Experimental investigation of water jets under gravity
aa r X i v : . [ phy s i c s . f l u - dyn ] O c t Experimental investigation of water jets under gravity
Wellstandfree K. Bani, and Mangal C. Mahato ∗ Department of Physics, North-Eastern Hill University, Shillong-793022, India (Dated: October 23, 2018)
Abstract
The Plateau-Rayleigh theory essentially explains the breakup of liquid jets as due to growingperturbations along the length of the jet. The essential idea is supported by several experimentscarried out in the past. Recently, the existence of a feedback mechanism in the form of recoilcapillary waves was proposed to enhance the effect of the perturbations. We experimentally verifythe existence of such recoil capillary waves. Using our experimental setup we further show that thewavy nature of the jet surface appears almost right after the emergence of the jet from the nozzleirrespective of the recoil capillary wave feedback. Moreover, our experimental results indicateexistence of a sharp boundary, along the length of the continuous jet, beyond which gravitationaleffect dominates over the surface tension.
PACS numbers: 47.60.-i, 47.20.Dr, 47.35.Pq ∗ Electronic address: [email protected] . INTRODUCTION There have been many attempts theoretically as well as experimentally to understandthe physical mechanism behind the breaking up of continuous jets into drops. However, thestudy of this macroscopic phenomenon continues to be of current interest[1–4].The problem of instability of liquid jets was investigated by Plateau and then by Rayleighand developed a theory which came to be known as Plateau-Rayleigh theory. Rayleigh[5, 6],based on surface energy considerations of inviscid liquids, showed that perturbations ofwavelengths λ larger than π times the jet diameter d grow rapidly with time. However,it is the fastest growing perturbation ( λ ≈ . × d ) that ultimately makes the jet col-umn unstable against formation of droplets. Chandrasekhar[7] later extended the theoryto viscous liquids. Many experimental investigations have been conducted to examine thevalidity of Plateau-Rayleigh theory. The experiment of Goedde and Yuen, for example,applied external perturbations to study the length of the liquid jet before it breaks up[8].In some recent works, Umemura and co-workers[9–12] emphasized the idea that soonafter the jet breaks up the new tip of the remaining column contracts to make its shaperound once again to minimize its surface energy. The tip contraction (recoil) gives rise toupstream propagating capillary waves which upon reflection at the mouth of the nozzle movedownstream with Doppler modified wavelengths. These feedback perturbations superposewith the preexisting perturbations and move down along the jet as combined perturbations.Some of these combined perturbations with the right wavelength cause the liquid column tobreakup producing another contraction of the tip of the column and so on. II. EXPERIMENTAL SET UP AND EXPERIMENT
We set up and conduct an experiment to verify the existence and effect of the recoilcapillary waves on the length of continuous water jet. We achieve this by damping the recoilcapillary waves by bringing the jet in contact with a liquid surface beneath it. Moreover,when the continuous water jet smoothly merges into the water it creates ripples on thewater surface in the beaker. Surface waves are observed using photographic methods andinfer about the surface shape profile of the jet all along its length.Our experimental set up is similar in essentials to that of Goedde and Yuen[8]. The new2
IG. 1: A sketch of the experimental set up. and important addition is the intervening water containing beaker, Fig. 1. The details aregiven in Ref.[13]. The transparent rectangular beaker with a level outlet on one of its verticalsides is placed vertically below the nozzle so that the water jet falls directly on (or smoothlymerges into) the water kept in the beaker. The water level in the beaker is maintained fixedby letting the excess water flow out through the level outlet. The beaker is placed on ahorizontal platform fitted to the vertical stand of a travelling microscope (vernier scale leastcount = 0.001 cm) so that the beaker can be smoothly moved vertically and its positionmeasured. A vertical-height adjustable laser-pointer-and-detector arrangement is also fittedto the platform so that the horizontal laser beam is incident normally on the vertical surfaceof the beaker and passes through the path of the water jet and then through the oppositesurface of the beaker before it is collected by the detector. A digital counter[14] with a clockis connected to the detector to count the number of discontinuities in the water jet over aperiod of time.Distilled water is issued vertically downward through a long glass nozzle (of length largerthan