an experimental investigation of the convective instability of a jet

Chemical Engineering Science 58 (2003) 2421–2432www.elsevier.com/locate/ces

An experimental investigation of the convective instability of a jet

Anuj Chauhana ;∗, Charles Maldarellib, David S. Rumschitzkib, Demetrios T. Papageorgiouc

aDepartment of Chemical Engineering, University of Florida, Florida, FL 32611, USAbDepartment of Chemical Engineering, The Levich Institute, City College of CUNY, New York, NY 10031, USAcDepartment of Mathematics, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology,

University Heights, Newark, NJ 07102, USA

Received 30 August 2001; received in revised form 26 August 2002; accepted 13 November 2002

Abstract

This paper is an experimental study of the convective instability of a jet. It is well known that a jet issuing forth from a nozzle isunstable due to surface tension forces that cause it to break downstream into drops. We apply a disturbance of a given frequency at thenozzle tip. This applied frequency determines the wavelength and the growth rate of the growing disturbances and, thereby, the drop size.We measure the wavelength and the growth rate by 6tting the entire digitized image of a jet to the functional form suggested by thelinear theory. Thus, it makes use of the entire pro6le instead of the small number of points used in previous studies. Also, in contrastto previous work, we independently measure the jet velocity and the wave speed. At high non-dimensional jet velocity, the experimentalresults for the growth rates and the wave numbers agree with the linear stability theory of an in6nite jet in the absence of gravity. At verylow velocity (low Froude number) gravity is important and the agreement is not good.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Hydrodynamic stability; Jet; Spatial instability; Temporal instability; Satellite drops

1. Introduction

A liquid jet issuing from an ori6ce is unstable to ax-isymmetric interfacial disturbances due to destabilizingcircumferential capillary forces, represented by the in-terfacial tension �. Drop breakup from jets is impor-tant in the technological processes of atomization, inkjet printing, fuel injection, particle sorting and polymer6ber spinning (where it is desirable to arrange for a re-duced growth rate so that the jets can polymerize beforebreakup). Due to these technological applications, theproblem of jet stability has been widely studied, bothexperimentally and theoretically. Rayleigh (Rayleigh,1879) analyzed the temporal stability of this base state byimposing, at t = 0, an interfacial disturbance with Fourierwave numbers k and n in the z and � directions, respectively.He determined the evolution of this initial disturbance toan inviscid jet using normal modes (ei(kz+n�)+s(k;n)t , wherez is along the thread axis and s(k; n) is complex). Rayleigh

∗ Corresponding author. Tel.: +1-352-292-2592; fax: +352-292-9513.E-mail address: [email protected] (A. Chauhan).

established that only axisymmetric (n=0) disturbances withwavelengths (2=k) larger than the undisturbed thread cir-cumference (2a) grow in time (sr(k; n= 0)¿ 0), and thatthere is a maximum in the growth rate at a wave number(kmax) equal to 0:696=a. The disturbances grow without trav-eling, (si (k; n=0)=0). This temporal stability of the staticthread was reconsidered for the case in which the threadBuid is viscous (Chandrasekhar, 1961), and is surroundedby a second immiscible viscous liquid (Tomotika, 1935).Since capillarity drives the instability, the range of unsta-ble wavelengths remains the same when viscous eCects areincluded, but growth rates are reduced and the maximallygrowing waves are shifted to longer wavelengths.In all the above-mentioned references, the disturbances

were modeled as growing in time. In most applications,the jet issues from an ori6ce and breaks downstream, i.e.,disturbances continually imposed at the nozzle tip (eitherintentionally through periodic oscillations, or through tip im-perfections) grow by the destabilizing action of capillarityas they are convected downstream until they cause the jet tobreak up into drops. Keller, Rubinow, and Tu (1973) 6rstnoted this fact. They suggested a base state in which thejet is modeled as a circular (doubly z → ±∞) in6nite jet

0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0009-2509(03)00076-9

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2422 A. Chauhan et al. / Chemical Engineering Science 58 (2003) 2421–2432

moving with uniform velocity upon which interfacial distur-bances with Fourier frequency modes ! are imposed locallyin space (z = 0, for example) and periodically in time. Foran inviscid jet and axisymmetric disturbances, Keller et al.(1973) determined the spatial growth for the harmonic re-sponse (eik(!)z+i!t ; k(!) complex, ki ¡ 0 for spatial growth)as a function of theWeber number (W=�V 2a=�), essentiallythe scaled velocity V squared. � is the Buid density. Kelleret al. demonstrated that for asymptotically high W , the spa-tial and temporal theories coincide for kiV equal to sr andkr equal to !=V . (In this limit, this conversion just amountsto a change from the lab frame to a frame moving with thejet velocity V .) In addition, for moderateW and dimension-less frequencies or Strouhal numbers (!a=V ) between zeroand approximately one (the exact cutoC being a function ofW ), localized periodic disturbances still grow axially, butwith wave lengths and linear growth rates that increasinglydeviate from the temporal theory as W is reduced. For fre-quencies in the unstable range, the drop size and distanceto breakup can be predicted (assuming the linear theory isvalid up to break-up; see the experimental veri6cation dis-cussed below) since the drops which are formed then have asize which scales as 1=kr(!), and the distance downstreamfrom breakup is of order 1=ki(!). Leib and Goldstein (1986)found that below a critical value of W of approximately 3.1(for the inviscid or a slightly smaller value that dependson the Reynolds number for the viscous theory) a new un-stable branch dominates Keller et al.’s convectively unsta-ble solution, and leads to a fast growth in time as well asin space, commonly known as an absolute stability. Otherstudies were performed on the spatial instability by Lin andLian (1989) and others. Leib and Goldstein examined theeCect of Buid viscosity on growth rates and Lin and Lian(1989) included eCects of a surrounding gas phase on thespatial instability.A large body of experiments studying the breakup of jets

