe next goes along with �2,2� oscillations. In the �2,2� mode,a bubble is no longer axisymmetrical but the shape remains

a) b)

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FIG. 6. NPSs of tangential velocity and shape. See Table I. Upper plot: NPS of tangential velocity. Lower plot: �—� NPS of d1 /d2: �------� NPS of d1d2.

FIG. 7. Bubble along a zigzag path. The arrows give the direction of forcedue to pressure differences.

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ellipsoidal. Quoting Lunde and Perkins,11 “An interpretationis of the bubble as an ellipsoid rotating about its minor axis,which is consistent with the bubble being a spheroid under-going mode �2,2� oscillations.” Undoubtedly, other nonaxi-symmetric modes will be present but not with significantdetectable intensity. They have higher frequencies. For ex-ample, for a bubble with Deq=4 mm and with �=2,4, the�3,1� mode has a frequency, 77 Hz, see Meiron.12 Modes ofthis type are not present to a significant intensity in our mea-sured spectra.

The measured frequencies for the bubbles listed in TableI are collected in Table II. The more the path resembles a

zigzag, the closer the path frequency is to a multiple of thefrequency of the �2,2� oscillation.

B. With Schlieren optics

The dynamics in the wake of a bubble performing azigzag or helical motion are very important but not visible inthe images as in Fig. 5. Lunde and Perkins11 were aware ofvortex shedding but could not directly determine the associ-ated frequencies. Instead, they derived the vortex sheddingfrequency from measurement of the velocity componentsnormal to the mean path. To obtain more insight into thewake frequencies and their origin, we carried out a series ofexperiments with Schlieren optics. To this end, a slight tem-perature gradient is applied in the vertical direction, as de-scribed in Sec. II. Because of the associated viscosity varia-

TABLE II. Measured frequencies of shape oscillations for the bubbles inFig. 6. The dashes denote that no distinct frequency was detected.

Deq

�mm�f2,0

�Hz�f2,2

�Hz�fpath

�Hz� f2,2 / fpath Path

3.0 61.8 40.6 6.7 6.0 Zigzag

3.4 50.7 34.0 7.0 4.9 Zigzag

3.6 45.5 – 5.5 – Spiral

4.0 39.0 20.5 6.5 3.2 Flattened spiral

4.5 – 17.5 6.2 2.8 Tilted flattened spiral

5.2 – 14.0 5.2 2.7 Chaotic

TABLE III. Overview of experiments in clean water with Schlieren optics.

Subfigure Deq �mm� � Re Path

�a� 2.8 2.1 1062 Spiral

�b� 3.5 2.2 1197 Zigzag

�c� 3.9 2.3 1289 Zigzag

�d� 5.5 2.7 1674 Chaotic

a) b)

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FIG. 8. �Color online� Stereoscopic Schlieren images of bubbles with the bubble path indicated by the dot-dashed line and the bubble shape indicated by theclosed contours with an interval of 0.016 s. The Reynolds numbers are �a� 1062, �b� 1197, �c� 1289, and �d� 1674. For further details, see Table III. �Thefigures in the upper left-hand corner give the time elapsed from the appearance of a bubble in the image.�

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tion, the equivalent diameter and the aspect ratio fortransition from the regime without shape oscillations to theone with such oscillations slightly differ from the valuesgiven in Sec. III A. Data of four typical experiments to showthe wakes are collected in Table III.

Figure 8 shows for these bubbles their wake. In cases�a�–�d�, there are surface oscillations and there is vortexshedding clearly visible, in contrast to the corresponding pic-ture �Fig. 1�a�� without surface oscillations.

Similar to Fig. 6, the relevant spectra are presented inFig. 9. For each of the bubbles in Table III, the spectra of thevelocity �upper graph� and those of the shape oscillations�lower graph� are given.

The first one �Fig. 9�a�� performs a spiraling motion.Although both the velocity and the d1d2 oscillation are ratherweak, they are coupled, a phenomenon which we also saw inSec. III A. This coupling is also clearly visible in Figs. 9�b�and 9�c�, which correspond to a zigzag motion. In Fig. 9�d�,belonging to a bubble with more or less chaotic shape oscil-lations, no prominent frequencies are to be seen. In Figs.9�a�–9�d�, a frequency of about 14 Hz is visible. This is aspurious one caused by the bubble moving in and out offocus of the high speed camera and bears no relation withsurface oscillations. Due to the Schlieren method, we couldnow directly measure the frequency in the wake of thebubbles. From images as in Fig. 8, we obtained the valuescollected in Table IV.