about ten times its internal diameter d ) in to the water in the beaker. The waterflow rate is measured manually by collecting the jet water on a measuring cylinder for twominutes and calculating the mean value. As long as the jet remains continuous, at the levelof the laser beam, the detector remains quiescent. However, a discontinuity in the jet after3reakup allows the laser beam to pass through unobstructed and detected as a water dropcount. We call the vertical distance between the mouth of the nozzle and the position ofthe laser beam as the jet-length, l . The vertical distance between the nozzle exit and theposition of breakup of the jet, as detected by the laser-beam-counter, gives the breakuplength, l = l B .Initially, the laser beam is made to face the continuous jet by moving the platform upcloser to the nozzle, l ≈
0, and then the platform is gradually lowered in small steps so that l increases. For each value of l , the number of counts is recorded for two minutes each forseveral times and their average calculated to obtain the mean drop-count rate. Naturally, thecount rate begins from zero (at a threshold value of l ) and then gradually keeps increasingas l is increased in small steps. The jet-length l at the very threshold point is termed hereas the first breakup length l F B of the jet. The process is continued (by gradually increasing l ) till the count rate reaches a saturation value.Throughout the above process the water flow rate is kept fixed. The same process is thenrepeated for several values of flow rates. Note that after each change of flow rate, the flowand the jet are allowed to become steady before the measurement process is begun. Thesame experiment is repeated for nozzles of various internal diameters d .For our purpose, we perform two distinct sets of experiments. In the first set (set 1), byadjusting the height of the laser beam arrangement, we let the laser beam pass just about0.2 cm above the water surface on the beaker. In the other set (set 2) the beam is kept ata height of about 1.5 cm above the water surface. Note that for the same position of thebeaker, l = l for the first set is larger by 1.3 cm than l = l for the second set of experiments.Crucially, as explained below, the first breakup lengths l F B need not be the same for boththe sets.Consider a situation wherein the jet begins to break up just about 2 mm above the watersurface on the beaker. In the first set of experiments the counts just begin, that is, l = l F B .However, if the moving tip of the remaining continuous jet touches the water surface beforeit gets the chance to recoil, the recoil capillary wave will get damped. On the other hand,consider a situation wherein the jet begins to breakup at about 1.5 cm above the watersurface. This is the threshold point for the second set of experments, that is l = l F B . Inthis case the tip of the remaining jet will have ample opportunity to recoil before it touchesthe water surface and hence recoil capillary waves will propagate up the jet undamped. The4ame will be the case even for the first set of experiments if the breakup were to take placeat a somewhat larger height than 2 mm. Therefore, if the effect of recoil capillary waves onthe jet breakup length is to be a reality, l F B for the two sets of experiment must be differentbut the mean values of l B should be the same in both the sets of experiments.Next, we observed the effect of water jet merging smoothly into the water in the beakerwith the help of an ordinary (Nikon D5300) camera. We call the vertical distance betweenthe mouth of the nozzle and the point at which the continuous jet touches the water surfaceagain as jet-length but denote by the upper case L . We have taken photographs of the wavescreated on the water surface keeping the water surface at various positions (values of L )with respect to the stationary nozzle. From a submerged position of the nozzle, the beakerarrangement was gradually lowered in stages and photographs taken. The photographsat various L values show the nature of surface waves travelling towards the walls of thebeaker. We measured the wavelengths of the waves (that is the mean separation betweenthe successive crests of the waves) using the digitally stored photographs. III. EXPERIMENTAL RESULTS
All the measurements are done at the temperature of (25 ± .5) ◦ C and at relative humidityof (80 ± ρ w udµ ) and the Webernumber We (= ρ w u σ d ) we have used the tabulated values of surface tension σ = 72 × − Nm − , coefficient of dynamic viscosity µ = 8 . × − kgm − s − and density ρ w = 997 . − of water. The issuing jet speed u is calculated as the ratio of the flow rate and theinner area of the nozzle exit.Figure 2 shows the average number of counts (drops) per second, for a water flow rate of35.0 cc/min and the nozzle inner diameter d = 0 .