into drops have been undertaken, usually with a sinusoidalperiodic disturbance of frequency ! applied at the nozzle tipusing acoustic, electromagnetic, or piezo-electric-inducedpressure vibrations or a vibrating impinging needle. SeeDonnelly and Glaberson (1966)—W ∼ 8:5–297 Goeddeand Yuen (1970)—W ∼ 400, Taub (1976)—W ∼ 360,Pimbley and Lee (1977)—W ∼ 95, Chaudhary andMaxworthy (1980)—V not given, Kowalewski (1996)—W ∼ 360, and the review monograph by Yarin (1993).In all the above studies the distance between two succes-sive peaks determines the wavelength of the disturbances.The (spatial) growth rate was measured by measuringthe breakup length or by measuring the amplitude of thedisturbance at two successive peak/troughs and assumingexponential spatial growth. In addition the waves were as-sumed to convect with the jet velocity and thus that thetemporal theory applies in the moving frame. Other than theexperiment at W ∼ 8:5, all of the others quoted are at highenough W to clearly justify this assumption. As such, thejet velocity was calculated as n� where n is the frequency of

the applied disturbance and � is the measured wavelength.The temporal growth compared well with sr = |ki|V whereV is the jet velocity and ki is the imaginary part of the wavenumber, i.e., the spatial growth rate.Our study diCers from the above in the following re-

gards. First, we digitize the entire jet pro6le and 6t it to thetheoretical form predicted by the linear stability analysis,�nozzle exp(−kiz) cos(krz + !t) for small amplitude distur-bances, to extract the wave number and the growth rate fora given frequency. Since, we make measurements from thedigitized image and utilize the entire pro6le, which corre-sponds to about 250–300 pixels, our method is expected toyield more reliable and robust measurements. Secondly, wemeasure the undisturbed jet radius and since we know theBow rate supplied by the pump, we calculate the jet veloc-ity. We independently measure the traveling velocity of thewaves by tracking a maximum in time and compare the wavevelocity to the jet velocity. Lastly, we compare the experi-mental results with the predictions of the convective theoryand assess when the temporal theory should be adequate.

2. Procedure

We perform all our oil experiments at a room tempera-ture of about 20◦C using FC40 Buorocarbon oil supplied bySigma Aldrich. The density and viscosity of FC40 oil are1:85 g=cm3 and 4:4 cp, respectively and the surface tensionof the oil–air interface is 14 dynes=cm.The experimental setup to study the convective growth of

disturbances in a jet is shown (not to scale) schematicallyin Fig. 1. A Cavro Scienti6c digital pump pushes the Buidthrough a needle to create a jet and a periodic disturbanceis imposed on the jet as it issues out of the needle (innerradius R0 =0:0275 cm) in an assembly shown in Fig. 2. Os-cillations of a piezo crystal introduce periodic perturbationsin the Bow-rate and consequently in the jet velocity at theneedle tip. The magnitude and frequency of the oscillationdepend on the frequency and amplitude of the input signalto the piezo crystal and are in turn controlled by a functiongenerator and an ampli6er, respectively. The assembly pic-tured in Fig. 2 used to superimpose the piezo-generated dis-turbance is clearly not axisymmetric but the needle lengthis more than 100 times the diameter, and this ensures thatthe Buid coming out of the needle as a jet is axisymmetric.A collimated beam of light generated by a lamp and a lensassembly illuminates the jet from behind and a digital cam-era records the images of the jet as it Bows vertically down-wards and breaks into drops. In order to capture the growingdisturbance, we utilize a KODAK Fast Video Camera withspeed up to 12,000 frames per second. The captured framesare stored in the memory of the camera and are later down-loaded to a computer and digitized to detect the jet interface,i.e., r = r(z). We de6ne the interface location as the pixelat which the interpolated intensity is 122.5, which is the av-erage of completely black (0) and completely white (255).

A. Chauhan et al. / Chemical Engineering Science 58 (2003) 2421–2432 2423

ComputerCamera

Jet

Lenses

Pin Hole

Lamp

PumpFunctionGenerator

1000 Hz

Amplifier

PiezoCrystal

Needle

Fig. 1. Experimental setup to create the jet, 6lm it and impose periodic disturbances to the jet before it exits the needle.

Fig. 2. The Plexiglass assembly used to introduce the disturbance to the jet using a piezo-electric crystal.

A linear interpolation of intensity is done between the pixelsto get a more accurate representation of the interface. Thedigitized image is then converted to real dimensions by ap-plying the calibration factors, which are determined a prioriusing a precision bearing.Upon exit from the nozzle the undisturbed jet initially

contracts quickly due to the jet’s adjustment from a no slip toa zero stress boundary condition. Conservation of mass andmomentum suggests (Harmon, 1955) a contraction to

√3=2

of the nozzle diameter R0. Beyond this point, if there wereno gravity, the base state of the jet would be a cylindricalthread of constant and uniform radius a. In the presence ofgravity, as in all of the referenced studies, the jet continuesto thin, but more slowly. In our experiments (see below) thetransition (break in slope) from fast to slow thinning occursat roughly two needle diameters, where the radius is also∼ √

