Whether a thin filament of liquid separates into two or more droplets or eventually condenses lengthwise to form a single larger drop depends on the liquid's density, viscosity, and surface tension and on the initial dimensions of the filament. Surface tension drives two competing processes, pinching-off and shortening, and the relative time scales of these, controlled by the balance between capillary and viscous forces, determine the final outcome. Here we provide experimental evidence for the conditions under which a liquid filament will break up into drops, in terms of a wide range of two dimensionless quantities: the aspect ratio of the filament and the Ohnesorge number. Filaments which do not break up into multiple droplets demand a high liquid viscosity or a small aspect ratio.This work was supported by EPSRC (RG53364 and RG55605

Castrejón-Pita, Alfonso A

Castrejón-Pita, JR

Hutchings, IM

Physical Review Letters

English

Apollo (Cambridge)

Breakup of Liquid Filaments Alfonso A. Castrejón-Pita, J. R. Castrejón-Pita, and I. M. Hutchings Phys. Rev. Lett. 108, 074506 (2012) The breakup of liquid lamentsA.A. Castrejon-Pita, J.R. Castrejon-Pita, and I.M. HutchingsDepartment of Engineering, University of Cambridge,17 Charles Babbage Road, Cambridge, CB3 0FS, U.K.Whether a thin lament of liquid separates into two or more droplets or eventually condenseslengthwise to form a single larger drop depends on the liquid's density, viscosity and surface tensionand on the initial dimensions of the lament. Surface tension drives two competing processes,pinching-o and shortening, and the relative timescales of these, controlled by the balance betweencapillary and viscous forces, determine the nal outcome. Here we provide experimental evidencefor the conditions under which a liquid lament will break up into drops, in terms of a wide rangeof two dimensionless quantities: the aspect ratio of the lament and the Ohnesorge number. Longlaments which do not break up into multiple droplets demand a high liquid viscosity or a smalllament aspect ratio.PACS numbers: 47.55.D-, 47.55.db and 47.80.JkThe dierence in behaviour between thin streams ofvarious liquids is familiar in the formation of dropletsfrom a thin water stream trickling from a tap, and inthe quite dierent behaviour of the more viscous threadsof honey or syrup owing from a spoon [1]. The break-up of liquid jets is important in several practical con-texts, including the dispersion of liquid drugs into res-pirable droplets, microuidics, crop- and paint-spraying,and ink-jet printing [2{4]. There are also biological sys-tems in which either long laments remain intact, ormany droplets are formed [5, 6]. In some cases, suchas the water jets generated by the archersh [7], inter-mediate behaviour is seen and an initially continuous jetbreaks up eventually into drops. A geological example ofliquid threads that do not break up is provided by thevery ne laments of glassy lava, widely known as Pele'shair, produced by certain volcanic eruptions [8, 9].FIG. 1: (Colour online) Geometry of a free cylindrical liquidlament with spherical end caps as assumed in the models ofSchulkes [10] and Notz and Basaran [11].Theoretical models for the collapse of a thin lament ofa Newtonian liquid (i.e. in which viscosity is independentof shear rate) of nite length, as shown in Fig. 1, havebeen proposed by Schulkes [10] and Notz and Basaran[11], but experimental evidence for their predictions hasso far been very limited. This is largely because mostexperimental techniques can cover only a small range ofuid properties and lament length. Furthermore, it isnot easy in practice to generate the free cylindrical la-ments that these models assume, and most studies havetherefore explored the quite dierent geometries of a con-tinuous jet or of liquid dripping from a tube [1]. We haveused an apparatus which was originally designed to repli-cate jet and droplet formation in an ink-jet printer, buton a larger scale [12], to produce free cylindrical lamentsof Newtonian liquids and explore their collapse behaviourover a wide range of aspect ratios and liquid properties.The results provide experimental conrmation