78 mm, as a function of jet-length l . Thecounts range from zero to a saturation value and l B are essentially distributed over a rangeof values due to the absence of any fixed external perturbation and presence of unavoidablenoise in the laboratory as remarked by Donnelly et al [15]. The mean break-up length isthus calculated using the distribution of breakup lengths l B and it is plotted in Fig. 3.The magnified picture of the graph for low values of count rates is shown in the inset ofFig. 2. The inset clearly shows that l F B as measured in the first set of experiments is largerby about 5 mm compared to l F B measured in the second set of experiments. Recall that5 A ve r ag e c oun t s / s ec l (cm) Set 1Set 2 l FB (Set 1)l FB (Set 2) FIG. 2: Average count rate (s − ) as a function of l at the flow rate of 35.0 cc/min and d = 0 .
78 mmfor the two sets of experiments. The magnified graph (inset) show the first (jet) breakup points inthe two sets of experiments. A ve r ag e c oun t s / s ec ρ l (cm)Set 1Set 2 ρ Set 1 ρ Set 2
FIG. 3: Average count rate (s − ) and probability distribution as a function of l at the flow rate of35.0 cc/min and d = 0 .
78 mm for the two sets of experiments. The distribution, ρ , of breakup lengthis calculated as the derivative of the count rate with respect to l and the normalized distribution ρ n = ρ /100.
30 40 50 60 70 80 90 100 110 2 2.5 3 3.5 4 M e an l B / d , l FB / d √ We Set 1Set 2Set 1Set 2
FIG. 4: The first breakup length l F B /d (lower set of two curves) and the mean breakup length l B /d (upper set of two curves) as a function of √ W e for d = 0 .
78 mm.FIG. 5: Photograph of the water surface in the beaker for d = 0 .
95 mm and water flow rate of 40.0cc/min. When the jet-length a) L = 0 .
168 cm; b) L = 0 .
568 cm; c) L = 3 .
668 cm; and d) L = 4 . λ s ) on the water surface just appear at a) then decrease with L b)and c) and then increase till it reaches d) (i.e., just before the jet breaks into droplet). IG. 6: Photograph of the water surface in the beaker when the jet-length L = 0 .
942 cm for d = 0 .
95 mm and water flow rate of 50.0 cc/min. in the first case the recoil capillary wave is damped whereas in the latter case it propagatesfreely up the jet length. The delay in the process of first breakup of the jet in the first setof experiments indicates that the effect of recoil capillary waves do exist. Or, equivalently,it shows that the tension of the water surface drags the jet down by about 5 mm before itallows the contact between them to breakup. Obviously, the effect of recoil capillary waveis small and its mere absence cannot delay the breakup indefinitely.In Fig. 4 the mean l B and the l F B for the two sets of experiment are plotted as a functionof water flow rate (or, equivalently, as a function of √ W e ) for a nozzle of inner diameter 0.78mm. The difference between l F B for the two sets persists for all flow rates. The experimentwas repeated for various other nozzles with internal diameters, d= 0.69 mm,0.72 mm, 0.82mm, 0.95 mm, 1.04 mm, 1.14 mm, 1.26 mm, 1.34 mm and 1.54 mm. In all cases the resultsare consistently similar to Fig. 4 and we arrive at the same conclusion about the existenceof recoil capillary waves.As mentioned earlier, Savart’s pioneering experiment together with the Plateau-Rayleightheory stimulated further investigations on surface profile of the jet and its breakup, forexample[15, 16]. We capture the waves produced on the surface of the water in the beakerphotographically as the continuous water jet merges into the water body. Figure 5 shows a8
IG. 7: Photograph of the water surface in the beaker when the jet-length L = 1 .
295 cm for d = 1 .
26 mm and water flow rate of 72.5 cc/min. sequence of photographs at different L = 0 . , . , . .
