Alberto Tufaile

Chaotic behavior in bubble formation dynamics

We constructed an experimental apparatus to study the dynamics of the formation of air bubbles in a submerged nozzle in a water/glycerin solution inside a cylindrical tube. The delay time between successive bubbles was measured with a laser-photodiode system. It was observed bifurcations, chaotic behavior, and sudden changes in a periodic regime as a function of the decreasing air pressure in a reservoir. We also observed dynamical effects by applying a sound wave tuned to the fundamental frequency of the air column above the solution. As a function of the sound wave amplitude, we obtained a limit cycle, a flip bifurcation, chaotic behavior, and the synchronization of the bubbling with sound wave frequency. We related some of the different dynamical behaviors to coalescent effects and bubble sizes.

The circle map dynamics in air bubble formation

Abstract We studied the air bubbles formation in a submerged nozzle in a water/glycerol solution inside a cylindrical tube, submitted to a sound wave perturbation, whose amplitude is a parameter of control. It was experimentally observed quasiperiodicity, transition from quasiperiodicity to chaos, routes to chaos via period doubling cascade according to the values of Ω=f s /f b , where fs is the sound wave frequency and fb is the bubbling rate. Our data can be explained by a two-dimensional circle map dynamics. We simulated some bifurcation diagrams as well as some reconstructed attractors with amazing results.

Hénon-like attractor in air bubble formation

Abstract We studied the formation of air bubbles in a submerged nozzle in a water/glycerol solution inside a cylindrical tube, submitted to a sound wave perturbation. It was observed a route to chaos via period doubling as a function of the sound wave amplitude. We applied metrical as well as topological characterization to some chaotic attractors. We localized a flip saddle, and we also could establish relations to a Henon-like dynamics with the construction of symbolic planes.

Simulations in a dripping faucet experiment

Abstract The profiles of two experimental attractors were simulated by using a simple one-dimensional spring-mass model. Some peculiar behaviors observed in experimental bifurcation diagrams (in short ranges of dripping rate variation) were emulated by combining two quadratic maps (a kind of coupling) in two different ways: parallel combination with non-interacting maps; and series combination with strongly interacting maps. The choice of each kind of combination was suggested by the own characteristics of each experimental bifurcation diagram.

Period-adding bifurcations and chaos in a bubble column

We obtained period-adding bifurcations in a bubble formation experiment. Using the air flow rate as the control parameter in this experiment, the bubble emission from the nozzle in a viscous fluid undergoes from single bubbling to a sequence of periodic bifurcations of k to k+1 periods, occasionally interspersed with some chaotic regions. Our main assumption is that this period-adding bifurcation in bubble formation depends on flow rate variations in the chamber under the nozzle. This assumption was experimentally tested by placing a tube between the air reservoir and the chamber under the nozzle in the bubble column experiment. By increasing the tube length, more period-adding bifurcations were observed. We associated two main types of bubble growth to the flow rate fluctuations inside the chamber for different bubbling regimes. We also studied the properties of piecewise nonlinear maps obtained from the experimental reconstructed attractors, and we concluded that this experiment is a spatially extended system.

Hyperbolic kaleidoscopes and Chaos in foams and Hele-Shaw cell

Liquid foams have fascinating optical properties, which are caused by the large number of light refractions and reflections by liquid films and Plateau borders. Due to refraction and reflection at the interfaces, the direction of the rays leaving a Plateau border can vary greatly for the same incident angle and a small positional offset. A close look in some configurations of the Plateau borders or liquid bridges reveals the existence of some triangular patterns surrounded by a complex structure, and these patterns bear a resemblance to those observed in some systems involving chaotic scattering and multiple light reflections between spheres. Provided the optical properties of the sphere surfaces are chosen appropriately, fractals are natural consequences of multiple scattering of light rays in these cavities. The cavity acts as a hyperbolic kaleidoscope multiplying the scattering of light rays generating patterns related to Poincare disks and Sierpinski gaskets in comparison to linear kaleidoscopes. We present some experimental results and simulations of these patterns explained by the light of the chaotic scattering.

Explosion of chaotic bubbling

We have studied a saddle-node bifurcation/explosion of air bubble formation driven by a sound wave, whose amplitude is the control parameter. The bubbles are formed in a nozzle submerged in a water/glycerol solution inside a cylindrical tube, and the sound wave is tuned to the air column above the fluid. The nonlinear interaction between sound wave and the fluid oscillations, caused by the air bubbles passage through the liquid, results in a route to chaos via quasi-periodicity, with some resonant states characterized by the rational winding numbers W=fs/fb, where fs is the sound wave frequency and fb is the bubbling rate. We also have shown that the bubble dynamics is similar to the one observed in the two-dimensional circle map.

Light Polarization Using Ferrofluids and Magnetic Fields

We are presenting an experimental setup based on polarized light, enabling the visualization of the magnetic field of magnetic assemblies using a Hele-Shaw cell filled with ferrofluids. We have simulated the observed patterns with hypergeometric polynomials.

An Interdisciplinary Approach to Teach Geoscience

Music is the central theme of the interdisciplinary approach presented in this chapter. The interdisciplinary approach of this teaching proposal focuses on using the construction of a lithophone (from Greek lithos = stone), a percussion instrument with rock plates, for teaching geoscience. Lithophones are similar to the more common xylophones, but with unique musical characteristics which reflect their peculiar design. The main aims of this chapter focus on how to build a lithophone in order to use it as education tool and as an appealing educational resource. When designing a lithophone, the teacher and his/her students should characterize the chosen lithotype (rock type) and the relationship between the length of the plates and the vibration frequency in order to select the musical notes. When the lithophone has been constructed, the teacher should use the sound of the rock plates as a tool to discuss the elastic characteristic of materials. The amazing results obtained through this interdisciplinary approach illustrate the importance of this educational resource in teaching and disseminating geosciences.

Random Matrix Thermodynamics

We have performed a study of the statistical mechanics of correlated spectra first introduced by Dyson and Mehta some 40 years ago. We have derived a modified thermodynamical statistics (a number and its variance) for linear spectra. This approach was used to analyze the statistical properties of the eigenvalues of random matrices of Gaussian ensembles and of the experimental eigenfrequencies of acoustic resonances of an aluminum plate cut in the shape of a chaotic billiard. The results obtained suggest that this statistics provides a robust tool for the investigation of spectral properties.