BBlowing Big Bubbles
Christopher A.E. Hamlett, a Dolachai N. Boniface, b Anniina Salonen, b Emmanuelle Rio, b ConnorPerkins, a Alastair Clark, a Sang Nyugen, a and David J. Fairhurst ∗ a a Nottingham Trent University, Nottingham, NG11 8NS, UK b Université Paris Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France.
February 16, 2021
Abstract
Although street artists have the know-how to blow bubbles over one meter in length, the bubble width istypically determined by the size of the hoop, or wand they use. In this article we explore a regime in which, byblowing gently, we generate bubbles with radius up to ten times larger than the wand. We observe the big bubblesat lowest air speeds, analogous to the dripping mode observed in droplet formation. We also explore the impact ofthe surfactant chosen to stabilize the bubbles. We are able to create bubbles of comparable size using either Fairyliquid, a commercially available detergent often used by street artists, or sodium dodecyl sulfate (SDS) solutions.The bubbles obtained from Fairy liquid detach from the wand and are stable for several seconds, however thosefrom SDS tend to burst just before detachment.
To blow a soap bubble sparks joy in people of allages, irrespective of their scientific knowledge. Thereis no need to understand Empress Dido’s isoperimet-ric inequality [17] to appreciate the simplicity of theirperfectly spherical shape, arising from the minimisationof surface energy. Nor is there a requirement to haveread Newton’s
Opticks [8] to be enchanted by the swirlingbands of colours indicating how the thickness of the soapfilm changes with drainage and evaporation [6]. Streetartists know that making stable giant bubbles is not easy:preparing the ideal soap solution for given environmentalconditions (humidity, temperature, air speed) requiresyears of know-how. And although recent work [5] hashighlighted how the addition of high-molecular weightpolydisperse polymers can improve stability, there is nofundamental model combining film formation and growththat can predict the optimal surfactant solution. Theanalysis is further complicated when film rupture is alsoconsidered, a singular event [11] that is nonetheless es-sential for the bubble to detach from the original film:rupture controls when the blown film pinches off to pro-duce a detached closed bubble[1, 2]. To successfully blowindividual bubbles they must not rupture until they haveleft the wand to drift away.The simple question of what determines the size atwhich the bubble detaches was first addressed by Plateauin the late 19 th century [9] and 80 years later Boys [1] pre-sented a beautiful overview of early experimental mea-surements. However, it is only much more recently thatSalkin et al. [13] designed a controlled experiment inwhich a bubble is formed by blowing a jet of air withcontrolled flow-rate through a film of falling surfactantsolution. At low air speeds the bubbles are generated one by one, while at higher air speeds the bubble sizeand frequency are set by the Rayleigh-Plateau instability[12]. Their results are applicable to both "contained" airjets, which fall entirely within the soap film, and "uncon-tained" jets which are larger than the film. They showthat the minimum velocity v for which a bubble can beinflated is given by a simple expression found by equat-ing the inertial pressure due to the jet of gas (density ρ g ) exiting from a nozzle of radius R with the Laplacepressure exerted by the expanded air flow at a distance d from the nozzle: v = (cid:115) γρ g R (cid:18) dR (cid:19) . (1)Turbulent jets, such as these, have a universal openingangle [10, 7] of 11.8 ◦ ≈ / radians, so the jet diameterincreases linearly with distance between nozzle and film.In this regime, the bubble size is set by the Rayleigh-Plateau instability and equals R . Su et al. systemati-cally varied the nozzle diameter and also found that bub-ble size was correctly predicted by Rayleigh-Plateau in-stability [15]. To generate large bubbles in the Rayleigh-Plateau regime, it is thus necessary to use a correspond-ingly large wand or a wide air flow.In this article, we explore the possibility of making"giant" bubbles, significantly larger than the wand, byblowing downwards in the low air-speed regime. Thisfollows on from preliminary work mentioned briefly inRef. [13], where the existence of a dripping