Giant bubble pinch–off Raymond Bergmann1, Devaraj van der Meer1, Mark Stijnman1, Marijn Sandtke1, Andrea Prosperetti1,2, and Detlef Lohse1 1 Physics of Fluids Group and J.M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands 2 Department of Mechanical Engineering, The Johns-Hopkins University, Baltimore, Maryland 21218, USA 6 Self-similarity has been the paradigmatic picture for the pinch-off of a drop. Here we will show 0 throughhigh-speedimagingandboundaryintegralsimulationsthattheinverseproblem,thepinch- 0 off of an air bubble in water, is not self-similar in a strict sense: A disk is quickly pulled through 2 a water surface, leading to a giant, cylindrical void which after collapse creates an upward and a downward jet. Only in the limiting case of large Froude number the neck radius h scales as n h(−logh)1/4 ∝τ1/2,thepurelyinertialscaling. ForanyfiniteFroudenumberthecollapseisslower, a and a second length-scale, the curvatureof thevoid, comes into play. Both length-scales are found J toexhibitpower-lawscalingintime,butwithdifferentexponentsdependingontheFroudenumber, 4 signaling thenon-universality of thebubblepinch-off. 2 PACSnumbers: 47.55.df,47.55.db,47.20.Ma ] n y d The pinch–off of a liquid drop is a prime example of nects the disk with the linear motor. This arrangement - a hydrodynamic singularity and has been studied exten- generates giant voids in a very controlled fashion. The u sively in recent years [1, 2, 3, 4, 5, 6, 7]. It has become advantage of this setup is that the velocity is a control l f paradigmatic for self–similar behavior in fluid dynamics: parameterandnotthe responseofthe objecttothe fluid . s After appropriate rescaling, the shapes of the pinching forces upon impact. Secondly, due to the large scale of c i neckatdifferenttimescanbesuperimposedontoasingle the experiment, viscosityandsurfacetensionplaya neg- s shape [4,5, 6, 7]. With the exceptionofsomepioneering ligiblerole[24]. Thereforetheonlyimportantdimension- y h work [8, 9], the inverse problem of the collapse of a gas– less parameter is the Froude number Fr = V2/(hdiskg), p fillednecksurroundedbyaliquidhasnotattractedmuch the ratio of kinetic to gravitationalenergy, which ranges [ attention until very recently, with the analysis of the from 0.6 to 90. The large scale of the experiment is also 1 pinch–offofabubblerisingfromaneedleandthebreak– advantageous for the observation of details during the v upof agasbubble ina strainingflow [10, 11, 12,13, 14]. impact and collapse process, which is imaged with dig- 8 The time–evolutionofthesecollapsinggas–fillednecksis ital high-speed cameras with frame rates up to 100,000 8 found to follow a power law. If the dynamics near the frames per second. 1 singularityaresolelygovernedbyliquidinertia,thenthe 1 A typical series of events is seen in Fig. 1a-d. The im- 0 radius of the neck h expressed in the time τ remaining 6 until collapse scales as h τ1/2 [8, 9, 10, 11], or, with a pactofthe disk creates anaxisymmetricvoidwhichfirst 0 logarithmic correction, as∝h( logh)1/4 τ1/2 [14]. De- expands until the hydrostatic pressure drives the walls / − ∝ inward. The inward moving walls collide and cause a s viations from this exponent of 1/2 are reported to occur c only due to the inclusion of other effects. The collapse pinch-off at some depth below the undisturbed free sur- i face. The energy focusing of this violent collapse cre- s may be slowed down by viscosity (h τ [10, 11, 12]) or y surfacetension(h τ2/3[13]),oracce∝leratedbytheiner- ates a strong pressure spike on the axis of symmetry h tiaofthe gasflowi∝nginsidethe neck,leadingtoh τ1/3 which releases itself in a downward and an upward jet p ∝ [15, 16]. The latter reaches heights exceeding 1.5 m for : [14]. v the higher impact speeds in this experiment. It is this i In this paper we focus onanother example of this “in- dominating role of inertia that makes our