Dripping faucet in extreme spatial and temporal resolution Achim Sack and Thorsten Poschel€ Institute for Multiscale Simulations, Friedrich-Alexander-Universitat€ Erlangen-Nurnberg,€ Erlangen 91052, Germany (Received 5 February 2016; accepted 22 March 2017) Besides its importance for science and engineering, the process of drop formation from a homogeneous jet or at a nozzle is of great aesthetic appeal. In this paper, we introduce a low-cost setup for classroom use to produce quasi-high-speed recordings with high temporal and spatial resolution of the formation of drops at a nozzle. The visualization of the process can be used for quantitative analysis of the underlying physical phenomena. VC 2017 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4979657] I. INTRODUCTION depending on the parameters of the experiment, one or more much smaller satellite drops may appear for each drop (see The formation of drops of liquids is an everyday phenom- Fig. 1 below); and (ii) small perturbations of the surface of a enon, yet obtaining a detailed understanding of the process fluid cylinder grow over time if the wavelength of the pertur- is not straightforward from an experimental or a theoretical bation exceeds a certain value. The latter observation allowed point of view. In engineering, drop formation is of impor- for the systematic investigation of the instability of fluid surfa- tance for a number of processes such as ink jet printing, ces and gave rise to a large body of related experimental work spray drying, spray coating, and others. by Hagen,8,9 Magnus,10 Rayleigh,11,12 and many others. To observe the process of drop formation at a faucet in The formation of satellite drops attracted much attention as detail, the time and length scales have to be resolved appro- it is a non-linear effect and therefore not covered by the linear priately. The duration of the pinch-off process of about 1 ms theory developed by Rayleigh in 1879.13 Although in static or and the required spatial resolution of about 1.5 megapixels quasi-static situations the physics of fluid surfaces is governed (see Sec. III below for an estimate) determine the require- by an equilibrium of surface tension and gravity, Rayleigh ments for the apparatus. Most modern experiments use high- noticed that in rapid processes the relationship between sur- speed cameras with advanced technical specifications that face tension and inertia dominates.12,13 Thus, when fluid flows are far too costly for typical educational budgets. In order to slowly out of a faucet, the characteristic shape of the slowly bring to the classroom the phenomenon of drop breakup, forming pendent drop14 is determined by the balance of grav- which is of high educational value and great aesthetic appeal, ity and surface tension and the dynamics is quasi-static. As we exploit the idea of stroboscopic recording that was devel- the drop grows, the situation becomes unstable and the drop 1 oped by Lenard 130 years ago. develops a neck that grows in length and shrinks in diameter The aim of this paper is to describe a low-cost experimen- accordingly. At the moment of breakup, the elongated neck tal setup capable of recording periodic processes at time res- quickly recedes. This highly dynamic process, which is of 5 olutions of up to approximately 3  10 frames per second very short duration, was first experimentally observed in 1887 (fps) and almost any spatial resolution, limited only by the by Lenard.1 As a result of this pinch-off phenomenon, we resolution of an ordinary CCD camera. We will demonstrate observe complex oscillations that ultimately result in one or that this setup allows high-quality videos to be produced of more satellite drops (which were investigated in detail by the pinch-off process of a drop falling from a faucet. The Guthrie in 1863 (Refs. 15 and 16). experimental setup is used in undergraduate lab courses on There is a large body of literature following this pioneer- experimental physics, fluid dynamics, nonlinear dynamics, ing work, meaning that we now have a good understanding and measurement technology. of the process. Eggers and Dupont17,18 proposed a self- The formation of drops from a homogeneous jet, at a drip- similar description for the time-dependent shape of the neck ping faucet, or in splashes has been considered by some of whose single parameter contains the density, viscosity, and the most eminent scientists in history. It was Mariotte who surface tension of the fluid. This result was confirmed by first described the disintegration of a homogeneous flow of many experimental investigations for both liquid jets (e.g., water into drops under the influence of gravity in his seminal Ref. 19) and dripping faucets (e.g., Ref. 