Drop formation in a falling stream of liquidVladimir Grubelnik, and Marko Marhl

Citation: American Journal of Physics 73, 415 (2005); doi: 10.1119/1.1866100View online: https://doi.org/10.1119/1.1866100View Table of Contents: https://aapt.scitation.org/toc/ajp/73/5Published by the American Association of Physics Teachers

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Drop formation in a falling stream of liquidVladimir Grubelnik and Marko Marhla)

Department of Physics, Faculty of Education, University of Maribor, Koroska cesta 160,SI-2000 Maribor, Slovenia

~Received 6 September 2004; accepted 14 January 2005!

The narrowing of a falling stream of liquid is a well-known demonstration of the equation ofcontinuity. We consider the behavior of the bottom of a falling liquid stream where the stream ceasesto narrow, and swells and forms droplets. Drop formation is demonstrated by detailed photos ofexperiments. A simple mathematical description of the observations is given, including the keyprocesses responsible for drop formation. ©2005 American Association of Physics Teachers.

@DOI: 10.1119/1.1866100#

I. INTRODUCTION

Drop formation in a falling stream of liquid is an interest-ing and commonly observed phenomenon that has been asubject of much research from the beginnings of hydrody-namic theory. It soon was realized that surface tension is thedriving force in drop formation, because it tends to reducethe surface area by decreasing the radius of the stream. In the19th century the phenomenon was described by Rayleigh.1–3

By using a stability analysis of an infinite cylinder of liquidwith radius r 0 , Rayleigh studied perturbations of differentwavelengths and calculated their growth rates. Although longwavelength perturbations tend to form large drops with thesmallest possible surface area, large mass transport betweenthe drops is required. Surface tension and inertia strike abalance at the wavelengthl59r 0 . In other words, drop for-mation depends crucially on the ratio between inertia andsurface tension, which is known as the Weber number.4

Subsequently, the theory of drop formation was refined byincluding the viscosity,5 higher order nonlinear effects,6 andsurface charges.7 Recently, drop breakup studies have beendevoted to the idea of scale invariance.8–11 The break upprocess becomes scale invariant if the dynamics is governedby proximity to the thinnest point in the neck, where the dropeventually breaks apart. Here the fluid decreases in diameteruntil its thickness goes to zero, and, at some point, it hasbecome so much smaller than any other macroscopic lengthin the system that the larger lengths no longer matter for adescription of the neck itself. The dynamics depends only onthe thickness of the neck and does not depend on the size ofthe nozzle or the size of the drop. Therefore, all memory ofthe initial and boundary conditions is lost. Some importantexceptions to this type of behavior have recently beenanalyzed.11

The importance of visualization of drop formation hasbeen understood from the outset of studies in this field. In1833 Savart investigated the decay of fluid jets by illuminat-ing a jet with sheets of light. He observed undulations grow-ing on a jet of water that cause the breaking of the jet andsubsequent drop formation.8 The development of photogra-phy and the recent use of computers gave us new ability tovisualize the dynamics of drop formation. The developmentof visualization from early photographs taken by Rayleigh12

to the first high-resolution photographs of water falling froma faucet taken by Peregrineet al.13 are reviewed in Ref. 8.

Although drop formation can be experimentally observedand analyzed, its quantitative mathematical description isdifficult. The theory of drop formation based on the Navier-

Stokes equations3,8,14,15 is not accessible to undergraduates.Simplifications of the theory using one-dimensional approxi-mations of the Navier-Stokes equations14 are also too diffi-cult.

In this paper we suggest a semi-quantitative approach todrop formation. The quantitative description involves onlythe key processes involved in drop formation in a fallingstream of liquid, which require understanding about the con-cepts of surface tension, pressure in liquids, and conservationof volume flux in noncompressible liquids~the equation ofcontinuity!. Our study concerns Newtonian fluids for whichprevious experimental observations10,16 have shown that thefree-surface shapes of the fluid are very similar when ap-proaching the pinch point. The dynamics become universal,that is, independent of initial conditions such as the nozzleradius and independent of the experimental method of form-ing the droplets,8 whether it is by a jetting, liquid bridge, ordripping experiment~as performed in this study!.

As a starting point, we consider the narrowing of a streamof water coming from a faucet. The narrowing of the streamof a nonviscous liquid represents an analytically solvableproblem.17,18We are interested in the bottom part of the fall-ing stream of water, where the transition from a thin streaminto drops takes place.