system differ- X verse pinch-off”, namely the violent collapse of the void ent from other pinch-off processes in the literature. At r created at a fluid surface by the impact of an object. a higherrecordingspeeds the pinch-offcanbe investigated Here we findexponents whichdeviate substantially from inmoredetailasinFig.1e-h. Thereisaclearlossofboth 1/2, even though the dynamics are shown to be purely azimuthal and axial symmetry in Figs. 1f and 1g, which governed by liquid inertia, without significant contribu- can be attributed to a combination of the same conver- tions from the effects mentioned above. The self-similar genceeffectthatcausesaninstabilityinacollapsingbub- behaviorh( logh)1/4 τ1/2 appearstoholdonlyinthe − ∝ ble [17, 18, 19], and a Kelvin-Helmholtz instability due asymptotic regime of very high impact velocities. to the rapid air flow in the neck. The latter increases Inourexperiment,alinearmotorisusedtodragmetal with increasing Froude number and limits the range of disks with radiih between10and40mm throughan ourexperiments. AnotherfactorwhichlimitstheFroude disk air/waterinterfacewithawell-controlled,constantveloc- number range is the so-called surface seal, in which the ityV between0.5and3m/s(seeFig.1a). Arodrunning void closes at the water surface as the crown-like splash through a seal in the bottom of a large water tank con- is entrained by the air flowing into the expanding void 2 FIG. 1: Snapshots of the formation and collapse of a surface void for the plunger experiment: A linear motor pulls down a disk of radius h = 30 mm through the water surface at a constant velocity V =1.0 m/s (Fr = 3.4). (a-d) The collapse of disk the void imaged at a 1000 frames per second. The white lines (overlay) are the void profiles obtained from boundary integral simulations with thesame initial condition, and without theuse of any free parameter. (e-f) Details of thecollapse imaged at 12800framespersecond. (g-h)Detailsofthecollapseimagedat48000framespersecond. Notethatthefieldofviewdecreases with increasing frame rate. In the verylast stages of thecollapse (f-g) there is a Kevin-Helmholtzinstability that complicates the determination of the time of collapse. Immediately after the collapse air is entrapped, both in the form of a large bubble above disk (d) and as a cloud of microbubbles at thecollapse point (h). [20,21]. Thisprocess,whichoccursatlargeFroudenum- ical calculations, again showing excellent agreement for bers,changesthepinch-offconsiderablysinceinthiscase different Froude numbers. In this graph, a straight line the gas pressure inside the void differs appreciably from corresponds to the power law behavior h = h0ταh. The that of the ambient air. exponent α is plotted as a function of Fr in Fig. 2c. h Clearly, there are large deviations from the suggested In view of these experimental limitations, we per- behavior α = 1/2. Can these be explained by a log- formed numerical simulations using a boundary integral h method based on potential theory without ambient gas. arithmic correction as proposed in [14]? Letus firstestablishthe originofthis logarithmiccor- There is an excellent agreement between the numerical rection in our system. Near the neck, the flow induced calculations and the experiments, as seen in Fig. 1a-c. by the collapsing voidlooksvery much like that of a col- Here, the numerical void profiles (the solid white lines) lapsing cylinder, while it must look like that of a sink, coincide very well with the experimental profiles in the plus its image in the free surface (i.e., a dipole) in the pinch-offregionwithouttheuseofanyadjustableparam- farregion. Inthelanguageofsingularperturbations,the eter, either in space or in time. formerwouldbetheinnerregionandthelattertheouter To further quantify the pinch-offprocess,we nowturn region;acompletedescriptionswouldrequirethe match- to the time evolution of the neck radius