3=2 of the nozzle diameter. We measure the jet radiusat this point without imposing any disturbance and take thisto be the undisturbed base jet radius a. Next, we de6ne thedisturbance to the jet interface as � = r − a and determineit from the digitized image. Since, the acquired image istwo-dimensional, we obtain the location of the interface for

the two diCerent sides of the jet’s projection, correspondingto the two azimuthal angles 0◦ and 180◦. We perform thebest 6t for each angle separately and ensure that the growthrate and wavelength for the two cases is within 5% of eachother. If the diCerence is bigger we reject the image. Asstated earlier, in the linear convective instability regime (ne-glecting gravitational thinning), the disturbances introducedat the needle exit grow as �needle exp(−kiz) cos(krz + !t)where �needle is the amplitude of the periodic disturbance atthe needle exit located at z = 0. In our experiments, it isnot possible to measure �needle from the digitized data be-cause the amplitude of the oscillation at the needle tip isextremely small. Thus we arbitrarily shift the origin andde6ne z = 0 as the point closest to the needle exit in thedigitized image and 6t the interface shape to the function�0 exp(−ki(z + z0)) cos(kr(z + z0)) where �0 is the ampli-tude at some location z = −z0. Thus, there are four 6ttingparameters, kr; ki; �0, and z0. The main aim of this paper isto utilize the procedure described above to obtain the wavenumber, kr and the spatial growth rate, ki, for various fre-quencies of the applied disturbance and at diCerent jet ve-locities. The details of the four-parameter 6t along with the


procedure followed to determine the initial guesses are givenin the appendix. This best 6t procedure in principle workseven if we have less than one full wavelength in the digitizedimage but the accuracy and reliability are lower. Such a sit-uation arises when the growth rate is so large that soon afterthe disturbances are visible, they grow very fast and breakthe jet within one wavelength, but when the linear theorynevertheless holds until close to break-up. In such situationsthis best 6t method is the only reliable way to measure thewavelength and the growth rate. The method to obtain theinitial guesses in such situations is diCerent than the situa-tion in which a few waves are visible in the digitized image.We also note that sometimes the disturbance to the jet is sosmall that it is not visible to the eye or in an analog imagebut it can still be detected in the digitized image. This is an-other advantage of using the high-speed image analysis tostudy jet stability.In the next section, we discuss the results, i.e., kr and ki

for diCerent frequencies and Weber numbers and comparethe experimental results with the predictions of the linearstability theory for a doubly in6nite viscous jet in the absenceof gravity.

3. Results and discussion

As noted above, upon exit from the nozzle the jet thinsquickly due to the change from no-slip at the needle wallto zero traction at the jet-air interface and then more slowlydue to gravity. Fig. 3 is a representative curve for a relativelyslow Bow. The change in jet radius R(z)=R0 with distancez in units of jet diameter from the nozzle tip shows theinitial thinning to a ∼ √

3=2R0 at za ∼ 2 jet diameters andthe subsequent gravitational thinning. We take an averageexperimental value of 0:0245 cm (compared with the value

0 2 4 6 8 10 12 140.7

0.75

0.8

0.85

0.9

0.95

1

z/D

R/R

0

Fig. 3. ECect of gravity on the jet diameter. The solid line is the jet radiusas it exits the needle and travels down. The jet 6rst relaxes to a plugBow in a distance about twice the needle diameter and then thins dueto gravity. The dashed line shows the theoretical prediction for the jetthinning on the basis of a simple 1D model (V =40:5 cm=s, inner needleradius R0 = 0:0275; a = 0:0245 cm; W = 5:3; Re = 41:7; J = 328; D =jet diameter = 2a).

0:0238 cm from the above simplistic theory) for a for thisand subsequent 6gures. In this 6gure, we have adjusted thepump Bow rate Q so that V =Q=(a2) ∼ 40:5 cm=s, givingRe ∼ 41:7, slow enough that the gravitational thinning is no-ticeable over several jet diameters, unlike subsequent caseswith higher Re. The experimental data of Middleman andGavis (Middleman & Gavis, 1961) agree with Harmon’s(1955) prediction (a ∼ √

3=2R0; Va := V (za) = (4=3)× themean velocity V0 at the nozzle tip) which neglects viscousdissipation, above a Reynolds number of 80, because vis-cous eCects are negligible at high Re (Goren & Wronski,1966; Middleman, 1995). In contrast, they showed that atlow Reynolds number, a jet expands as it relaxes to a plugBow pro6le. Since the Reynolds numbers in our Buorocar-bon oil experiments below are over 100, we neglect viscousdissipation.We estimate the slow gravitational jet thinning from

Bernoulli’s equation, which equates the change in kineticenergy to the gravitational acceleration. (Although the cap-illary pressure in the jet is comparable to the gravitationalpotential energy per unit volume, the diCerence in the cap-illary pressure with z is not comparable to the latter diCer-ence.) Using conservation of mass R(z)2V (z) = a2Va gives

R(z)R0

=aR0

R(z)a

=

√32

√VaV (z)

=

√32

(1

1 + 2g(z − za)=V 2a

)1=4

=

√32

(1− g(z − za)=2V 2a + O(g(z − za)=V 2

a )2): (1)

The dashed, nearly linear curve in Fig. 3 is Eq. (1), butwhere the constant term (

√3=2) has been adjusted to 6t the

experiment, and agrees reasonably well.In the 6rst set of experiments, for an undisturbed jet ra-

dius taken as a = 0:0245 cm, the pump Bow is set so thatV = Q=(a2) ∼ 100 cm=s. The average of V , as calcu-lated from the average undisturbed radius of 0:0245 cm, was105:6 cm=s. Since the actual value of a Buctuated ca. 5%,there is an uncertainty in V of this order. As such, we simplyuse V=100 cm=s in the calculation of dimensionless groups.The Reynolds number Re is 103 and the Weber number Wis 32. The critical frequency, i.e., the frequency above whichthe jet is stable is Va=(2a) where Va is the dimensional jetvelocity and a is the jet radius is 650 Hz. To scan the en-tire unstable frequency range, we vary the frequency of theimposed disturbance from 100–600 Hz.In the absence of any imposed disturbance the jet breaks

due to the perturbations arising naturally at the needle tip.In this situation, the magnitude of the disturbance to the jetas it exits the needle is small and consequently, the breakuplength is large. Since there is no speci6c disturbance im-posed, the mode k with the largest spatial growth ki dom-inates and causes the jet breakup. In our experiments, wewish to study the frequency dependence of the wavelength


0 100 200 300 40010

15

20

25

30

35

40

z (z-pixels)

f (x

-pix

els)

Fig. 4. A photograph of the jet (inside) and a digitized image (out-side) on diCerent scales. The frequency of the applied disturbance is600 Hz (V = 100 cm=s, a = 0:0245 cm, W = 32; Re = 103; J = 328).