In Fig. 10, we have collected all our experimental dataregarding measured frequencies as a function of the equiva-

lent bubble diameter. These measurements have all beendone with purified water. In addition to the shape oscillationsand velocity oscillations, we were able to also measurethe wake frequencies, see Table IV. These are indicatedwith * in Fig. 10.

In the next section, we will discuss these results andcompare them to other works.

IV. DISCUSSION OF RESULTS OBTAINEDIN CLEAN WATER

First we compare the measured frequencies of the �2,0�and �2,2� modes with available theoretical values. Whereasthese frequencies have been already calculated by Lamb17

2.5 3 3.5 4 4.5 5 5.

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FIG. 10. Frequencies obtained from experiments in purified water: �o� ve-locity; �*� wake; �·� �2,0� mode shape oscillation; �� �2,2� mode shapeoscillation.

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NP

S

FIG. 9. NPSs. See Table III. Upper plot: NPS of tangential velocity. Lower plot: �—� NPS of d1 /d2; �------� NPS of d1d2.

040705-8 Veldhuis, Biesheuvel, and van Wijngaarden Phys. Fluids 20, 040705 �2008�

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for a spherical shape, an exact analytical calculation for anellipsoid is much harder to do and has hitherto not beendone. Meiron12 numerically calculated a number of modesfor some equivalent diameters. A very useful analytical ap-proximation was made by Lunde and Perkins.11 We shall usethis here for the �2,0� and �2,2� modes. Their idea is as fol-lows: The wavelength of the �2,0� mode is one-half of theperimeter of the ellipsoid measured in the lateral direction ��direction with reference to Eq. �1��. With perimeter Pl, thewavelength �2,0 is

�2,0 = P1/2. �4�

Lunde and Perkins11 argued with the wave being a cap-illary wave, the wave speed can be approximated by thewave speed c on a plane interface,

c = � 2��

�2.0�1/2

, �5�

where � denotes the surface tension and denotes the liquiddensity. By using a known expression for the perimeter, forthe frequency, they find

f2.0 =1

2�� 162�2

��2 + 1�3/2�1/2� �

req3 �1/2

. �6�

In the same way, Lunde and Perkins11 obtained an expressionfor the frequency of the �2,2� mode, now with one-half of thelength of the equator of the ellipsoid as wavelength, giving

f2.2 =1

2�� 8

��1/2� �

req3 �1/2

. �7�

In Fig. 11, nondimensionalized forms of Eqs. �6� and �7�are brought together with the numerical results of Meiron12

and our experimental results for the frequencies of the shapeoscillations. This figure shows that the results of Lunde andPerkins11 results �Eqs. �6� and �7�� agree very well with theexact result of Meiron.12 More importantly, it shows that our

measured frequencies are indeed �2,0� and �2,2� modes andthat the measured values agree well ��2,0� mode� and satis-factorily ��2,2� mode� with the results obtained by theory.

The wake frequencies that we measured, indicated with* in Fig. 10, agree very well with both the velocity oscilla-tions and the �2,0� shape oscillations. The coupling betweenvelocity oscillations and the �2,0� mode is, as mentioned be-fore, due to changes of added mass going with the shapeoscillations. The velocity oscillations are strong enough to beaccompanied by the shedding of vorticity. This view is cor-roborated by the good agreement between the wake frequen-cies reported in Table IV and f2,0 as given by Eq. �6�. Forcomparison, some values of the latter are included in TableIV. Lunde and Perkins11 observed in other experiments withdye injection that the vortex shedding frequency is twice thepath frequency. They assumed the same vortex sheddingmechanism for their experiments in tap water without dyeinjection. Therefore, they measured, albeit indirectly, wakefrequencies in the experiments by doubling the path fre-quency. They extracted the path frequency by looking at theoscillation of velocity components normal to the direction ofthe mean path.

In Fig. 12, taken from Lunde and Perkins,11 the mea-sured wake frequencies, or vortex shedding frequencies, arealso given. In contrast to our findings, these do not agreewith the �2,0� frequencies but are equal to twice the pathfrequency, that is, about 14 Hz. It is extremely difficult toobtain water completely devoid of surfactants, and sinceLunde and Perkins11 observed the wake frequency by inject-

FIG. 12. Various frequencies as a function of equivalent radius �from Lundeand Perkins, Ref. 11�. Note the vortex shedding frequencies are indicatedby �.