20). The scenario of 2 book “The Flow of Water and other Fluid Bodies” in 1686. satellite droplet formation follows a highly complex hierar- He understood that the flow becomes thinner due to gravity chical scheme.21 and explained the phenomenon of drop formation by stating A comprehensive review on the physics of drop formation that the “elements of the fluid” cannot be too distant from due to instability of surfaces, including many historical one another. As a qualitative explanation, this is not far from remarks, can be found in Refs. 22–24. For an introductory text 3 what we today call surface tension, described by Laplace discussing many important physical arguments, see Ref. 25. and Young4 more than a century later. It was Plateau5,6 who explained the instability of a homogeneous jet by the action II. PHOTOGRAPHIC IMAGES OF DRIPPING of surface tension. FAUCETS Systematic experimental investigations of drop formation from a jet were first performed by Savart in 1833.7 Among The dynamics of drop formation can only be observed by others, Savart made two important observations: (i) the jet the naked eye to a limited extent.7 Much of the dynamics and, breaks into approximately identical drops, but additionally, thus, of the physics of the problem would remain unrevealed 649 Am. J. Phys. 85 (9), September 2017 http://aapt.org/ajp VC 2017 American Association of Physics Teachers 649 Fig. 1. In 1887, using a stroboscopic technique, Lenard obtained sequences of drops at high time resolution. Shown here is a section of Fig. 3(a) from Ref. 1. without sophisticated imaging techniques. Given that the col- Therefore, the idea of stroboscopic recording, introduced by lapse of the neck is a very fast process that typically takes a Lenard,1 is applied with some modifications. To enable pre- millisecond or less, mechanical shutter cameras are not suit- cise timing, the free running interruptor was replaced by a able for recording this process. Therefore, Worthington26 illu- light barrier that is broken just before the pendent drop falls. minated splashes of water drops with electrical sparks of very The light barrier triggers a sequencer that, after a certain short periods of time (typically 1 ls) in an otherwise dark delay, ignites the flash and thus exposes the picture. By incre- room. When impacting the target plate, the drop interrupted menting the delay between breaking the light barrier and fir- an electrical circuit, thus igniting the spark through its ing the flash, a series of pictures can be recorded that, when own weight. Using this technique, which he called instanta- played back at a fixed frame rate, reproduce the process of neous photography, Worthington obtained splashes of great detachment of the drop in slow motion. Special attention 27–29 beauty; his paper of 1897 (Ref. 28) contains more than has to be paid to the flash bulb; in order to obtain sharp pic- 150 pictures of splashes and he later published an entire tures, no object should move by more than one pixel during 30 book containing mainly photographs of splashes. The beauty the time of illumination. We assume a typical neck length of of splashes created by the impact of an object on a liquid still 4mm(seeFig.4) corresponding to about 400 pixels and a col- fascinates scientists today (e.g., Refs. 31–33). lapse time of about 1 ms. If we assume constant velocity of The first true dynamic recording of a dripping faucet was 1 the neck during collapseÀÁ for an estimate, the maximum expo- obtained in 1887 by Lenard, who noted that the time lag sure time is thus 10À3=400 px  1px¼ 2:5 ls. between successive drops is highly constant, provided that the inflow of fluid is constant. Therefore, he used a strobo- scopic method1 where a drop was used to trigger the ignition IV. SPECIFICATION OF THE SYSTEM of the flash for the subsequent drop. By adjusting the vertical Figure 2 shows a CAD drawing of the setup, and Fig. 3 position of the camera, he was able to record the drop at dif- shows the actual apparatus. The fluid is initially contained in ferent stages of evolution after it was released from the fau- cet. In this way, he recorded a stroboscopic film of the entire sequence at high time resolution (see Fig. 1). True high-speed films allowing non-periodic processes like splashes to be investigated became available in the 1930s mainly due to the pioneering work of Edgerton,34 who recorded dripping faucets at up to 1200 fps.35,36 Edgerton’s technique and several applications are explained in an impressive video that is available online.37 Due to the great progress in the field of imaging technology in recent deca- des, it is now possible to produce high-speed recordings with high optical resolution and large frame rates.