0 50 100 150 200 2505

10

15

20

25

30

35

z (z-pixels)

f (y

-pix

els)

Fig. 5. A digitized image along with the best theoretical 6t tothe form predicted by the linear spatial instability analysis. Thefrequency of the applied disturbance is 600 Hz (V = 100 cm=s,a = 0:0245 cm; W = 32; Re = 103; J = 328).

and the growth rate. We thus need to provide a disturbanceat the needle tip of the desired frequency and of suPcientlyhigh amplitude so that the imposed frequency dominates andcauses the breakup even though its growth rate is smallerthan the maximally growing wave. However, we also needto ensure that the applied disturbance is suPciently small inamplitude so that the jet does not immediately go into thenon-linear regime.Fig. 4 shows an image of the jet captured by the camera

along with the digitized image. Note that the real image isat a diCerent scale and is shown inside the digitized image.The frequency of the applied signal for the image shown inFig. 4 is 600 Hz. Fig. 5 shows the best 6tted interface shapesfor azimuthal angle 0◦ and 180◦ along with the digitizedimages. As can be seen from the 6gure, there is a very goodmatch between the digitized shape and the best 6tted curve.From the best-6t curve we obtain kr and ki and comparethese with the predictions from the linear stability theory.Similarly, we obtain the wave numbers and the growth ratesby applying disturbances at 300, 400, and 500 Hz.We also repeat this procedure at a frequency of 100 Hz.

However, in this case there is a signi6cant diCerence

between the digitized shape and the best-6tted curve. This isdue to the non-linear interaction of the principal frequencyand the 6rst harmonic of the imposed frequency. At highfrequencies in the 400–600 Hz range, this 6rst harmoniclies in the stable regime, but at a frequency of 100 Hz the6rst harmonic grows much faster than the imposed fre-quency. This harmonic has a frequency that is double thatof the imposed frequency and hence, half the wavelength.Thus, growth of the secondary harmonic results in an ob-servable growth around the minima in the pro6le of theprincipal disturbance resulting in the formation of smallerdrops that are called satellites. This eCect is shown in Fig.6. The white curve in Fig. 6 represents the sum of the inten-sities at all the pixels located at diCerent radial coordinatesbut at the same axial location. We reiterate that the insideof the jet is black—intensity zero and the background whitecolor has an intensity of 255. At a minimum in the jetdisturbance, most of the pixels are white in color and thusthe physical minima in interfacial deBection correspond tointensity maxima in the white curve. In the 6gure, nearthe breakup region, there is clear evidence of the harmonicthat leads to satellite formation. Because of the inBuenceof this harmonic of the imposed frequency, we cannot usethe best-6t method described in the appendix to determinethe wave length and growth rates at low frequencies suchas 100 Hz. In this regime we only rely on all the maximaand calculate wave length as the average distance betweenthem. The growth rates are calculated for each pair ofmaxima by using Eq. (10) in the appendix.As we remarked in the introduction, for high enough We-

ber numbers the spatial theory, when placed in a frame mov-ing with the undisturbed jet velocity V , coincides with thetemporal theory. As derived by Leib and Goldstein (Leib &Goldstein, 1986) and Lin and Lian (Lin & Lian, 1989) thedispersion equation (in our notation) for the spatial analysisis

J (k2 − 1)kl1(k)l1(�)− 4k3�l1(k)(l0(�)− l1(�)

�

)

+2k2(l0(k)− l1(k)

k

)((s+ ik

√W )

√J + 2k2)l1(�)

+ (s+ ik√W )

√J l0(k)l1(�)

×((s+ ik√W )

√J + 2k2) = 0; (2)

where �2 = k2 + (s + ik√W )

√J . Fig. 7 solves this equa-

tion and compare the spatial and temporal solutions for thegrowth rate and wavelength as functions of the applied fre-quency. As they show, for Weber numbers of order 25 orhigher, the theories coincide, but for W close to the criti-cal value Wcrit for the onset of the absolute instability (forour Buid with J = 328, Wcrit ∼ 1:47, corresponding to aRecrit ∼ 22, Leib & Goldstein, 1986), they deviate near themaximum growth rate. Figs. 8 and 9 show the comparisonbetween our experimental measurements and the predictionsof the spatial linear stability theory for W = 32; Re = 103.


Fig. 6. A picture of the jet along with a plot (white) showing the mean intensity at all axial locations. A minimum in the plot corresponds to a maximumin the jet pro6le. The plot shows the satellite formation due to the harmonic of the applied frequency of 100 Hz (V = 100 cm=s, a = 0:0245 cm,W = 32; Re = 103; J = 328).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

1.2

1.4

2.14

10 25

100

ω a/V

kra

J= 328

Temporal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

2.1

410

25100

Temporal

ωa/V

-kia

W

J=328

(a)

(b)

Fig. 7. (a) Spatial and temporal solutions of wavenumber versus frequencyfor a single jet for J =328. The Weber numbers for the spatial solutionsare denoted on each curve. The large Weber number spatial solutionscoincide with the temporal solution. (b) Spatial and temporal solutionsof growth rates versus frequency for a single jet for J =328. The Webernumbers for the spatial solutions are denoted on each curve. The spatialgrowth rates are multiplied by

√W to show that the large Weber number

spatial solutions coincide with the temporal solution.