2.5 3 3.5 4 4.5 5 5.5

20

40

60

80

Deq

/mm

f/H

z

FIG. 13. Frequencies obtained from experiments in tap water as a functionof Deq: �0� velocity; �·�, shape oscillation �2,0� mode; �� oscillation �2,2�mode.

TABLE IV. Frequencies for several bubble diameters.

� 2.1 2.2 2.2 2.1 2.1 2.1 2.2

Deq �mm� 2.8 2.9 3.1 3.1 3.3 3.4 3.5

fwake �Hz� 74 65 65 54 48 45 46

f2,0 �Hz� �according to Eq. �6�� 71 69 61 64 57 58 50

1 1.5 2 2.5 30

2

4

χ

f/(σ/

ρr eq3

)1/2

FIG. 11. Nondimensionalized frequencies f �in rad/s� as a function of aspectratio �: �·� experimental values for �2,0� mode; �� experimental values for�2,2� mode; �o-----o� Meiron’s �2,0� mode: �o—o�, Meiron’s �2,2� mode;�-.-.-� �2,0� mode Eq. �6�; �—� �2,2� mode Eq. �7�.

040705-9 Shape oscillations on bubbles rising Phys. Fluids 20, 040705 �2008�

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ing dye into the flow and performed the rest of the experi-ments in tap water, we suspected that the origin of the dis-crepancy is in the quality of the water. Therefore, we decidedto also perform a series of experiments in tap water.

V. EXPERIMENTAL RESULTS WITH BUBBLESIN TAP WATER

The experiments were carried out in the same way aswith the purified water. The results for the various frequen-cies, that is, �2,0� and �2,2� surface modes, tangential veloc-ity oscillations, and vortex shedding frequencies, are col-lected in Fig. 13, which should be compared to the results forpurified water in Fig. 10.

Typical examples of Schlieren images are shown in Fig.14. Both bubbles shown perform zigzag oscillations.

VI. DISCUSSION OF RESULTS OBTAINEDIN TAP WATER

The differences between the data in Fig. 10 for purifiedwater and Fig. 13 for tap water can all be explained by thesurfactants with which tap water is contaminated. Surfactantsdiffuse from the bulk toward the interface where they are, byconvection, swept along the bubble surface in the directionfrom the forward stagnation point toward the rear. What re-sults �see Bel Fdhila and Duineveld,18 and Palaparthi et al.19�

is a front part free of surfactants, the remainder of thebubble, the so-called stagnant cap, consisting of an interfacewith a lower surface tension. Most importantly, even with thebulk concentration of surfactants as small as 10−3–10−2, theeffective boundary condition changes from no shear to norelative velocity. As a result, the surface tension gradientbetween the stagnant cap and the surfactant-free front partcauses a nonzero tangential stress, resulting in a Marangonieffect which yields a no-slip condition on the stagnant cap.This dramatically increases the drag and reduces the rise ve-locity of a bubble. With the help of these effects, producedby surfactants, we can explain the observed differences.

�i� Because of the reduced rise velocity, surface oscilla-tions set in at a larger effective diameter. Figure 13shows oscillations starting at Deq=3 mm, whereas inpure water, they start at Deq=2.8 mm.

�ii� Both by the phenomena described above and by thealso occurring shedding of surfactants from bubblesurfaces by the oscillations, the concentrations of con-taminants considerably vary. This explains the greatscatter displayed in Fig. 13.

�iii� The most striking difference is the completely differ-ent vortex shedding frequency in tap water. We haveseen in Fig. 10 that in purified water, this agrees withthe �2,0� frequency. As is clear from the images inFig. 14, with the zigzagging bubble in tap water, vor-ticity shedding takes place each time the velocitychanges direction. The frequency is therefore equal totwice the path frequency, or about 14 Hz. This agreeswith the observations by Lunde and Perkins.11 Similarresults were reported by Brücker10 and by Lindt.9 Ourexplanation is that this is also due to the activity ofsurfactants. We will clarify this with the help of amodel calculation. If we leave out the lift force for thesake of simplicity, the forces exerted on a bubble arethe buoyancy, drag, and reaction forces by the liquid.The latter is the change of impulse I, where, withvelocity with rise U and virtual mass M,

I = MU . �8�

In this simplified situation, the path is vertical. The trans-verse motion and the change in added mass due to the rota-tion of the bubble are left out.