II. NARROWING OF A FALLING STREAMOF LIQUID

The cross-section of a stream of liquid coming from afaucet becomes smaller due to the Earth’s gravitation. For afalling stream of liquid with cross-sectionS0 and velocityv0

at the faucet~see Fig. 1!, we can calculate the cross-sectionSand the velocity of waterv at a timet later. According to theequation of continuity, the conservation of the volume fluxΦV , we can write

ΦV5S0v05Sv. ~1!

Due to gravity, the velocityv changes with distance from thefaucet,h:

v25v0212gh. ~2!

The substitution of Eq.~2! into Eq. ~1! gives

S5S0

v0

Av0212gh

. ~3!

415 415Am. J. Phys.73 ~5!, May 2005 http://aapt.org/ajp © 2005 American Association of Physics Teachers

The radius of the stream of water,r , follows directly fromEq. ~3! by taking into account the relationsS5pr 2 and S0

5pr 02 ~see Fig. 1!:

r 5r 0A4 v02

v0212gh

. ~4!

It is impressive how well Eq.~4! predicts the form of areal stream of water. Figure 2 shows a photo of a fallingstream of water and the superimposed solution obtained fromEq. ~4!. The latter closely matches the form of the realstream.

To obtain the results in Fig. 2, the initial velocity of thestream,v0 , has to be measured very accurately. We proposean effective procedure in whichv0 can be accurately deter-mined by measuring the volume flux of waterΦV . We sim-ply measure the volumeV of water flowing into a glass andthe timet taken to fill it. BecauseΦV5const, it follows thatΦV5V/t. From the relationΦV5S0v0 , we can expressv0as:

v05V

S0t. ~5!

For the experimental system shown in Fig. 2, the initialvelocity is v0511.060.5 cm/s. If we use this value in Eq.~4!, the radius as a function ofh of the stream of water canbe calculated, and the results compared with experiment asshown in Fig. 2.

Fig. 2. Photo of a falling stream of water with the analytical solution of theform of the stream. Experimental measurements:r 050.5060.02 cm, v0

511.060.5 cm/s; the calculations were done withr 050.50 cm andv0

511.0 cm/s.

Fig. 3. Drop formation at the bottom of a stream of water. The drops growas swellings in the stream.

Fig. 1. Narrowing of a stream of liquid falling from a tap.

416 416Am. J. Phys., Vol. 73, No. 5, May 2005 Vladimir Grubelnik and Marko Marhl

III. DROP FORMATION

The narrowing of a falling stream of liquid discussed inSec. II closely models the form of a real stream. However,this model holds only for the upper part of the stream. Thebottom part of the stream where drops are formed requiresadditional explanation. In Fig. 3 we see that at the bottom ofthe stream, the stream does not become narrower; rather, itswells again.

At the top of the stream, perturbations in the stream causewaves of very small amplitude that cannot be easily ob-served. The undulations grow in time and cause drop forma-tion at the bottom of the stream. This process is favored bysurface tension because the surface area of the liquid tends tobe reduced. It would be desirable to collect all the fluid into

one sphere, corresponding to the smallest surface area. How-ever, the surface tension has to work against inertia, whichopposes fluid motion over long distances. The two effects,surface tension and inertia, strike a balance and smaller dropsare formed.

To understand the growth of undulations in a stream ofwater, we recall the expression for the pressure in a stream ofliquid. We locally approximate the stream of water by a cyl-inder. Due to surface tension, the pressure in the cylinder islarger than in the surrounding air. The pressure differencebetween the pressure in the stream,p, and the surroundingair pressure,p0 , Dp5p2p0 , acting on the surface of thecylinder,S, is equal to the surface tension forceF ~see Fig.4!:

DpS22F cosw50. ~6!

We write S52rl , wherel is the length of the cylinder, andF5g l , whereg is the surface tension, and expressDp as

Dp5gcosw

r. ~7!

Becauser 5R cosw, Dp is given by

Fig. 4. Cross-section of the cylindrical stream of liquid. The pressure dif-ferenceDp acting on the surfaceS of the cylinder is in balance with theforce of surface tensionF.

Fig. 5. Early phase of drop formation:~a! schematic presentation and~b!photograph. The liquid flows from a location with higher pressurep1 into aswelling with lower pressurep2 .

Fig. 6. End phase of drop formation:~a! schematic presentation and~b!photo. The liquid flows from an area of higher pressurep1 into a sphericaldrop with lower pressurep2 .