SinceW ¿ 25 this corresponds to the highW limit. In theseand subsequent plots, each data point is a mean of 20 im-ages obtained from typically 6ve experimental runs and the

0 100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FREQUENCY (Hz)

kr*a

Fig. 8. Comparison of the experimental wave number versus frequencyresults with the predictions of the spatial linear stability analysis. Theexperimental results are the mean of 20 images and the error bars inthis and subsequent 6gures mark the standard deviation in the data(V = 100 cm=s; a = 0:0245 cm; W = 32; Re = 103; J = 328).

ki*a

ki*a

0 100 200 300 400 500 600 700-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

FREQUENCY (Hz)FREQUENCY (Hz)

Fig. 9. Comparison of the experimental spatial growth rate versus fre-quency results with the predictions of the spatial linear stability analysis(V = 100 cm=s; a = 0:0245 cm; W = 32; Re = 103; J = 328).


error bars represent the standard deviation in the experimen-tal data. At lower frequencies the non-linear eCects are morepronounced, and this led to the highest standard deviationin the measured values of the spatial growth rate at 100 Hz(Fig. 9). Even at higher frequencies, our results are aCectedby the non-linear aCects but the good agreement betweenthe interface shape predicted by the linear theory, and theexperimental measurements (Fig. 4) suggest that these ef-fects are minimal in the region where we measure the spa-tial growth rates. The agreements with theory in Figs. 8 and9 are adequate. Our results are about 20% lower than thoseof Donnelly and Glaberson’s (1966) Fig. 5 and Goedde andYuen’s (1970) Fig. 7. Although these authors worked in thehighW regime, their J values were much higher (order 105)and thus in the inviscid limit of Chandrasekhar (1961). Theviscous theory naturally gives growth retarded relative tothe inviscid case. As noted by other authors (see Introduc-tion), the agreement between theory and experiment here israther remarkable because we are clearly making measure-ments in non-linear regime and comparing the results withthe linear theory.The experiments described above are conducted in the

high-velocity regime and in this regime linear stability anal-ysis predicts that the disturbances are just convected withthe jet velocity. To verify this, we measure the velocityof the growing disturbance by tracking a particular maxi-mum and measuring the distance traveled by this crest ina given time (see Fig. 10). The jet velocity as mentionedabove is obtained simply by dividing the pump Bow rate by

Fig. 10. Images of the jet at a 6xed spatial position and at various instances in time. The slope of the solid lines represents the velocity of the waves(V = 100 cm=s; a = 0:0245 cm; W = 32; Re = 103; J = 328).

the jet cross-sectional area. The measured wave velocity atdiCerent frequencies is compared with the jet velocity inFig. 11. Again, there is good agreement between the wavevelocity and the jet velocity.The capillary growth of disturbances that grow convec-

tively is periodic in time at a 6xed point in space. To verifythis periodicity, Fig. 12 takes the average of 20 frames at thesame spatial location separated by 1 time period (1=300 sfor 300 Hz). If the disturbances are indeed periodic, eachframe should be identical and the average frame should lookvery similar to each of the individual frames. In particularthe wavelength of the superimposed frame will match thoseof the individual frames and should be easily discernable.This is indeed the case in Fig. 12.In most industrial applications involving jet break-up, it

is important to control the size of the drops breaking fromthe jet. We use the frequency of the imposed disturbanceto accomplish this. One might expect that each wavelengthof the disturbance should eventually produce one drop, andthe volume of the drop equals the Buid volume containedin one wavelength, i.e., a2�. Fig. 13 compares the mea-sured volume of the drop with the volume contained in onewavelength at diCerent frequencies. As seen in the 6gure,the comparison is reasonable at high frequencies but breaksdown at lower frequencies. The reason for this discrep-ancy at lower frequencies is the non-linear interaction ofthe 6rst harmonic of the imposed frequency that leads tothe formation of satellite drops. The non-linear eCects areimportant because we are making measurements very close


0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

V/V

jet

V/V

jet

FREQUENCY (Hz)FREQUENCY (Hz)

Fig. 11. A comparison of jet velocity and wave velocity at dif-ferent applied frequencies. The ratio is close to one at all fre-quencies implying the waves are convecting with the jet velocity(V = 100 cm=s; a = 0:0245 cm; W = 32; Re = 103; J = 328).

to the breakup. However, at frequencies close to and largerthan half the critical frequency, the next harmonic is stable,and thus the non-linear eCects are not dominant. To takeinto account the satellite formation we measure a decreas-ing volume of the satellites as a function of increasing theapplied frequency (Fig. 14). Rutland and Jameson (1971)also observed a decrease in satellite-drop size with increas-ing frequency. The vertical line in Fig. 14 marks the end ofthe region in which satellite formation is expected based ona non-linear theory by Chaudhary and Redekopp. This the-ory is consistent with our experiments because the satellitevolume to the right of the vertical line is small. If a satel-lite forms then one would expect the sum of the volumesof a parent drop and a satellite drop to equal the Buid vol-ume contained in one wavelength of disturbance. To ver-ify this, we add the satellite volumes to the drop volumesand the results are shown in Fig. 15. The comparison of thecombined volume with the volume contained in one wave-length is reasonable at all frequencies. This con6rms thateach wavelength gives rise to one drop at high frequenciesand a drop and a satellite at lower frequencies. We can usethe experimental system described above to produce a largevariety of drop sizes at a broad range of rates. Let us con-sider a case when we need to produce N drops of diameterD per unit time. In industrial applications one wants to avoidsatellite formation so we want to use a frequency close tothe critical frequency so that the next harmonic is stable. Inaddition, a higher frequency also implies a higher produc-tion rate of drops. However, we do not want to choose afrequency very close to the critical frequency. With such achoice the growth rate of the imposed disturbance would betoo low and hence we would need to impose a much higherinitial amplitude to make sure that the imposed disturbance,and not the fastest growing mode, causes the break up. Basedon the above argument, let us choose a frequency of 0:9ncwhere nc = Va=(2a) is the critical frequency. In order tohave a high production rate one would like to work at high