268 cms, respectively,for nozzle diameter d = 0 .
95 mm and a flow rate of
F R =40 cc/min. Similar photpgraphscan be obtained for other nozzle diameters ( d ) flow rates F R at different jet length L aswell. Figures 6-8 show representative photographs for d = 0 .
95 mm and
F R = 50 cc/minat L = 0 .
942 cm, d = 1 .
26 mm,
F R = 72 . L = 1 .
295 cm, and d = 1 .
54 mm,
F R = 120 . L = 1 .
806 cm, respectively.We contend that the waves are produced on the water surface due to the time periodicvariation of cross-section of the jet, that is, due to the periodic crossings of necks and bulgesof the jet, at the position of the water surface. We vary L from zero till the jet-breakupbecomes imminent and take phographs of the water surface for various L . We find that nowaves are produced on the water surface when L was zero. On increasing L we could discernthe appearance of the circular waves for the first time when L was 0.221 cm for d = 1 .
54 mmand
F R = 120 . L the surface waves begin to wane. At a particular L = L the waves become the least sharp. However, on further lowering the surface, thewaves reappear with increased sharpness and the waves persist till ultimately the jet breaksup before touching the water surface. Figure 5 exhibits the above mentioned behavior for9 IG. 8: Photograph of the water surface in the beaker when the jet-length L = 1 .
806 cm for d = 1 .
54 mm and water flow rate of 120.0 cc/min. nozzle diameter d = 0 .
95 mm and a flow rate of
F R =40 cc/min. Similar behavior is alsoobserved for other nozzle diameters ( d ) flow rates F R at different jet length L as well.We measure the wavelengths λ s of the waves on the water surface using the photographs.Figures 9-11 show the measured wavelengths ( λ s ) as a function of L , respectively, for d =0 . , .
26 and 1.54 mm. The length scales of the waves are much smaller than the depth( ≈ λ s we calculate the group velocities v g = q g λ s π of thesewaves, where the acceleration due to gravity g =9.8 m/s . From these data we calculate thetime difference (∆ t ) between two consecutive crests of the waves. The ∆ t ( ∝ √ λ s ), are alsoplotted in Figs. 12-14 together with λ s , for the same d values as in Figs. 9-11. This ∆ t canalso be taken as the time difference between two bulges of the jet ’hitting’ the water surface.The measured λ s initially decreases as L increases and reaches a minimum value at L = L and thereafter, the λ s increases with L . L = L thus marks a sharp boundary inthe nature of the water jets. We conjecture that the minimum λ s corresponds to the fastestgrowing Rayleigh perturbation in the jet and not the one at the breakup point L = l B .This is partially supported by roughly similar values of ∆ t . For instance, in the inset ofFig. 14, ∆ t ≈ .
038 s at the minimum of λ s (at L = L ) is roughly equal to the time scale10 λ s ( c m ) Flow rate = 40.0 cc/min0.054* √ (1+980*( L -3.668)/70.0) λ s ( c m ) L (cm)
Flow rate = 45.0 cc/min0.052* √ (1+980*( L -3.558)/200.0) Flow rate = 50.0 cc/min0.051* √ (1+980*( L -3.442)/200.0) L (cm)
Flow rate = 57.0 cc/min0.047* √ (1+980*( L -3.442)/285.0) FIG. 9: Average wavelength ( λ s ) on the water surface of deep water gravity waves as a functionof L for d= 0.95 mm. τ = πω ≈ .