of impor-tant aspects of theoretical models and also demonstratesome limitations.The relative importance of surface tension and viscos-ity in the behaviour of a liquid lament is expressedthrough the Ohnesorge number Oh = pR0, where , and  are the viscosity, density and surface tension ofthe liquid respectively and Ro is the initial radius of thelament. The initial geometry of a lament of length 2~Locan be described by the aspect ratio Lo = ~Lo=Ro.The models of Schulkes and Notz and Basaran usedcombined Lagrangian-Eulerian methods (Galerkin niteelement model, G/FEM) to solve the dynamics of in-compressible Newtonian liquid laments surrounded byan inert ambient uid, [10] and [11]. Due to computa-tional limitations, Schulkes' simulations could not followthe dynamics to breakup and therefore the results aremostly based on neck contraction rates. Despite its lim-itations, this pioneering study provided the rst insightsinto the existence of a critical Oh number (Ohc) separat-ing regimes in which a lament would breakup or not.Notz and Basaran performed a more detailed and exten-sive exploration of the parametric space to conrm theprevious hypotheses that the initial lament aspect ra-tio Lo is a crucial parameter that, together with the Ohnumber, determines the fate of liquid laments. Schulkes[10] predicted that a lament for which Oh  O(1) willcontract into a droplet regardless of its initial aspectratio Lo, while a suciently long lament with Oh O(0.01) will form bulges separated by thinner lamentsand ultimately break up into separate droplets. More-over, Schulkes predicted that Ohc lies within the rangeof 0:005 < Oh < 0:01 for 8  Lo  15. Notz and Basaran[11] went further, by concluding that for low Ohnesorgenumbers, Oh  O(0.1), the nal breakup is governed byboth a critical value (Ohc) and a critical initial aspectratio (Lo;c). A lament with Lo > Lo;c and Oh > Ohcwill break up, whereas a lament with Lo < Lo;c and Oh< Ohc will not. They established explicit values for thecritical Ohnesorge number and aspect ratio by computa-2tional modelling but experimental data is not availablefor the (Lo, Oh) parameter space. Previous studies inthe context of ink-jet printing have provided only verylimited experimental evidence [13, 14], and in any casetypical ink-jet print heads, which can be used only withuids with a narrow range of viscosity (typically 5 - 30mPa s), are restricted to the range  0:1 < Oh <1. Incontrast, here we have explored the behaviour over therange 0.003 < Oh < 10, for laments with 1 < Lo < 70.We used a liquid jet generator modied from a de-sign described elsewhere [12, 14] and driven by an electri-cal waveform to produce jets of water-glycerol mixtureswhich were imaged with a high-speed camera. Dierenttypes of drive waveform were required to produce freesymmetrical laments for dierent regimes of Oh. In allcases, the lament was formed as a secondary product ofjetting, behind a primary drop or sequence of drops.FIG. 2: (Colour online) Apparatus with a nozzle of variablediameter used to generate liquid jets.The jet generator (schematically shown in Fig. 2) con-sists of a cylindrical chamber with an outlet nozzle in itslower side and closed at the top by a exible rubber mem-brane which is mechanically coupled to an electromag-netic actuator (V201, LDS Instruments, UK) driven byelectrical pulses from a waveform generator under com-puter control. The pressure history within the chamberis monitored by a transducer mounted in its side. Inthis work we used a variable iris diaphragm as the noz-zle which could be continually varied from 0.2 to 5 mmin diameter to allow us to achieve a wide range of Oh.We used mixtures of water and glycerol (including purewater and pure glycerol) with the following values of vis-cosity, measured at 22 Celsius: 1, 2, 5, 6, 20, 30, 50,100, 200, 430 and 1400 mPa s. The surface tensions laybetween 72 mN m 1 (for pure water) and 63 mN m 1(for pure glycerol). Jet ejection speeds were between 0.5and 2.4 m s 1. We recorded images of the jet after ithad emerged from the nozzle with a high-speed camera(Phantom V310, Vision Research, USA) in shadowgraphmode at a frame rate of 10,000 frames per