velocities. We know that in the high Weber number limit,the disturbance simply convects with the jet velocity. Thus,� = Va=n= Va=(0:9nc) = Va=(0:9Va=(2a)) = (2a)=0:9. Inthe absence of satellites each wavelength will break into onedrop of volume (a2�) = a2(2a)=0:9 = 22a3=0:9. Thevolume of drop is also (D3=6) where D is the drop diame-ter. Equating the two volumes gives 22a3=0:9=(D3=6) ⇒a = (0:9=12)1=3D = 0:288D. Since each wavelength pro-duces one drop the applied frequency is same as the numberof drops that need to be produced per unit time. Thus, n=N .The jet velocity is n�=N (2a)=0:9=N (2)(0:288D)=0:9=1:81(ND) and the volumetric Bow rate is, N ((D3=6). Oneprovides a fairly uniform size distribution of particles andcan be very useful in systems in which one needs to producedrops of a given size. The system can change the productionrate simply by changing the frequency and the Bow rate bythe same ratio. There is also some Bexibility in the drop sizebecause the satellite formation is small for kr ¡ 0:7 (Chaud-hary and Redekopp) which corresponds to n¿ 0:85nc. Sothere is a possibility of almost 10% change in the drop sizewithout satellite formation by reducing the frequency. Onedownside of this setup is the fact that the size of the nozzlescales with the drop size. Thus this system will not be veryuseful for producing extremely small drops because then thenozzle will need to be extremely small and may clog. Insuch situations a system based on much higher frequency ismore useful (Tsai, Childs, & Luu, 1996).As discussed above, at high Weber numbers the tempo-

ral and spatial theories agree, because the jet velocity is thesame as the wave velocity, and our experimental results arein close agreement with both. Also, the results show thatthe temporal analysis can be used to analyze the stabilityproblem at high Weber numbers. At intermediate Webernumbers linear stability theory predicts that the waves im-posed at the needle tip exhibit dispersion relative to the jet asthey grow downstream. The predictions of spatial theory be-gin to diCer from the temporal theory on reducing the Webernumber to close to the critical W for the absolute instabil-ity (order 2). We now examine a case of W ∼ 5:3 with thesame Buid (J = 328) and V ∼ 40:5 cm=s. Our experimentsshow that at Weber numbers below about 6, the changes inthe jet radius due to gravity become important; neglectingthis eCect leads to erroneous predictions that do not matchthe experiments. Fig. 16 shows the comparison of the the-oretical wave number and the experimental wave numberas a function of frequency. As can be seen the comparisonis not good. The same holds true for the comparison of thegrowth rates in Fig. 17. Let us investigate if this disagree-ment between the experimental results and the theoreticalpredictions is due to the neglect of gravity in the linear sta-bility analysis. At high velocities the eCect of gravity is notsigni6cant. Let l be the length of the jet. Gravitational eCectsbecome signi6cant when Froude number (Fr = V 2=gl), theratio of inertia to gravity, is close to one. We note that thetotal breakup length of the jet is the appropriate length scalein the de6nition of the Froude number because the stability


AVERAGE OF 20 IMAGES SEPARATED BY 1 PERIOD (1/300s)

Fig. 12. Superposition of 20 frames separated by 1 period at a frequency of 300 Hz. The average frame looks similar to an individual frame provingperiodicity of disturbances at a given spatial location (V = 100 cm=s; a = 0:0245 cm; W = 32; Re = 103; J = 328).

200 300 400 500 600 7000.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

FREQUENCY (Hz)

DR

OP

VO

LUM

E (

cu m

m)

Fig. 13. The solid line is the plot of the volume of Buid con-tained in one wavelength of Buid when the imposed disturbanceis negligibly small and the experimental data represents the vol-ume of Buid contained in one drop formed upon the jet break up(V = 100 cm=s; a = 0:0245 cm; W = 32; Re = 103; J = 328).

200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

FREQUENCY (Hz)

SA

TE

LL

ITE

VO

LU

ME

(cu m

m)

Fig. 14. Volume of satellite drops as a function of the fre-quency of the applied disturbance. The solid vertical line marksthe edge of the region in which satellite formation is predictedby a non-linear stability theory of Chaudhary and Redekopp (1980)(V = 100 cm=s; a = 0:0245 cm; W = 32; Re = 103; J = 328).

200 300 400 500 600 7000.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

FREQUENCY (Hz)

(SA

TE

LLIT

E+

DR

OP

) V

olum

e (c

u m

m)

Fig. 15. The solid line is the plot of the volume of Buid containedin one wavelength of Buid when the imposed disturbance is negligi-bly small. The experimental data represent the sum of the volume ofBuid contained in one drop and one satellite formed upon jet break up(V = 100 cm=s; a = 0:0245 cm; W = 32; Re = 103; J = 328).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

kra

ωa/Va/V

Fig. 16. Comparison of the experimental wave number versus fre-quency results with the predictions of the spatial linear stability anal-ysis (solid) and the temporal theory (dashed) at the lower velocity(V = 40:5 cm=s; a = 0:0245 cm; W = 5:3; Re = 41:7; J = 328).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.35