417 417Am. J. Phys., Vol. 73, No. 5, May 2005 Vladimir Grubelnik and Marko Marhl

Dp5g1

R. ~8!

Equation~8! shows that the pressure in the stream is largerwhere the stream has a smaller radius and vice versa. Ac-cording to Eq.~8!, the liquid flows from an environment ofhigher pressurep1 into the swelling of the stream with lowerpressurep2 ~see Fig. 5!. The driving force,p12p2 , for theflow of liquid into the swelling can be approximated by us-ing Eq. ~8! for both regions 1 and 2~see Fig. 5!:

p12p25gS 1

R12

1

R2D . ~9!

Figure 5 shows that region 2 is not an ideal cylinder, andhence Eq.~9! is just a rough approximation for the driving

force for the flow of liquid into the growing drop. For a moreaccurate estimation of the pressure in the growing drop, bothradii of curvature have to be taken into account. In analogyto the derivation of Eq.~8!, the pressure in the growing dropcan be expressed more accurately by

p22p05gS 1

R2,I1

1

R2,IID , ~10!

Fig. 7. Drop formation showing the dependence on the surface tension ofliquid: ~a! fresh water with surface tension 0.073 N/m~drops form ath5561 cm) and~b! soapy water with surface tension 0.03 N/m~drops format h51562 cm). The experiment was done forr 052.060.2 mm andv0

55.060.5 cm/s.

Fig. 8. Drop formation showing the dependence on the irregularity of astream of water:~a! no external forcing and~b! additional external pertur-bations.

418 418Am. J. Phys., Vol. 73, No. 5, May 2005 Vladimir Grubelnik and Marko Marhl

where R2,I and R2,II are the principal radii of the surfacecurvature.

At the beginning of drop formation, where the streamswells slightly, Eq.~9! is a good approximation for the driv-ing force becauseR2,II@R2,I and Eq.~10! simplifies to Eq.~8!. However, in the later process of drop growth,R2,I in-creases andR2,II decreases. Finally, when the drop is close toits spherical form~see Fig. 6!, R2,I5R2,II5R2 , and the pres-sure in the drop takes the value

p22p052g

R2. ~11!

Equation~11! also can be obtained directly by reasoningsimilar to that leading to Eq.~6!. The difference between thepressure in the drop,p2 , and the surrounding air pressure,p22p0 , acting on the surface of the sphere equals the sur-face tension F5g2pR2 , which gives (p22p0)pR2

2

5g2pR2 , and hence Eq.~11! follows.The driving force for the liquid flowing into the spherical

drop is given by

p12p25gS 1

R12

2

R2D . ~12!

During the growth of a drop,R1 becomes smaller andR2larger. According to Eq.~12!, the driving force for the flowof liquid into the drop becomes even larger. The liquid issucked into the drop, and, finally, the drop is separated fromthe stream.

The dynamics of drop formation depends crucially on thetype of fluid. For a more comprehensive treatment of thedynamics, see Ref. 19. It is shown that water does not pro-duce long necks, but that oil can.16

IV. DISCUSSION

Because of gravity, a stream of liquid becomes narroweras it falls. At the bottom of the stream, it swells again anddrops are formed. Here we discuss the influence of the sur-face tension and external perturbations on the decay of aliquid stream.

According to Eqs.~9! and ~12!, a larger surface tensioncauses a larger driving force for the influx of liquid intodrops, and drop formation is accelerated; hence, the dropsare formed earlier in the stream, that is, closer to the nozzle.The influence of surface tension on drop formation can betested by comparing drop formation in a stream of fresh wa-ter and in a stream of soapy water~see Fig. 7!, which has asmaller surface tension than fresh water. We see that dropsare formed much earlier in a fresh water stream (h5561 cm) than in a soapy water stream (h51562 cm). Bothstreams have the same initial cross-sectionS0 and velocityv0 .

Another important issue is the influence of perturbations

on the decay of a liquid stream, which can be simply dem-onstrated by a slight shaking of the nozzle. If the externalperturbation is larger, measurable swellings in the stream ap-pear earlier, and therefore the swellings also grow earlier intodrops, that is, the drops appear closer to the nozzle~Fig. 8!.A more comprehensive discussion of the influence of pertur-bations on the decay of liquid streams is given in Ref. 8.

More viscous liquids can be studied experimentally. How-ever, for highly viscous liquids, viscous dissipation can bemore important than inertia as the primary resistance againstsurface tension, and, therefore, drop formation is mostly theresult of the balance between surface tension and viscousdissipation. In general, a combination of all these effects,surface tension, inertia, and viscous dissipation, is importantfor drop formation in viscous liquids.

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419 419Am. J. Phys., Vol. 73, No. 5, May 2005 Vladimir Grubelnik and Marko Marhl