second withan exposure time of 5 s. The region of interest was il-luminated from behind with a 500 W tungsten lamentlamp through a 30 30 cm2 optical diuser. The camerawas used with two dierent lenses to cover dierent eldsof view: a Nikkor 18-55 mm for 1515 cm2, and a macroTamron SP AF90 for 55 cm2.In drop-on-demand ink-jet printing, the speed of thejet is mainly controlled by the amplitude of the rst volt-age peak, and its diameter by the duration of this rstpulse [13, 15]. Detachment of the jet from the nozzleand the characteristics of the resulting liquid lamentare, in contrast, governed by later peaks in the driv-ing waveform. At low Ohnesorge number surface tensionFIG. 3: (Colour online) Examples of drive voltage waveforms(upper traces) and resulting pressure waveforms (lower traces)used to produce symmetrical laments for: (a) Oh = 0.01 andLo = 2.9; (b) Oh = 0.04 and Lo = 9.0; (c) Oh = 0.18 and Lo= 29.2; (d) Oh = 1.01 and Lo = 59.0.dominates the behaviour of the liquid [16], and in theejection of a jet from a nozzle this results on the forma-tion of a large droplet followed by an attached lament.When the lament detaches from the nozzle its free endrapidly condenses into a smaller tail droplet. The la-ment then breaks up into separate droplets, usually pre-ceded by pinching-o at the two ends, close to the twoend droplets. In order to form a symmetrical cylindricallament under these conditions, we adjusted the dura-tion of the single-pulse driving waveform to ensure thatthe two ends of the central lament pinched o simulta-neously, giving a free lament preceded and followed byseparate drops. Figure 3(a) shows a typical drive wave-form, together with the associated pressure pulse withinthe liquid chamber. At high Ohnesorge number, in con-trast, viscous forces dominate and a dierent approach isneeded. In this case, very long laments (up to 150 mmlong) were ejected by the use of long drive pulses (Fig.3d), and their behaviour was followed until they brokeup into a series of shorter laments. We then analyzedthe behaviour of these laments.At intermediate values of Oh, both surface tension andviscosity play important roles and we used two dier-ent methods to create symmetrical laments. The rstwas similar to that used at high Oh, and involved theproduction of long liquid laments. However, as thepinching-o process occurs more rapidly, we used a pulland push waveform (Fig. 3b) to form the liquid la-ment more quickly: the second pulse ensured that thelament became detached from the nozzle. The secondmethod used two positive pulses (Fig. 3c) to produce two3FIG. 4: Images of laments generated by the drive waveformsshown in Figure 3, together with their shapes after the timesshown, for the following liquid viscosities and lament half-lengths: (a)  =2 mPa s and Ro = 0.95 mm; (b)  =5 mPa sand Ro= 0.19 mm; (c)  = 20 mPa s and Ro = 0.22 mm (d) =100 mPa s and Ro = 0.13 mm.large droplets separated by a continuous lament, which,with the correct delay between the two pulses, then be-came detached at both its ends simultaneously. In allour experiments the movement and development of thelaments was followed by analyzing the video recording,to ensure that they collapsed symmetrically and to es-tablish whether they condensed axially into a single dropor broke up into two or more separate drops. Exampleswhere collapse was not symmetrical, for example in whichthe initial conditions led to signicant dierences in thesizes of the drops forming at the ends of the lament,were discarded. Careful experimentation, involving netuning of the drive waveforms, was needed to produce l-aments which contracted symmetrically, and to explorea wide range of values of Oh and Lo.The behaviour of a liquid lament moving freely inair would be expected to be inuenced by the pres-ence of the air, but this is ignored in the models un-der consideration. We can show that its inuence wasalso negligible in our experiments. Viscous drag fromthe surrounding air would be expected to become sig-nicant only for laments with a radius smaller than a0airOh2= = air=We where air is the viscosityof air, We is the Weber number dened as We= v2R0=and v the speed of the lament [16, 17]. This critical ra-dius is < 1m for all the liquids studied, and thus belowthe resolution of the imaging system we used. Inertialinuences from the air would become signicant whenthe gas Weber number [18], dened by Weg = air=We(where air is the density of air), is greater than 0.2.In our work Weg lay between 0.018 and 0.003, so thataerodynamic inuences from this source would also havebeen negligible. Higher ejection speeds would involveother type of breakup mechanisms induced by aerody-namic and viscous eects not considered in the numericalmodels being compared here [10, 11].FIG. 5: (Colour online) Results plotted in terms of Oh andLo. Solid symbols represent cases where the lament brokeup into two or more droplets and open symbols where thelament contracted into a single droplet. The error bars rep-resent uncertainty associated with the measurements of la-ment length and diameter. The shaded area shows the pre-dictions of the model of Notz and Basaran [11]. The hatchedregion represent the range of critical Oh predicted by Schulkes[10]. Stars represent the points explored by Schulkes in whichbreakup does not occur. The broken line corresponds to theprediction of the linear instability analysis.Figure 4 shows examples of typical liquid lamentswith a range of dierent aspect ratios produced in thiswork by the drive waveforms shown in Fig. 3, in eachcase with a series of images showing their subsequent be-haviour. The shortest laments (e.g. Fig. 4a) alwayscondensed into single drops for all values of Oh, whilefor the highest values of Oh the laments never brokeup, whatever their aspect ratio. The fates of all the l-aments are plotted in Fig. 5. Each point represents theresults of a large number (up to 40) of experimental ob-servations. Within this parameter space, there is a cleardivision between laments which broke up into two or4more droplets and those which collapsed to form a singledroplet.Within the range of Oh explored here, the results showthat there is a tendency towards a critical initial aspectratio of 6  1 below which a liquid lament will notbreak up, irrespective of the value of Oh. In addition, nolament breakup was observed for Oh> 1, regardless ofits initial aspect ratio. The results obtained in a morelimited regime by Schulkes together with the existence ofa critical value of Oh for break-up as predicted in [10] arein full agreement with our work. The predictions by Notzand Basaran [11] of the critical aspect ratio for breakupfor 10 3 < Oh < 5  10 2 (shown by the shaded areain Fig. 5) also agree quite well with our observations forthe lowest values of Oh, but underestimate the tendencyof the laments to break up as Oh tends towards 0.05.However, it should be noted that the shaded area is basedon only ve numerically-generated points.As shown in Fig. 5, the breakup transition predictedby Notz and Basaran diers from our observations in therange 0.7 < Oh < 0.05. This discrepancy may be at-tributed to computational factors and to dierences be-tween the theoretical model and the conditions in the ex-periments. Notz and Basaran noted that in their G/FEMsimulations, lament ratios exceeding L0 15 requiredmore axial mesh elements and substantially longer com-putation times that may compromise the accuracy of thenal result. In practice, there are also factors that dif-ferentiate these experiments from the simple conditionsassumed by the models. Surface eects (e.g. contam-ination, surface vibration, laments not at initial rest,temperature driven eects, etc.) will aect the dynamicsof the liquid surface and ultimately the breakup of real l-aments, [19]. In the experiments, these factors will havea dierent eect in dierent regimes. Internal dynam-ics are unlikely to be important for highly viscous uidssince they are quickly damped by viscous forces: the timescale over which the lament contracts is large comparedwith the typical time scale on which initial disturbanceswill attenuate, [11]. In contrast, for low Oh, the sur-face pinch o occurs fast and disturbances will have notenough time to grow. Surface defects will be more evi-dent for long and thin laments which is precisely wherethe predicted transition