WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.1 Drops Vibrating in a Circular Corral A.D. Skowronski, Glenn Patterson, Tuan Le, and Dr.Madsen Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated: October 20, 2014) Droplets can be made to stay and walk on a vibrating surface through the resonance of the surface wave and the drop itself. By tracking the trajectory of droplets on a vibrating bath, we show a dependence of the trajectory on the Faraday threshold. We expect to show wavelike statistics throught a probability distribution; however, we find a leveling problem in our current setup.
WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.2 I. INTRODUCTION Quantum mechanics suggests the idea that nature is inherently probabilistic. The wave equations used in quantum mechanics have no physical quantities, only probabilities until they are observed or measured. However, the pilot wave theory, proposed by De Broglie and David Bohm, challenges that idea. Indeed, the theory is now fueled with fluid dynamics, in which a walking, bouncing droplet, which is driven by a pilot wave, has quantum-like properties. Studying the droplet can lead us to a simpler model to explain the pilot wave theory by using the analogy of fluid dynamics. In 1978, Walker achieved the first floating droplet[9]. He observed bouncing for several minutes on a vibrating dish of the same fluid. Furthermore, in 2005, Couder, Protiere, Fort and Badouad, and in 2013, Molaek and Bush, studied the walking oil droplet, which bounces along the surface of a silicon oil that vibrates at a driven frequency [3][4]. The droplet is driven by the pilot wave, which is caused by the interaction between the oil droplet and its own ripples. Consequently, this pilot wave gives the quantum-like properties that were thought to be unique to quantum-mechanical particles. We first create a long-lasting droplet using soapy fluid, and study the Faradaywave on the surface of the fluid bath. We then use silicon oil in a circular corral, which is designed to test the quantum statistical properties of the pilot wave. II. SETUP Our initial set up consists of an old speaker with cone corral, soapy water (0.9M), and an amplifier with range f = 0 1000Hz. This setup was first used by Walker in 1978 [9] in order to suspend drops of liquid on a vibrating bath. Our circular corral apparatus consists of an old speaker with a circular corral Figure 1 c, which we fill with silicon oil. To keep the entire speaker and corral apparatus level, we constructed a three-point leveling system made of two blocks of wood and three screws. However, our approach towards designing the leveling system proved to be faulty. The speaker was fit into a hole on the top block of wood, which stood on three screws in a triangular-fashion. The three screws drove upward with the heads of the screws fitting into holes on the bottom block of wood. Because dissembly of the leveling system was required in order to drive the screws and level the plane, we essentially created a new variable in our experiment. Therefore, a leveling system that provides the
WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.3 least amount of uncertainty in our data is needed. We used a strobe light with a cylindrical lens to illuminate the bouncing droplet in order for it to be tracked on video. III. BASIC MECHANISM The wave on the vibrating fluid bath can be described as a Faraday wave. We follow Brady and Anderson by using a bessel function to model the height of the wave [8]. Robert Brady and Ross Anderson point out that the height of the surface h above the ambient lever is a solution of the wave equation[8]: 0 = 1 d 2 h c 2 dt d2 h 2 dx d2 h 2 dy 2 The standing waves on the vibrating bath are Faraday standing waves. One solution for the wave equation is h = h 0 Cos(ω 0 t)j 0 (ω 0 r/c) where h 0 is the maximum height, J 0 is a first kind Bessel function,and ω 0 is the driving frequency[8]. The interaction of the wave with the droplet can be seen in Figure 2 b. The resonance that occurs with the drop and the subsequent vibrating bath allows for the droplet to reach a walking mode. However, as we will see later, the drop resonating on non-peak elements of the wave is what gives it wavelike statistics. A. Faraday Threshold The Faraday threshold give us the upper and lower bound for angular frequency of the Faraday wave. If we exceed those values, chaotic motion appears. In 2002, Ugawa calculated the Faraday threshold for angular frequency for oil. 2ω 0 ( γω 0 γω0 2 )2 4µ 2 < ω < 2ω 0 + ( 2 )2 4µ 2 ) Where µ = 2ν 4π2 λ 2 = 2νk 2 is the damping rate and γ = A g is peak-non dimensional bath acceleration. In our experiment, the peak-non dimensional bath acceleration is (3.06 ± 0.16) 10 5 m/s 2 (95% CI). On the other hand, we can find the damping rate by finding the Faraday wavelength and using the kinematic viscosity of silicon oil, ν = 10cSt (25 C)[11].
WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.4 a) Droplet Vibration Exciter b) d 1 h 1 c) d 2 h 2 FIG. 1. a.) This is a simplified model of our apparatus. We use a vibration exciter in order to drive a corral containing a fluid. We then suspend a droplet of the same fluid on the bath. b.)we began using a cone shaped corral on top of a speaker, where d 1 = 35.38 ±.05 mm 95% CI, h 1 = 9.95 ±.05 mm 95% CI. With this apparatus, we used a soapy mixture of water in order to observe the dynamics of the droplet on the surface. c.) With the addition of silicone oil, we evolve our apparatus into a circular corral in order to confine the droplet to a finite geometry. The dimensions are d 2 = 4.63 ±.05 mm 95% CI, h 1 = 10.02 ±.05 mm 95% CI
WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.5 FIG. 2. a.) The wave shown models the propagating surface wave incident from the droplet. We observe that the droplets moves with twice the period of the wave. In the simple case, the droplet with interact with the top of the wave. b.)this is a 3-D representation of a droplet interacting with the pilot wave. The wave dynamics are those of a bessel function of the first kind.
WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.6 However, we have not found the Faraday wavelength in our experiments and therefore cannot provide a damping rate. We use a particle tracking software in order to track the position of the droplet with respect to time. Following from Couder and Forts paper in 2013, we move to show a dependence of the droplets trajectory on the faraday threshold. dimensionless parameter Γ, where We do that through a Γ = (γ f γ) γ f [7]. In this case, γ f is equal to the upper bound of the Faraday threshold divided by 2π. Following off of Couder and Fort, as this parameter approaches zero, the path memory of the droplet interacting with the pilot wave increases[7]. As seen in Figure 3, the circle represents a situation with low path memory. The elipsoid represents one of greater path memory. As Γ approaches zero, we expect the circular orbit to become more and more unstable, until we enter an interesting regime. Using our detergent solution, we show confirmation of the path memory increasing by showing an eliptical orbit at f = 143Hz. FIG. 3. a.)the circle represents a low path memory limit whilst the ellipsoid represents a droplet with a higher path memory limit. We expect the situation with low path memory to follow a circular orbit, whilst the higher path memory limits will yield a more chaotic orbit[7]. b.) We show confirmation of an eliptical orbit with a drop of detergent with an intermediate path memory, where the outer circle depicts the confinement of our first apparatus. After showing a dependence of the droplets trajectory on the Faraday threshold, we expect
WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.7 to show wavelike statistics in our circular corral by showing a wave like radial dependence. According to Couder and Fort, we can understand the probability distribution through the underlying dynamics of the corral. In the confined circular geometry, the pilot wave seems to drive the droplet along circular orbits with radii corresponding to maxima in the cavity modes. As the path memory increases, the circular orbits become unstable and we see drifts between the Faraday orbits, which are given by the Faraday standing wave troughs[7]. Following, in a recent paper from the University of Liege, researchers find that if we examine the fundamental forces that are at play, we can begin to explain the wavelike statistcs that appear in this pilot-wave situation. The research team at Liege explains the behavior through two forces: the hydrodynamic force and the quantum force. This team shows that the hydrodynamic force is proportional to the slope of the wave at the point that the droplet interacts with the wave[7][10] Therefore, at points with a high slope, or points in between the peaks and troughs of the waves, the hydrodynamic force will be relatively large and point toward the trough of the wave. This research team also shows that the hydroynamic force is larger in magnitude than the quantum force in these areas. Futhermore, the inherent force that should give us wavelike statistics is caused by a force different than the quantum force because it is the hydrodynamic force that contains the droplet to the troughs of the standing waves[10]. IV. DATA We used particle tracking software to track our walking droplet in the circular corral. We calculated the average position of the droplet in both the x and y direction. From the histograms in Fig. 4, the average position in the y position is 0.78 ± 0.52mm (95% CI) while the average position in the x direction is 0.68 ± 0.61mm (95% CI) Due to the system not being level, we cannot make any conclusions about the radial probability. However, we expect the probabilty function to follow the same form as the equation for the height of the wave due to the hydrodynamic force that pushes towards the troughs of the Faraday standing waves, noted by the research team at the Univeristy of Liege[10].
WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.8 a) # of Frames # of Frames b) c) R FIG. 4. a) Histogram of y position of two-minute video of walking droplet. We find the average position to be 0.78±0.52mm (95% CI).b) Histogram of x position of two-minute video of walking droplet. We find the average position to be 0.68±0.61mm (95% CI). c) Expected probability density following from the shape of the height of the wave function. Histograms show that the average position is non-zero, therefore the experimental setup is non-level.
WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.9 V. CONCLUSION Thus, beginning with an initial crude setup, we were able to isolate a bouncing droplet on the surface of a detergent bath. The results from this experiment allowed us to see dependence of the particle s trajectory on the the Faraday threshold through the low path memory. Moving onto our advanced aluminum apparatus with silicone oil, analysis of the bouncing droplet s position over time should give us a probability distribution which displays wave-like statistics, indicating that a macroscopic system can be related to quantum mechanics through a physical wave. It is also important to note that according to the research team at the University of Liege, two separate forces are modeled: the hydrodnamic force and the Bohmian force. When looking closer into this quantum analogy, differences arise due to the inherit differences between the two [10]. [1] Oza, A., Wind-Willassen, O., Harris, D. M., Rosales, R.R. and Bush, J. W. M., 2014. Pilotwave dynamics in a rotating frame: Exotic orbits, Phys. Fluids, 26, 082101. [2] Oza, A. U., Rosales, R. R. and Bush, J. W. M., 2013. A trajectory equation for walking droplets: hydrodynamic pilot-wave theory, J. Fluid Mech., 737, 552-570. [3] Molacek, J. and Bush, J.W.M., 2013. Droplets bouncing on a vibrating bath, J. Fluid Mech., 727, 582-611. [4] Molacek, J. and Bush, J.W.M., 2013. Droplets walking on a vibrating bath: Towards a hydrodynamic pilot-wave theory, J. Fluid Mech., 727, 612-647. [5] Gilet, T. and Bush, J.W.M., 2012. Droplets bouncing on a wet, inclined surface, Physics of Fluids, 24, 122103: 1-18. [6] Blanchette, F., Messio, L. and Bush, J.W.M., 2009. The influence of surface tension gradients on drop coalescence, Physics of Fluids, 21, 072107. [7] Daniel M. Harris, Julien Moukhtar, Emmanuel Fort, Yves Couder, and John W. M. Bush, 2013.Wavelike statistics from pilot-wave dynamics in a circular corral, Phys. Rev. E 88. [8] Brady, R. and Anderson, R. 2014. Why bouncing droplets are a pretty good model of quantum mechanics, University of Cambridge.
WJP, PHY381 (2014) Wabash Journal of Physics v1.3, p.10 [9] Walker, J.,1978. Drops of liquid can be made to float on the liquid. What enables them to do so?, Scientific American, 238-6:15 [10] Richardson, C D.,Schlagheck, P.,Martin, J., Vandewalle,Thierry Bastin.,2014.On the analogy of quantum wave-particle duality with bouncing droplets Departement de Physique, University of Liege, 4000 Liege, Belgium. [11] A. Ugawa, O. Sano. Dispersion Realtion of Standing Waves on a Vertically Oscillated Thin Granular Layer. J. Phys. Soc. Jpn., 71, 2815-2819 (2002).