-0.3

-

-

0.25

0.2

-0.15

-0.1

-0.05

0

Temporal

InviscidTemporal

kia

ωa/V

Fig. 17. Comparison of the experimental spatial growth rate versus fre-quency results with the predictions of the spatial linear stability analysisand the viscous and the inviscid temporal theories at the lower velocity(V = 40:5 cm=s; a = 0:0245 cm; W = 5:3; Re = 41:7; J = 328).

analysis requires the base velocity of the entire jet to be thesame. As noted the Weber number is 5.3, the Reynolds num-ber (Re=�Va=#=

√�aV 2=� �a�=#2=

√WJ ) becomes 41.7,

the breakup length is about 10 times the diameter, and theFroude number is 3.4. Eq. (1) shows that the reduction in jetradius is larger at smaller velocities. This explains why ourresults at the lower jet velocity of 40:5 cm=s do not matchthe predictions of the linear stability theory that neglects theeCect of gravity.To explore the lower W , high Fr regime, we reduce the

Weber number at high velocity by choosing water, a Buidwith a high interfacial tension with air and with a low den-sity. Water–air surface tension is approximately 6ve timesthe surface tension of FC40 oil–air and water’s density isapproximately half that of the oil. This reduces the Webernumber by a factor of 10 without any reduction in velocity.For a water jet velocity of 100 cm=s,W =3:4; J=17; 640

and Re = 245. For J = 17; 640Wcrit ∼ 2:45, correspond-ing to Recrit ∼ 210 (Leib & Goldstein, 1986). In theseexperiments, the jet always breaks at the wavelength cor-responding to the fastest growing mode irrespective of thefrequency imposed by the piezo crystal. We ascribe this tothe presence of air bubbles in the hydrophobic Plexiglaspiece where the base Bow from the pump is modulated bythe disturbance from the piezo-electric crystal. The applieddisturbance compresses the bubbles and is thereby com-pletely damped. Thus, in the experiments with water, weonly compare the fastest growing mode with the theoreticalanalysis. The experimental wave number kr and growth ratesare 0:58± 0:02 and −0:35± 0:043 and the theoretical pre-dictions from the solution of the dispersion equation givenabove for the spatial analysis for kra, −kia√W are 0.79and −0:407, respectively. The viscous temporal predictionsfor this experiment are krmaxa=0:69 and smax =−0:338 and

the inviscid temporal predictions are krmaxa = 0:70 andsmax = −:343. Apparently, except at Wcrit the diCerencebetween the temporal and spatial predictions for the growthrate lie within the error of measurement. This shows that thetemporal theory is accurate enough in most circumstancesand can be used instead of the more complicated spatialtheory unless the Weber number is very close to or belowthe critical W for the absolute instability.

4. Conclusions

A jet is unstable due to capillarity. As the jet issues from anozzle, it picks up disturbances, either speci6cally imposedor simply due to noise, and these disturbances grow downstream. The growth of these disturbances is accurately pre-dicted by linear stability theory. We independently measurethe wave speed and the jet velocity. We also use the entirejet pro6le to obtain the wave length and the growth rates ofthese disturbances by 6tting the digitized jet pro6le to thetheoretical form given by the linear stability analysis. The6t is very good, which is remarkable because our experi-ments are clearly in the non-linear regime. Fitting the entirejet pro6le would appear to be more robust than either usingjust the pro6le’s extrema or the breakup length to determinethe growth rate.Growing disturbances cause the jet to break into drops and

the diameter of the drops is proportional to the wavelengthof the growing disturbance. In certain situations these bigdrops are separated by smaller satellite drops that Chaud-hary and Redekopp’s non-linear stability analysis predicts.Satellite formation takes place at frequencies less thanapproximately 0.7 times the critical frequency because athigher frequencies the harmonics of the applied frequencyare stable. Our experimental results are consistent with these6ndings. In most industrial applications, satellites are notdesirable. Thus, if one wants to use the setup described inthis paper to generate uniform size drops, the imposed dis-turbances must be of frequencies higher than 0.7 of the crit-ical frequency. Controlling the frequency of the disturbanceat the nozzle tip controls the size of the drops produced onbreakup. This system can be used to produce a continuoussupply of uniform size drops, the production rate can bewidely varied and the size can be varied on line by changingthe frequency within a range of approximately 10%.The eCect of gravity on a jet issuing vertically downward

from a nozzle is to accelerate it, which, by continuity, causesthe jet to thin. The shape of the interface can be modeledaccurately by considering a uniform acceleration of a thin-ning jet in the vertical direction. The jet is in Poiseuille Bowinside the needle and quickly relaxes (over a distance of ap-proximately two needle diameters for Re=62) to plug Bowas it exits. Over this distance the jet’s radius decreases toapproximately 86% of the needle diameter and its velocityincreases by approximately 40%. These numbers in generaldepend on the jet’s Reynolds number.


The spatial analysis of the jet yields three diCerent regimesin Weber number. At very high Weber number, disturbancesintroduced at the nozzle convect with the jet velocity andgrow downstream without dispersion. Thus, by shifting to areference frame moving with the jet velocity, the predictionsof the spatial theory match those of the temporal theory,both without gravity. On lowering the jet velocity, the spatialanalysis predicts that the wave speed deviates from the jetvelocity (but remains positive-downstream) and the growthrate can diCer from the predictions of the temporal theory. Asit turns out, only for Weber numbers very close to the criti-cal value for the emergence of the absolute instability is thisdistinction predicted to be larger than the experimental error.In this regime, however, gravity is important, and the con-sequent jet thinning occurs over length scales short enoughto put the validity of the uniform undisturbed jet radius the-ory in question and to make imprecise any critical Webernumber for absolute instability. Fortunately since most ap-plications require high velocities to maximize production,the simpler gravity-free temporal analysis is usually valid.

Appendix A.