regime at thelowest air-speeds was mentioned [3]Using different soapsolutions to stabilize our bubbles, we show that the maindifference between a pure surfactant and a commercial1 a r X i v : . [ phy s i c s . f l u - dyn ] F e b ishwashing liquid is the bubble lifetime rather than thebubble size. Figure 1: Experimental setup: a bubble is blown by pass-ing air at a controlled pressure through a wand dippedin soap solution. The distance d between the tube ori-fice and the wand is fixed such that the diameter of theair flow is equal to the diameter of the wand. Images ofthe bubble are recorded by a digital camera and anal-ysed with ImageJ to extract the radius of the circle ofequivalent cross-sectional area.The experiments presented here have been performedusing both high-purity research-grade chemicals and acommercially available, multi-component dish-washingliquid.The pure system was a sodium dodecyl sulphate(SDS)/Glycerol solution. SDS (Aldrich, France usedas received) was dissolved in distilled deionised water(MilliQ conductivity σ = 18 . M Ω .cm) to give a rangeof molar concentrations, generally above the critical mi-cellar concentration c C MC ≈ c = 15 mM. As SDS is known to hydrolyse over time producingdodecanol, all samples were used within a week of prepa-ration. The SDS powder itself was kept for up to onemonth before a fresh supply was used. Glycerol (SigmaAldrich, France, purity > . %) was added with a con-centration of 20 % by mass. The sample was mixed witha magnetic stirrer until thoroughly homogenised. Thesolution with 15 mM SDS and 20 % glycerol was char-acterized with commercial apparatus. The surface ten-sion γ = 37 . mN/m was measured using the pendantdrop method (Tracker, Teclis, France)and the viscosity η = 1 . Pa.s using a double-couette rheometer (AntonPaar).For the commercial dishwashing product, we usedFairy (also known as Dreft) (Procter and Gamble, Bel-gium), the exact composition of which is unknown, but it does contain 15-30 % anionic surfactants (e.g. sodiumlaureth sulfate, similar to SDS) and 5-15 % non-ionic sur-factants (e.g. lauramine oxide). We used a concentrationof 10 % by volume. For these solutions, we measured asurface tension of 25.3 mN/m. The viscosity is the sameas water.The experimental setup is illustrated in Figure 1 witha downward airflow directed onto the soap film. Theair flow was controlled using an Elveflow OB1 multi-portpressure controller connected to the lab compressed airsupply, with pressures between 320 mbar to 480 mbargiving air velocities between 6 m/s and 8 m/s. The airwas passed along R = d = 5 cmdownstream from the nozzle, the fully turbulent jet ex-pands to a diameter of 20 mm, where the size of the jetwas comparable to R w the diameter of the 3D printedwands [13]. This value of d was then fixed for all sub-sequent measurements. At this distance, the speed ofairflow was measured using a TPI 575 digital hot wireanemometer positioned in the centre of the airflow andfound to be constant across the surface of the wand.Different wands were trialled, including hand-builtwire loops of various diameters (using wire of diam-eter 1 mm) and commercially available plastic wandswith ridged edges (which act as a solvent reservoir) be-fore choosing to manufacture custom wands using a 3Dprinter, with radius R w of 9.5 mm, width of 4.5 mm,thickness e of 3.3 mm and patterned with 36 periodicridges fanning out from the inside to the outside. Ad-ditionally, the material used by the 3D-printer (ABS) isslightly porous. For these reasons, the effective lengthof the wand, on which the contact line is pinned, is un-known. We thus introduce a parameter r defined as theratio between the effective length and the measured wandperimeter.Complementary experiments were undertaken using aVortice VC1 electric blower to generate upwards airflow,controlled using a Griffin and George variable AC trans-former on the input voltage. This setup necessitatedlarger tubing, a nozzle of diameter 7 mm and hand-builtwire loops of diameter 15 mm. Air speed was also mea-sured using a hotwire anemometer. The wider nozzleallowed for measurements to characterise the air velocitywithin the expanding jetImages of the bubbles were recorded using digitalcameras (either Imaging Source DBK 41AF02 or UEyeU148SE) at frame rates of up to 30 frames per second.Bubbles were either illuminated with ambient