045 s obtained from the estimate of the frequency ω at the maximum of thedispersion curve of the Plateau-Rayleigh theory and measured by Goedde and Yuen, (Fig.7 of Ref.[8]). Moreover, the ∆ t at the breakup point ( L = l B ) of the jet is comparatively fardifferent from τ = 0 . λ s becomes a minimum. In other words, the jet-surfaceprofile changes along the length ( L < L ) of the jet satisfying the stability conditions ofRayleigh theory and affected very little by gravity. This is the only way we can explain thevariation of ∆ t , for example in Fig. 14, from about 0.09s to 0.04s keeping the constancy ofmean mass flow rate of water at any section of the jet.We contend that for L < L , the effect of surface tension dominates over the gravitationaleffect on the jet whereas at larger L > L the gravitational effect plays a dominant rolemaking the surface profile of the jet nonsinusoidal. The nonsinusoidal surface profile canalso be seen from the high speed photographs of Ref.[16].In the spirit of Ref.[17], and considering λ s to be proportional to the difference in jetlength between two bulgings, we numerically fit λ s ( L ) of the waves on the water surface ofthe beaker (Figs. 9-14) as λ s ( L ) = λ s ( L ) q g ( L − L ) u taking u as a fitting parameter.11 λ s ( c m ) Flow rate = 50.0 cc/min0.072* √ (1+980*( L -4.60)/50.0) λ s ( c m ) Flow rate = 57.5 cc/min0.067* √ (1+980*( L -4.813)/70.0) λ s ( c m ) L (cm)
Flow rate = 65.0 cc/min0.060* √ (1+980*( L -4.810)/130.0) Flow rate = 72.5 cc/min0.060* √ (1+980*( L -4.795)/220.0) Flow rate = 80.0 cc/min0.062* √ (1+980*( L -4.395)/450.0) L (cm)
Flow rate = 87.5 cc/min0.062* √ (1+980*( L -4.255)/610.0) FIG. 10: Average wavelength ( λ s ) on the water surface of deep water gravity waves as a functionof L for d= 1.26 mm. As can be seen the fit is reasonable. We could similarly fit the data for all other nozzles ina range of flow rates. Our contention of dominance of gravitational effect for
L > L thushas good experimental support. IV. CONCLUSION
In conclusion, our experiment verifies the existence of recoil capillary waves and its ef-fect on jet breakup and points out the relative importance of Rayleigh perturbations andgravitational effects on the jet surface profile. The photographs of surface waves created bythe jets on the water surface helps us measure the wavelegths of the surface waves. Thebehavior of the surface wave lengths show a sharp transition as a function of the jet length.The jet length L at the transition point is different for different nozzle diameters and flow12 λ s ( c m ) Flow rate = 70.0 cc/min0.061* √ (1+980*( L -5.873)/65.0) λ s ( c m ) Flow rate = 90.0 cc/min0.057* √ (1+980*( L -5.470)/200.0) λ s ( c m ) L (cm)
Flow rate = 100.0 cc/min0.056* √ (1+980*( L -6.270)/200.0) Flow rate = 110.0 cc/min0.058* √ (1+980*( L -5.560)/440.0) Flow rate = 120.0 cc/min0.056* √ (1+980*( L -5.806)/544.78) L (cm)
Flow rate = 130.0 cc/min0.056* √ (1+980*( L -5.660)/900.0) FIG. 11: Average wavelength ( λ s ) on the water surface of deep water gravity waves as a functionof L for d= 1.54 mm. rates. For L < L Rayleigh perturbations dominate whereas for