h(τ), measured ingofthesetworegions. Ifwedisregardtheouterregion, atthedepthatwhichthevoideventuallycloses. Because we can use a two-dimensional version of the Rayleigh- both length and time scales become very small close to equation,whichdescribesthecollapseofaninfinitecylin- collapse, it is not feasible to experimentally observe the drical cavity under uniform pressure [9, 16, 22] collapse with only one high-speed camera recording [25]. Due to the reproducibility of the experiment, we over- d(hh˙) h 1 log + h˙2 =gZ . (1) came this difficulty by matching several data sets im- " dτ # h∞ 2 agedatdifferentframerates,increasinglymagnifyingthe region around the pinch-off. Figure 2a contains a dou- The pressure difference driving the collapse has been bly logarithmicplot of h(τ) (compensatedwith τ1/2) for equated to ρgZ, where Z denotes the depth below the boththe high-speedimagedexperiments andthe numer- fluid surface, which implies that the system is composed 3 FIG. 2: (a) The radius of the void at the depth of closure h, compensated with τ1/2, as a function of the time τ remaining untilcollapse inadoublylogarithmic plot,for threedifferentvaluesof theFroudenumberFr. Experiment(bluesymbols) and numerical simulations (red symbols) are seen to agree very well for Fr = 3.4 and 10.1. For Fr = 163 only numerical data are presented, because for this Froude numberexperiments are hindered by the surface seal (see text). The error bars, indicating the error in the experimental data, are usually small, but occasionally become very large for frames very close to the collapse time. (b)Doublylogarithmic plot of theradiusofcurvatureofthevoid profileR compensated withτ1/2 asafunction ofτ for the numerical simulations of (a). Both h and R are well described by power laws for up to four orders of magnitude in τ. (c) Power-lawexponentsα fortheradiusofthevoidatclosuredepthh(τ),α fortheradiusofthevoidincludingthelogarithmic h y correction y(τ) = h(−logh)1/4, and α for the radius of curvature of the void at closure depth R, all as a function of the R Froudenumber. of non-interacting horizontal layers of fluid, with a neg- asymptotic regimes ligible vertical velocity component [26]. Although the quantity h∞ must in principle be determined by the h˜ log(h˜∞) 1/4 = √2τ˜1/2 for h˜ 1/h˜∞ , and(4) matching process alluded to before, it is expected to be ≫ of the order of a typical length scale of the process, such h˜ (cid:16) log(h˜)(cid:17)1/4 = √2τ˜1/2 for h˜ 1/h˜∞ . (5) as the cavity depth. Thus, strictly speaking, h∞ is a − ≪ (cid:16) (cid:17) function of time and of the Froude number. However, From Eqs. (4) and (5) we conclude that the scaling de- nearpinch-off,thetimescalefortheneckmotionismuch pends crucially on the value of h∞: Initially, for the in- fasterthanthatfortheevolutionoftheotherpartsofthe cavity so that h∞ may be considered only a function of stetrrimctedpioawteerrleagwimhe hτ≫1/2h,2msinaxc/ehl∞og,(h˜w∞e)eixspceocntsttaonfit.ndFoar Fr. After an initial expansion of the void, the collapse ∝ starts from rest at a maximal radius hmax (of the order times closer to the pinch-off, when h ≪ h2max/h∞, loga- rithmiccorrectionsplayarole,andthepowerlawshould of h ). Using this as an initial condition, and treating disk be modified into Eq. (5). h∞ as a constant, the energy integral of Eq. (1) can be readily found: As h∞ ≈hmax in our experiments, the latter inequal- itycanbereadash h ,whichissatisfiedinmostof max ≪ theregionwhereh(τ)asymptoticallybehavesasapower 2 law (cf. Fig. 2a). We conclude that in our system the dh˜ 1 = 1 (1/h˜)2 , (2) logarithmic correction cannot be neglected. If we plot dτ˜! log(h˜/h˜∞) − the quantity y = h( logh)1/4 vs. time, we again ob- h i − serve a power law y ταy, but with a slightly different ∝ exponentα thantheonefoundforh. InFig.2cwecom- y where we have introduced the non-dimensional variables pare α and α as functions of the