laboratorylighting and blown in front of a dark background, or il-luminated from behind using a LED light panel (DORRLP-200LED 17.8 × As the air speed was increased from zero the soap filmbecame progressively more deformed. No bubbles wereobserved until a threshold air speed v was reached. Be-yond this threshold, we observed three different types ofbubble, depending on when they burst, depicted in Fig.2.Those which burst before detaching from the wand aredescribed as open bubbles and those which detach andfloat away from the wand are closed bubbles . Betweenthese, we identify almost closed bubbles which very nearlydetach, and show a narrow neck, but do not quite sepa-rate completely from the wand before bursting. As canbe seen in Fig.2, the bubbles are significantly larger thanthe wand diameter, 1.9 cm. In Fig.3 we plot the prob-ability of formation of each bubble type at varying airspeed for both SDS/Glycerol and Fairy solutions.With the Fairy solution, we observe no bubbles at airspeeds below 7 m/s. There is an abrupt transition fromopen to closed bubbles with a well defined closed-bubbletransition velocity v c = 7 . m/s, where the probabilityvalues for open and closed bubbles are equal. With thissolution, we observe hardly any almost closed bubbles.For the solutions of SDS/Glycerol, the first bubblesalso appear at a threshold velocity of around v = 7 m/s.However, most of these bubbles burst before detaching(open bubbles). On increasing the speed, we see a broadtransition region from open to closed bubbles with up to30 % almost closed bubbles. The closed-bubble transi-tion velocity is v c ≈ . m/s. For solutions at lower con-centrations (roughly below the cmc, data not shown) nobubbles of any type are seen. For solutions with higherSDS concentrations we find that v c decreases with con-centration (data not shown). In addition to characterizing the bubbles as open orclosed, we also used ImageJ to measure the equivalentradius of every bubble. Fig.4 shows the sizes of over 1500 Figure 3: The probability of obtaining open, closed andalmost closed (AC) bubbles plotted against air velocity,for both 15 mM SDS/20 % Glycerol and Fairy solutions.Almost closed bubbles are only seen in SDS solutions,in a narrow velocity range. The lines are guides for theeyes.bubbles blown using 15 mM SDS with 20% glycerol. Wesee significant scatter in the measured bubble sizes, pre-sumably caused by stochastic bubble rupture but also byfluctuations in air speed and in the thickness of the soapfilm on the wand. Despite the variability, we can makesome general observations. The largest bubbles, with ra-dius of almost 10 cm, are formed around v = 7 m/s andare typically open or almost closed. On increasing theair speed, the bubbles become smaller with an increas-ing probability of being closed. Around v ≈
10 m/s thereis a clear transition in bubble sizes, which now no longerdepend on v and exhibit a radius R around 2 cm. Thistransition corresponds to the the dripping to jetting tran-sition seen by liquids flowing through a narrow orifice [3],the existence of which Salkin also showed with bubbles[13].When comparing these downwards airflow results withthose from experiments using a larger nozzle and up-wards airflow, we found a systematic difference in thresh-old wind speeds. By converting wind speed to total airflux (by multiplying by the cross-section area of the re-spective nozzle) the data collapses, as shown in Figure 5.The high speed region is now extended to significantlyhigher fluxes.For bubbles made with the commercial solution, Fairyliquid, all the measured bubble radii as a function of windspeed are shown in Figure 6. As with the SDS bubbles,two populations of bubbles are seen: (i) at high air veloc-ity, small bubbles with an average radius of around 2 cmwhose size remains constant with changing air speed and(ii) at low air velocity large bubbles whose size decreaseswith air speed. Unlike with SDS, the Fairy solutions3igure 4: Radius of the bubbles obtained for differentair velocities using a solution of SDS at 15 mM with20% glycerol. The blue symbols represent the size of thebubbles which burst before detaching (open bubbles) andthe red symbols the size of the bubbles which burst afterdetaching from the wand (closed bubbles). The greenstars correspond to almost closed bubbles.show that at lower air speeds it is possible to create ei-ther small or large bubbles: at a given speed the twopopulations coexist, but are still distinct. Additionally,the large bubbles have a much higher probability of be-ing closed compared to those obtained with SDS (see alsoFig. 