High Dimensional Parameter Fitting of the Keller–Miksis Equation on an Experimentally Observed Dual-Frequency Driven Acoustic Bubble

A parameter identification technique of an underlying bubble model of an experimentally observed single bubble in a cluster under dual-frequency external forcing is presented. The measurements are carried out via high-speed camera recordings at a rate of 162750 frames per second. The used frequencies during the experiment are 25 kHz and 50 kHz. With a digital image processing technique, the measured bubble radius as a function of time is determined. The employed governing equation for the parameter fitting is the Keller–Miksis equation being a second order ordinary differential equation. The unknown four-dimensional parameter space is composed by the two pressure amplitudes, the phase shift of the dual-frequency driving and the equilibrium size of the bubble. In order to obtain an optimal parameter set within reasonable time, an in-house initial value problem solver is used running on a graphics processing unit (GPU). The error function measuring the distance between the numerical simulations and the measurement is based on the identification of the maximum bubble radii during each subsequent period of the external forcing. The results show a consistent estimation of both pressure amplitudes. The optima of phase shift and equilibrium bubble size are less significant due to a valley-like shape of the error function. Nevertheless, reasonable values are found that lead to estimations of pressure and temperature peaks during bubble collapse

Optimized Periodic Control of Chaotic Systems

In this work, we demonstrate the open-loop control of chaotic systems by means of optimized periodic signals. The use of such signals enables us to reduce control power significantly in comparison to simple harmonic perturbations. It is found that the stabilized periodic dynamics can be changed by small, specific alterations of the control signal. Thus, low power switching between different periodic states can be achieved without feedback. The robustness of the proposed control method against noise is discussed.Comment: 12 pages, uuencoded gzip-compressed postscript fil

Fast jets from bubbles close to solid objects: examples from pillars in water to infinite planes in different liquids

The dynamics of a single, laser-induced cavitation bubble on top of a solid cylinder and right at a plane solid boundary is studiedboth experimentally and numerically. The most intriguing phenomenon that occurs for a millimeter sized bubble right at a flatsolid boundary in water is the formation of a fast jet that is directed towards the solid with a speed of the order of 1000 m/s.Paradoxically, in this setting, fast jet formation causally is related to the viscosity of the liquid.Thus, results from numericalsimulations with varying liquid viscosity and bubble size are presented. Bubble dynamics and jet formation mechanisms arediscussed. It is shown, that fast jet formation persists for a wide range of liquid viscosities, including e.g. 50 cSt silicone oil. Forbubbles generated close to the flat top of a long, thin cylinder the parameter space of initial distance to the cylinder, bubble size andcylinder radius is scanned numerically and partly compared to experiments. When the maximum radius of the bubble exceeds theone of the cylinder the bubble collapses in the form of a mushroom or can resemble a trophy, depending on the values of thegeometry parameters. Complex patterns of jet formation with jet speeds ranging from the order of a few hundred m/s to severalthousand m/s are found.The results represent a contribution to understand the behavior of bubbles collapsing close to solid surfaces,in particular, how thin, fast jets are generated

Non-feedback technique to directly control multistability in nonlinear oscillators by dual-frequency driving

A novel method to control multistability of nonlinear oscillators by applying dual-frequency driving is presented. The test model is the Keller–Miksis equation describing the oscillation of a bubble in a liquid. It is solved by an in-house initial-value problem solver capable to exploit the high computational resources of professional graphics cards. During the simulations, the control parameters are the two amplitudes of the acoustic driving at fixed, commensurate frequency pairs. The high-resolution bi-parametric scans in the control parameter plane show that a period-2 attractor can be continuously transformed into a period-3 one (and vice versa) by proper selection of the frequency combination and by proper tuning of the driving amplitudes. This phenomenon has opened a new way to drive the system to a desired, pre-selected attractor directly via a non-feedback control technique without the need of the annihilation of other attractors. Moreover, the residence in transient chaotic regimes can also be avoided. The results are supplemented with simulations obtained by the boundary-value problem solver AUTO, which is capable to compute periodic orbits directly regardless of their stability, and trace them as a function of a control parameter with the pseudo-arclength continuation technique