Froude number. As h y h˜ ≡ h/hmax, h˜∞ ≡ h∞/hmax, and τ˜ ≡ τ gZ/h2max. discussedbefore,αh is substantiallylargerthan1/2,but Close to pinch-off, h˜2 1, such that h˜−2 1 h˜−2. even if the logarithmic term is included we continue to ≪ p− ≈ With this approximation, we can integrate Eq. (2) once find a slowercollapse for low Froude numbers. Although more to arrive at the logarithmic correctiondoes bring the result closer to the suggested value 1/2, it cannot account for all of the observed deviations. π Clearly,theobservedanomalouspowerlawbehaviorof 2τ˜=h˜2 log(h˜∞/h˜)+ h˜2∞erfc 2log(h˜∞/h˜) . 2 theneckradiusmustreflectitselfinthetime-evolutionof q r (cid:18)q (cid:19) (3) thefree-surfaceprofilesofthecollapsingvoid. Ifthepro- Forsmallh˜ thetermwiththecomplementaryerrorfunc- cesswereself-similar,thefree-surfaceprofilesatdifferent tion is always small compared to the first one and their times τ would superpose when scaled by any character- ratiovanishesforh˜ 0. Neglectingthistermwefindtwo istic length, e.g., the neck radius h. Actually, it is found ↓ 4 that the depth of minimum radius increases somewhat ThisworkispartoftheresearchprogramoftheSticht- as the collapse progresses and it is therefore necessary ing FOM, which is financially supported by NWO. to translate the profiles in the vertical direction so as to match the position of the minimum radius point before attempting this operation. Even if this is done, how- ever, the results fail to collapse onto a single shape. The [1] A. L. Bertozzi, M. P. Brenner, T. F. Dupont, and L. P. rescaledprofilesbecomemoreandmoreelongatedasthe Kadanoff, in Trends and Perspectives in Applied Mathe- pinch-off is approached which proves that the collapsing matics,editedbyL.Sirovich(Springer,NewYork,1993), void is not self-similar in a strict sense. Vol. 100, p. 155. The free-surface shapes near the minimum point [2] H. A.Stone, Ann.Rev.Fluid Mech. 26, 65 (1994). shouldthusnotonlybecharacterizedbyh(τ),butalsoby [3] M. P. Brenner, X. D. Shi, and S. R. Nagel, Phys. 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This holds when they are defined globally, i.e., justable parameter. Even when we exclude the effects of with respect to the impact velocity and the disk radius air, viscosity, and surface tension, the collapse is found (Re = hdiskV/ν and We = hdiskV2ρ/σ), but also when to be not self-similar in a strict sense, but governed by theyaredefinedlocallyusingtheneckradiusandvelocity at a specific time (Re=hh˙/ν and We=hh˙2ρ/σ). power laws with non-universal, Froude-dependent expo- [25] Weimagethepinch-offprocessoverfourordersofmagni- nents. Self-similarity is recovered only in the limit of tudeintimeandtwoinspace.Asthefieldofviewofthe infinite Froude number, where the influence of gravity camera corresponds to 103 pixels, this would leave only becomes negligible and the collapse is truly inertially 10pixelsforthelast stageof thecollapse. Moreover,the driven. 5 holesequenceshouldthenbeimagedataframeratecor- then leads to the scaling h(τ)R(τ) for the axial direc- respondingtothesmallesttimescale(10µs),i.e.,100kHz tion Z. The aspect ratio of the void is then given by requiringat least 10GB of fast storage capacity, greatly H/Z =(h(τ)/R(τ))1/2. p exceedingthe physical capabilities of our cameras. [28] Thefact that bothhand Raredescribedbypowerlaws [26] A similar equation is used in [14], without the term h∞ suggeststhatwemaybedealingwithself-similarityofthe and also without thehydrostatic drivingpressure gZ. secondkind,inwhichtheradialandaxialcoordinatesare [27] At the minimum 1/R(τ) = d2H/dZ2|Z=Zmin and the rescaled by different power laws of time [23]. At present shapeoftheinterfacecanbetakentobelocallyparabolic, thereishoweverinsufficientexperimentalandtheoretical which implies H = (δZ)2/R(τ)+h(τ) with δZ = Z − ground to substantiate such a claim. Z . The scaling of the radial direction H with h(τ) min