3). At very low air speeds the film hardly moves, but asthe speed increases the kinetic energy of the moving airbecomes sufficient to deform the film. The thresholdair speed v to overcome the curvature energy can becalculated from the maximum Laplace pressure requiredduring deformation of the soap film, which is the valuepredicted by Eq. 1 [12].Using relevant experimental values ( γ SDS = 39 mN/m, γ Fairy = 25 mN/m, ρ g = 1.2 kg/m , R w = 9.5 mm and d = 5 cm) gives v ≈ m/s for SDS/Glycerol and . m/s for Fairy, which is a little lower than the experimen-tal value for SDS/Glycerol but corresponds well to thethreshold air speed observed in Fig 3 for Fairy, where v ≈ m/s. One reason for the underestimation of thethreshold in the case of SDS/Glycerol may be that thebubbles obtained at such small velocities burst duringtheir formation before they have been measured. Figure 5: Comparing the bubble size using both up anddown air flow - the up direction used larger nozzles, sothe wind speed is normalised by calculating the flux (i.e.velocity × area).Figure 6: Radius of the bubbles obtained for differentair velocities using a solution of commercial detergent(Fairy). The blue symbols represent the size of the bub-bles which burst before detaching (open bubbles) andthe red symbols the size of the bubbles which burst afterdetaching from the wand (closed bubbles).. As the surface tension values of SDS and Fairy are sim-ilar, we observe a subsequent similarity in threshold ve-locities. However surface tension alone is not sufficientto fully understand and predict the bubble-blowing pro-cess of a given surfactant. Figure 3 illustrates that thereare significant differences between the two solutions weconsidered. For the solutions of Fairy, at air velocitiesbelow . m/s more than 50 % of the bubbles are open.Between 7.5 and 8.0 m/s, although most of the bubbles4re closed, some of them break before detachment, witha probability of breaking around 0.3. Above a velocity ofaround 8.0 m/s, the bubbles become almost exclusivelyclosed, with a probability approaching 1. At this veloc-ity, the bubbles are rather small and correspond to thejetting regime (Fig. 6).This is very different to what is observed for bubblesstabilized by SDS/Glycerol, for which only a few big bub-bles are ever closed. In particular, a probability closeto 1, corresponding to mostly closed bubbles, is reachedonly in the jetting regime when the bubbles are small.This emphasises that the big bubbles stabilized by SDSare much more prone to burst than the ones stabilized byFairy liquid. The poorer stability of the SDS bubbles isnot surprising (street artists never use SDS in their solu-tions), however we note that the creation of large bubblesis possible, despite their bursting before detachment.A better understanding of this observation is still anopen question (and is known to depend on added highmolecular weight polymers [5]) and is beyond the scopeof this paper in which we chose to concentrate on thedescription of the bubble size R , whether they burst ornot before detachment. In 1864, Tate was the first to consider the formation andsize of droplets dripping, for example, from a tap [16].His measurements were well described by a simple math-ematical model that balanced the droplet mass with thesurface tension. Even earlier than this, in 1833, Savart[14] investigated the instability of a falling liquid jet, ob-serving it to break up into small droplets.At high air speeds, our small bubbles, with a radiusaround 2 cm, correspond to the jetting regime observedand described by Salkin et al. [13]. The size is due to theRayleigh Plateau instability [12] and the bubble radiusis twice the radius of the wand (horizontal lines in Fig.4 and 6). In the following, we concentrate on the largestbubbles made at the lower air speeds, the existence whichwas noted by Salkin et al. [13]. We characterise thetransition and propose that the large bubbles do not formas a consequence of instabilities in a hollow soap tube,but are inflated while attached to the wand and detachat some criteria.We propose a mechanism similar to thatproposed by Tate for liquid droplets.