L > L gravitaional effectis more important. [1] J. Eggers, Rev. Mod. Phys. 69, 865 (1997).[2] J. Eggers, and E. Villermaux, Rep. Prog. Phys. 71, 036601 (2008).[3] S. P. Lin, Breakup of Liquid Sheets and Jets, Cambridge University Press, 2010.[4] B. Lautrup, Physics of Continuous Matter, Second Edition, CRC Press, Boca Raton, 2011.[5] Lord, J. W. S. Rayleigh, Proc. London Math. Soc. 10, 4 (1879).[6] J. W. S. Rayleigh, The Theory of Sound, Vol. II, pp. 351-375, Dover Publications, New York,1945. λ s ( c m ) ∆ t ( s ) L (cm) L √ (1+980*( L -3.442)/200.0) ∆ t ( s ) Flow rate (cc/min)
At breakupRayleighAt L FIG. 12: Average wavelength ( λ s ) on the water surface and time period (∆t) of deep water gravitywaves as a function of L for d= 0.95 mm and water flow rate of 50.0 cc/min. A numerical fit to λ s (for L > L ) is also shown. λ s ( c m ) ∆ t ( s ) L (cm) L √ (1+980*( L -4.795)/220.0) ∆ t ( s ) Flow rate (cc/min)
At breakupRayleighAt L FIG. 13: Average wavelength ( λ s ) on the water surface and time period (∆t) of deep water gravitywaves as a function of L for d= 1.26 mm and water flow rate of 72.5 cc/min. A numerical fit to λ s (for L > L ) is also shown. λ s ( c m ) ∆ t ( s ) L (cm) L √ (1+980*( L -5.806)/544.78) ∆ t ( s ) Flow rate (cc/min)
At breakupRayleighAt L FIG. 14: Average wavelength ( λ s ) on the water surface and time period (∆t) of deep water gravitywaves as a function of L for d= 1.54 mm and water flow rate of 120.0 cc/min. A numerical fit to λ s (for L > L ) is also shown.[7] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, NewYork, 1961.[8] E. F. Goedde, and M. C. Yuen, J. Fluid Mech. 40, 495 (1970).[9] A. Umemura, Phys. Rev. E 83, 046307 (2011).[10] A. Umemura, S. Kawanabe, S. Suzuki, and J. Osaka, Phys. Rev. E 84, 036309 (2011).[11] A. Umemura, and J. Osaka, J. Fluid Mech. 752, 184 (2014).[12] A. Umemura, J. Fluid Mech. 575, 665 (2014); ibid 797, 146 (2016).[13] W. K. Bani, and M. C. Mahato, arXiv:1608.04915v1 [physics.flu-dyn].[14] We have used a (3.5-4.99 mW) Taurus Series 635 nm (Class IIIa) red laser pointer, a photo-conductive LDR (1KΩ) sensor (detector), a UA741CN Op Amp, a SN74LS14N Hex InverterSchmidt Trigger, a 74ALS04BN Hex Inverter, HCF4033BEs and CD4026BEs decade coun-ters, a NE555P timer, HEF4082BPs AND Gate, Common Cathode 7-Segment single digitLED display in the experimental set up.[15] R. J. Donnelly, and W. Glaberson, Proc. Roy. Soc. A 290, 547(1966).[16] D. F. Rutland, and G. J. Jameson, J. Fluid Mech. 46, 267 (1971).[17] L. E. Scriven, and R. L. Pigford, A.I.Ch.E Journal 5, 397(1959).) is also shown.[7] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, NewYork, 1961.[8] E. F. Goedde, and M. C. Yuen, J. Fluid Mech. 40, 495 (1970).[9] A. Umemura, Phys. Rev. E 83, 046307 (2011).[10] A. Umemura, S. Kawanabe, S. Suzuki, and J. Osaka, Phys. Rev. E 84, 036309 (2011).[11] A. Umemura, and J. Osaka, J. Fluid Mech. 752, 184 (2014).[12] A. Umemura, J. Fluid Mech. 575, 665 (2014); ibid 797, 146 (2016).[13] W. K. Bani, and M. C. Mahato, arXiv:1608.04915v1 [physics.flu-dyn].[14] We have used a (3.5-4.99 mW) Taurus Series 635 nm (Class IIIa) red laser pointer, a photo-conductive LDR (1KΩ) sensor (detector), a UA741CN Op Amp, a SN74LS14N Hex InverterSchmidt Trigger, a 74ALS04BN Hex Inverter, HCF4033BEs and CD4026BEs decade coun-ters, a NE555P timer, HEF4082BPs AND Gate, Common Cathode 7-Segment single digitLED display in the experimental set up.[15] R. J. Donnelly, and W. Glaberson, Proc. Roy. Soc. A 290, 547(1966).[16] D. F. Rutland, and G. J. Jameson, J. Fluid Mech. 46, 267 (1971).[17] L. E. Scriven, and R. L. Pigford, A.I.Ch.E Journal 5, 397(1959).