The function is to be 6t to the experimental data at discretevalues of z = zi; i = 1:N . The parameters kr; ki; �0, and z0are the chosen by the best 6t as the values that minimize theobjective function: F(kr; ki; �0; z0) =

∑i (&

∗(zi)− &i)2.

F =∑i

&∗2(zi)− 2&∗(zi)&i + &2i

=∑i

&20 exp(−2ki(zi + z0)) cos2(kr(zi + z0))

− 2&i&0 exp(−ki(zi + z0)) cos(kr(zi + z0)) + &2i : (A.1)

At the minima of F(kr; ki; �0; z0),

@F@kr

= 0;@F@ki

= 0;@F@&0

= 0;@F@z0

= 0; (A.2)

where

@F@&0

=∑i

2&0 exp(−2ki(zi + z0)) cos2(kr(zi + z0))

− 2&i exp(−ki(zi + z0)) cos(kr(zi + z0)) = 0: (A.3)

Thus,

&0(kr; ki; z0)

=∑

i &i exp(−ki(zi + z0)) cos(kr(zi + z0))∑i exp(−2ki(zi + z0)) cos2(kr(zi + z0))

; (A.4)

@F@kr

=∑i

− &20zi exp(−2ki(zi + z0)) sin(2kr(zi + z0))

+ 2&i&0zi exp(−ki(zi + z0)) sin(kr(zi + z0))

= 0; (A.5)

@F@ki

=∑i

− 2&20zi exp(−2ki(zi + z0)) cos2(kr(zi + z0))

+ 2&i&0zi exp(−ki(zi + z0)) cos(kr(zi + z0))

= 0; (A.6)

@F@z0

=∑i

− 2&20ki exp(−2ki(zi + z0)) cos2(kr(zi + z0))

− &20kr exp(−2ki(zi + z0)) sin(2kr(zi + z0))

+ 2&i&0ki exp(−ki(zi + z0)) cos(kr(zi + z0))

+ 2&i&0kr exp(−ki(zi + z0)) sin(kr(zi + z0)): (A.7)

We solve Eqs. (A.3)–(A.7) simultaneously by NewtonRaphson to give the optimum values of (kr; ki; �0; z0). It isimportant to get a good an initial guess for (kr; ki; z0) forfast convergence. The initial guess for �0 is found by usingEq. (A.4). We adopt the following procedure to obtain theinitial guesses for the other variables:

A.1. Initial guess

It has been shown that at large jet velocities, the distur-bance introduced at the nozzle tip convect with the jet andgrows axially. Thus, the jet velocity Va is same as the wavevelocity. The wave velocity is &� where & is the frequency ofthe disturbance, � is the wavelength, &=!=2 and �=2=kr .Since Va=Q=(a2) we choose the initial guess kr=a2!=Q.If the digitized data has at least 2–3 peaks, the initial guessfor ki follows:

&∗(zi) = &0 exp(−ki(zi + z0)) cos(kr(zi + z0)); (A.8)

&∗(zi + 2=kr) = &0 exp(−ki(zi + 2=kr + z0))

× cos(kr(z + z0) + 2): (A.9)

Dividing (A.9) by (A.8),

&∗(zi + 2=kr)&∗(zi)

= exp(−ki(2=kr)) ⇒ ki

=− kr2

log[&∗(zi + 2=kr)

&∗(zi)

]: (A.10)

For the initial guess, we assume &∗(zi)=&i. The initial guessfor ki is the mean of all the ki’s calculated by Eq. (A.10).If the digitized data contains less than one cycle then thismethod cannot be used. The theoretical solution to the spa-tial instability yields kr and ki as a function of the applied


frequency. The theoretical solution is used as the initial guessfor ki in such situations.The initial guess for z0 is obtained as follows:

&∗(zi) = &0 exp(−ki(zi + z0)) cos(kr(zi + z0)); (A.11)

&∗(zj) = &0 exp(−ki(zj + z0)) cos(kr(zj + z0)): (A.12)

Dividing Eq. (A.12) by Eq. (A.11)&∗(zj)&∗(zi)

= exp(−ki(zj − zi))cos(kr(zj + z0)cos(kr(zi + z0)

: (A.13)

Rearranging Eq. (A.13) and assuming &∗(zi) = &i we get,

z0(zi; zj) =1kr

tan−1

×{&i cos(krzj) exp(krzi)− &j cos(krzi) exp(krzj)&i sin(krzj) exp(krzi)− &j sin(krzi) exp(krzj)

}:

(A.14)

Thus we get a value of z0 for each pair of points, i.e.,i; j = 1; : : : ; N : i¿ j. Using the repeated median theorem,the optimum value for z0 is given by

z0 =median(median(z0(zi; zj; i = 1; : : : ; N ; i = j));

j = 1; : : : ; N ): (A.15)

Using these initial guesses we solve Eqs. (A.3), (A.5)–(A.7)to get the optimum values of (kr; ki; �0; z0).

References

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Chaudhary, K. C., & Redekopp, L. G. (1980). Nonlinear capillaryinstability of a liquid jet. Part 1. Theory. Journal of Fluid Mechanics,96(2), 257–274.

Donnelly, R. J., & Glaberson, W. (1966). Experiment on capillaryinstability of a liquid jet. Proceedings of the Royal Society of LondonA, 290, 547–556.

Goedde, E. F., & Yuen, M. C. (1970). Experiments on liquid jet instability.Journal of Fluid Mechanics, 33, 151.

Goren, S. L., & Wronski, S. (1966). The shape of low-speed capillaryjets of Newtonian liquids. Journal of Fluid Mechanics, 25, 185.

Harmon, D. B. (1955). Drop sizes from low speed jets. Journal of theFranklin Institute, 259, 519.

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an experimental investigation of the convective instability of a jet

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