seems to be underestimated. Inaddition to these problems, the experimental conditionsof the imaging setup were demanding, as very large eldsof view and high framing speeds were needed to followlong moving laments from initial jetting to the point ofbreakup. Consequently, the symmetrical shape of the l-aments (or any initial surface perturbation) could only beassured down to the resolution of the camera system; theimage resolution here ranged from 40 to 16 pixels/mmfor the longest ligaments.Linear stability analysis oers an alternative approachto describe the pinching o of a liquid thread [2], [13] and[20]. This method assumes that the breakup is causedby the fastest growing disturbance; if the disturbancehas the longest wavelength (max) that ts in the nitelength of the thread (2~Lo) then the lament will breakup (Eq. 2 in [13]). Although this method has providedgood approximations for the breakup times of droplets,it does not predict the dynamics of contracting satellitesor ligaments that are formed once the thread separatesfrom both the uid and the nozzle, as is the case in thiswork, [13] [20].Our results signicantly extend the range of la-ment length and Ohnesorge number over which lamentbreakup has been explored, and provide experimentalvalidation for the main conclusions of the earlier model-ing work. It is now possible to predict whether a lamentwill break up or not. For a Newtonian liquid, long liquidlaments which do not break up into multiple dropletscan only be produced when the Ohnesorge number islarge enough (Oh > 1), which in turn demands a highviscosity or a small lament aspect ratio.This project was supported by the UK Engineering andPhysical Sciences Research Council and industrial part-ners in the GlassJet and I4T projects (Grant numbersEP/G029458/1 and EP/H018913/1).[1] J. Eggers and E. Villermaux, Rep. Prog. Phys. 710366601 (2008).[2] J. Eggers, Rev. Mod. Physics 69 865 (1997).[3] S. Limpanuphap and B. Derby, J. Mater. Sci. Mater.Medicine 12 1163 (2002).[4] O.A. Basaran, AIChE Journal 48 1842 (2002).[5] T.D. Sutherland, J. H. Young, S. Weisman, C. Y. Hayashiand D.J. Merritt, Annu. Rev. Entomol. 55 171 (2010).[6] T. Eisner and D. J. Aneshansley, Proc Natl Acad Sci USA96 69705 (1999).[7] T. Schlegel, C. J. Schmid and S. Schuster, Current Biol-ogy 16 R836 (2006).[8] D. Shimozuru, Bull Volanol. 56 217 (1994).[9] S. Moune, F. Faure, P.J. Gauthier, K. W. W. Sims, J.Volcanol. Geoth. Res. 164 244 (2007)[10] R.M.S.M. Schulkes, J. Fluid. Mech. 309 277 (1996).[11] P.K. Notz and O.A. Basaran, J. Fluid Mech. 512 223(2004).[12] J.R. Castrejon-Pita, G.D. Martin, S.D. Hoath, and I.M.Hutchings, Rev. Sci. Instrum. 79 075108 (2008).[13] H. Dong, W.W. Carr and J.F. Morris, Phys. Fluids 18072102 (2006).[14] J.R. Castrejon-Pita, N.F. Morrison, O.G. Harlen, G.D.Martin and I. M. Hutchings, Phys. Rev. E 83 036306(2011).[15] A.U. Chen and O.A. Basaran, Phys. Fluids 14 L1 (2002).[16] A.U. Chen, P.K. Notz and O.A. Basaran, Phys. Rev. Lett.88 174501 (2002).[17] J.R. Lister and H.A. Stone, Phys. Fluids 10 2758 (1998).[18] W. Hoeve, et.al., Phys. Fluids 22 122003 (2010).[19] M.A. Herrada, J.M. Montanero and J.M. Vega, Phys.Fluids 23 082102 (2011).[20] D. Henderson, H. Segur, L.B. Smolka, and M. Wadati,Phys. Fluids 12 550 (2000).

Breakup of liquid filaments.

Whether a thin filament of liquid separates into two or more droplets or eventually condenses lengthwise to form a single larger drop depends on the liquid's density, viscosity, and surface tension and on the initial dimensions of the filament. Surface tension drives two competing processes, pinching-off and shortening, and the relative time scales of these, controlled by the balance between capillary and viscous forces, determine the final outcome. Here we provide experimental evidence for the conditions under which a liquid filament will break up into drops, in terms of a wide range of two dimensionless quantities: the aspect ratio of the filament and the Ohnesorge number. Filaments which do not break up into multiple droplets demand a high liquid viscosity or a small aspect ratio

Breakup of liquid filaments.

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