There are three main forces that act during thedownward-blowing experiments, in which we observe thedripping mode: an inertial force, the weight of the bubbleand surface tension.The inertial force I is directed downward. It is dueto the moving air which expands the bubble at a rateof R directly beneath the wand so we can write I = ρ g (2 ˙ R ) πR w . To estimate the inertial force, we follow the method ofClerget et al. [4] and write the Bernoulli equation alonga stream line between the center of the wand and theexpanding edge of the bubble, which gives P + 4 γR + 12 ρ g (2 ˙ R ) = P + 12 ρ g v . (2)Now the inertial pressure I can be expressed in termsof the the inertia of the jet and the Laplace pressure as I = (cid:0) ρ g v − γR (cid:1) πR w .The weight of the liquid contained in the film sur-rounding the bubble also acts downwards and depends onthe bubble surface area and the thickness h of the soapfilm. Assuming a spherical bubble, this can be writtenas M = 4 πR ρ l gh where ρ l is the density of the liquid.The surface tension force acts upwards, keeping thebubble attached to the wand. It acts along the lengthover which the film is in contact with the wand, equalto wand’s perimeter multiplied by a parameter r whichquantifies both the macroscopic ridges and the micro-scopic porosity of the wand. So, finally, we write thesurface tension force as T = 4 πrγR w . If we neglect theroughness ( r = 1 ) we find that the surface tension isnever high enough to compensate for inertia.Both I and M act to detach the bubble, while T keepsthe bubble attached to the wand provided I + M ≤ T .If we estimate these three terms with a film thicknessaround 1 µ m, the weight is around 8 times smaller thaninertia. Additionally, the film thickness, although prob-ably micrometric is unknown. We thus choose to neglectthe weight which leads to v = (cid:115) γρ g (cid:18) R + rR w (cid:19) . (3)Equivalently, we can express this in term of dimension-less variables by introducing the Weber number W e = ρ g v R w γ and by normalizing the bubble radius by thewand radius. This leads to the simple expression of RR w = 1 W e − r (4)This prediction is plotted using r as an adjustable pa-rameter together with data in Figure 7.The data ob-tained with the Fairy and with the glycerol collapse ona master curve and their sizeis well described by thismodel using a value of r equal to 3.2 for both SDS andFairy/Glycerol solutions, showing that our model cap-tures the main physical parameters.Note that the agreement between data and model isvery sensitive to the value of r . This could explain whypeople use a rough wand to blow bubbles: as well asserving as a reservoir for the solution, it provides ad-ditional contact area to hold the bubble in place. Theagreement is also very sensitive to the surface tension,which is not surprising since the pendant drop experi-ment uses the dripping mode of droplets to measure this5uantity. An alternative interpretation of r could thusbe a larger surface tension, which would give a slightlydifferent equation since the surface tension acts in bothterms of Eq. 3. Both a different surface tension and aneffective wand length could also contribute in principle.We have shown that gravity is negligible in predictingthe maximum size of the bubbles in our dripping model.Thus we could expect that big bubbles can be formedby blowing up, which is not the case as shown in Figure5. We propose that gravity driven drainage cannot beneglected in this case. When blowing up, the film atthe top of the bubble is expected to thin and eventuallybursts. On the contrary, when blowing down, the film atthe top is fed by the liquid contained in the wand.Interestingly Zhou et al. [19], using an experimen-tal setup similar to Salkin’s, have observed the oppo-site of our findings: smaller bubbles at lower air speeds,larger at higher speeds and a regime between in whichno bubbles are observed. However, their air speeds (ascharacterised by the Reynolds number) are slower thanours, meaning that their no-bubble regime may corre-spond with our almost closed bubbles regime.Figure 7: Bubble radius normalized by the wand velocityplotted as a function of the Weber number. The dashedline corresponds to a jetting model, in which R = 2 R w and the solid line is the best fit by Equation 4 of theentire set of data. To conclude, we measured the size of bubbles generatedby controlled blowing on a film of soap solution sus-pended in a rough wand. The measured threshold ve-locity necessary to blow a bubble agrees with previouspredictions as do the bubble sizes observed at high ve-locities, in the Rayleigh-Plateau regime, which are thesame size as the wand diameter. At lower velocities,when blowing downwards, the bubbles are formed oneby one in a dripping mode. We create bubbles that are significantly larger than the wand by blowing at low ve-locity, very near the threshold.In this regime, the bubblesize is well described by a balance between inertia andsurface tension.We observed different types of bubbles: open, closedand almost closed. The latter are only observed forSDS/Glycerol solutions. Nevertheless, the bubble sizeseems independent of the bubble type so that it is possi-ble to blow big bubbles with SDS solutions but we neverobserve their detachment.In practice, big bubbles are actually blown by varyingthe air speed - high initial to overcome Laplace pressure,then gently inflating a bubble without detaching at lowspeeds, and then increasing sharply to detach. This ob-servation provides scope for future work.
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