The drag force is strongly dependent on the amount ofcontaminant in the bulk of the liquid. In purified water, thedrag D is

D = 12��reqU , �9�

where � is the viscosity of the liquid. Let us call the volumeconcentration of surfactant C. Above some critical value ofC, the drag is almost equal to the drag of a solid ellipsoid,which is much larger. For simplicity, we take here

D = F�C�U , �10�

although of course for a solid body, the drag is proportionalto U2 rather than U. For our qualitative argument Eq. �10�suffices. With acceleration of gravity g and bubble volume V,the force balance is approximately �lift forces are left out�

1cm

t=0.372s

(a)

1cm

t=0.348s

(b)

FIG. 14. Schlieren images of bubbles with �a� Deq=3 mm and �b� Deq

=3.5 mm. Both rise in a zigzag motion in tap water. Clearly visible are theperiodic bursts of vorticity at the outer positions of the zigzagging motion.In between the bursts one can observe a double-threaded vortex structure.�The figures in the upper left-hand corner give the time elapsed from theappearance of a bubble in the image.�

040705-10 Veldhuis, Biesheuvel, and van Wijngaarden Phys. Fluids 20, 040705 �2008�

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d

dt�MU� = gV − F�C�U . �11�

In equilibrium, the right-hand side of Eq. �11� is zero, result-ing to UT= gV /F�c� for the terminal rise velocity. This al-lows us to write Eq. �11� as

d

dt�MU� = F�C��UT − U� .

With UT−U=v, linearizing around U=UT where M =M0,and introducing the relaxation time � with

� =M0

F�C�, �12�

Eq. �11� becomes

dvdt

+v�

=UT

M0

dM

dt. �13�

The variation in the added mass M with time is a result of the�2,0� oscillation. We take

M = M0 + �M0 cos �t, t � 0.

After passage of the transient, inserting this in Eq. �13� gives

v = UT − U =UT�M0

M0

�2�2 cos �t − �� sin �t

1 + �2�2 . �14�

From this expression, it follows that if the drag is small,which means �see Eqs. �10� and �12�� that � is large, suchthat ���1,

UT − U UT�M0

M0cos �t, �� � 1. �15�

If, on the other hand, the drag is large �contaminated water�,and consequently � is small, it follows from Eq. �14� that

UT − U − ��UT�M0

M0sin �t, �� � 1. �16�

In purified water where Eq. �15� applies, the velocity varia-tions produced by the surface oscillations are strong enoughto be accompanied by a vortex shedding with frequency �2,0.In contaminated water, the velocity variations are �� timessmaller, which is not sufficiently strong for vortex shedding.These appear now only at the turn of the path, with zigzagmotion. If we compare the wake behind bubbles without sur-face oscillations �Fig. 1�a�� with wakes behind bubbles withsurface oscillations �Figs. 1�b� and 8�, we see that in thelatter case, vortices are shed. Apparently, the vortices areassociated with the surface oscillations. Moreover, as re-marked earlier, the measured wake frequencies correspond tothe velocity oscillations and with to �2,0� mode. Movingbodies shed vorticity in an unsteady motion. In clean water,the accelerations are, as predicted by the simple modelabove, much larger than those in tap water. We suggest there-fore that the significant accelerations in clean water are causeof the shedding of vortices. Because the Schlieren techniquecannot resolve the details of the vorticity production at thebubble surface, it is not possible to see at what particularinstant during an oscillation the vortices are formed. How-

ever, our suggestion is in line with the observation that dur-ing zigzag motion in tap water, vortices are shed at the sharpturn of the bubble into the opposite direction.

VII. CONCLUSIONS

In this paper we investigated surface oscillations onbubbles rising in water. We confirmed the findings of Lundeand Perkins11 that these oscillations are mainly of �2,0� and�2,2� modes �in terms of surface harmonics�. A feature re-ported in our work here is making the wake, trailing behindbubbles, visible with the help of Schlieren optics. This re-vealed that in clean water, the vortex shedding frequency iscoupled with the �2,0� mode, in contrast to the results re-ported by others. In agreement with those by others, repeat-ing the experiments in tap water gave the result that thevortex shedding frequency is coupled with twice the pathfrequency. We explain this discrepancy by analyzing the ac-tivity of surfactants.

ACKNOWLEDGMENTS

We enjoyed helpful discussions with Detlef Lohse. Wethank Peter van Oostrum for his contributions to the experi-ments, and Gert-Wim Bruggert and the late Henni Scholtenfor their invaluable technical support during the experiments.Thanks are due to Andrea Prosperetti for reading this manu-script. This work is part of the research program of the Stich-ting Fundamenteel Onderzoek der Materie.

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040705-11 Shape oscillations on bubbles rising Phys. Fluids 20, 040705 �2008�

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