## Deliverables

Laser induced bubbles serve as a very flexible model for cavitation bubbles in acoustic fields. In particular the high reproducibility and the possibility of exact timings are reasons for the employment of this method for cavitation bubble research. The bubbles are generated by an intense focused laser pulse in the liquid. The induced plasma creates a high temperature and high-pressure zone which serves as the germ for an expanding vapour bubble. Subsequently the vapour bubble collapses again with a few potential rebounds. In particular the first (strongest) bubble collapse has been investigated.
This heavy implosion is the most interesting part of a bubble life from the point of view of many applications, as it causes heat and pressure peaks along with shock waves. However, it is also the part, which is most difficult to observe and to model. In experiments, performed by the Göttingen team, collapse times and shock wave pressures have been measured. A theoretical model, which can reproduce the measured quantities well, has been developed by the Ufa team.
An accurate description of the interior and the near neighbourhood of a collapsing bubble is important for the relation of cavitation activity to chemical reactions and to damage or cleaning effects, because they are linked to temperature and shockwaves, respectively. Applications of the given results comprise estimates of this activity, which in turn can lead to better theoretical predictions of the effectivity and optimisation of ultrasonic devices. The results will be used by other researchers in the field of general cavitation, ultrasonic cleaning, sonochemistry, sonoluminescence, and sonofusion, for instance.
Experimental bubble behaviour could be determined very accurately by advanced high-speed imaging techniques, and shock wave pressures have been measured with a special fibre optic hydrophone, which is one of the fastest and smallest pressure probes available. The theoretical model accounts for the realistic equation of state for water (in liquid and vapour states) that is valid up to shock compression, the thermodynamics of evaporation and condensation phenomena at the bubble interface, and the influence of non-condensable gas.
The matching of experimental and theoretical results allows evaluating the accommodation (condensation) coefficient for water. This approach may be recommended as an effective method of the measurement of accommodation coefficients of various liquids. The investigation of the laser bubble dynamics refers to spherical symmetry. The experimentally produced laser focus has always some degree of asphericity. These deviations, however, could be kept small. Additionally, the scaling of bubble sizes is restricted (maximum radius not smaller than about half a millimetre) because the laser intensity has to be larger than the optical breakdown threshold (the pulse length is fixed to about 8 nanoseconds).
Very high temperatures and pressures are expected at the final stage of the bubble collapse. Complex short time scale physical and chemical effects (like dissociation and ionisation) are involved which are still hard to model accurately. This is currently an active and ongoing field of research. Investigations of cavitation bubble dynamics and erosion by laser-induced bubbles has proved to be an invaluable tool for certain problems. The details of experimental set-ups are relatively mature, and such experiments can be conducted rather easily in the meantime. In the future, different extensions of the standard will be further investigated, for example multiple laser bubble interaction and laser bubble dynamics near to complicated surfaces.
The basics of the theoretical model proposed and tested during this research grant are applicable for many technological and scientific applications, also apart from water treatment. Among them are the enhancement of drug delivery by ultrasound, sonochemistry, or production of nanophase materials. The essential plasma physics during the bubble implosion may be modelled and used for the analysis of sonoluminescence. Moreover, this model may be useful to estimate the potential for the attaining of nuclear fusion reactions in collapsing bubbles (sonofusion).

The outcome of the research activity in the context of the present project is an improvement of the understanding of the basic principles governing dynamic interaction of bubbles in the presence of acoustic disturbances. The nature and importance of different forces exerted upon interacting bubbles was clarified for the case of single, bubble-bubble or many-bubble dynamics. In particular, the Research Group from Patras focused on the theoretical and numerical approach on the bubble-bubble and many-bubble dynamics. A number of crucial conclusions were arrived at that will be essential for the implementation of the ideas that were put forward in this project on a more practical level. The most important results are summarised in the following:
- Viscosity breaks the symmetry of the secondary Bjerknes force between two bubbles, with the larger one experiencing a smaller drag force. For the particular case of air bubbles in water viscosity alone cannot cause sign inversion of the interaction force.
- When two bubbles of unequal size with radii in the order of 100 micrometres are subjected to a sound wave with amplitude P_A<1.0 bar and forcing frequency om_f=0.510.5bar, the two bubbles are seen to attract each other due to the growth of even higher harmonics that fall outside the range defined by the eigenfrequencies of the two bubbles.
- In the case of much smaller bubbles, radii in the order of 10 micrometres driven well below resonance, om_f/2Pi=20kHz, at very large sound intensities, P_A>1bar, numerical simulations show that the forces between the two bubbles tend to be attractive, except for a narrow region of bubble sizes corresponding to a nonlinear resonance related to the Blake threshold. As the distance between them decreases, the region of repulsion is shifted indicating sign inversion of the force between them.
- In agreement with the above findings, extensive numerical simulations with the two bubbles driven above resonance, om_f

For simulation of few or many bubbles in a cavitating field, computer codes have been created by the Göttingen and Ufa teams which implement a so-called 'particle approach'. The software created in Ufa computes coupled oscillation and translational motion of few (from single up to ten) interacting bubbles in an acoustic field. This software is able to model the formation of narrow bubble clusters when the distances between the adjacent bubbles become very small and the bubble oscillations couple to each other.
The code developed in Göttingen computes individual tracks of hundreds of bubbles in the liquid and is suited to model larger structures of bubbles ('streamers') in an acoustic field. Both models account for the influence of primary and secondary Bjerknes forces together with added mass and drag force. Features of the models include different and variable bubble sizes, a realistic calculation of the secondary Bjerknes forces, and an arbitrary spatial distribution of the external sound pressure. Strong nonlinear bubble oscillations are allowed. Coalescence and splitting of bubbles is included in the schemes.
The application of the result is to reproduce quantitatively and qualitatively the behaviour of cavitation bubbles under the influence of a standing wave sound field. Experimental observations and both numerical codes show local increase of bubble concentration due to structure formation in an acoustic field. Small impurities suspended in wastewater may stick to bubbles, move with them and can be removed with them from the water. Additionally, oscillating bubbles can destroy plankton and bacteria and create free radicals that attack chemical contaminations. Thus, the aim is to understand existing bubble structures in ultrasonic reactors (e.g. cleaning devices), and a future prospect is to predict bubble distributions in given geometries for optimisation purposes.
Potential users are scientists and designers in the field of cavitation and high-power applications of ultrasound like wastewater treatment, ultrasonic cleaning, and sonochemistry.
Other existing computer codes for simulation of cavitation typically rely on so-called continuous descriptions which model bubble density as a continuous variable in space and time. In contrast to these codes our approach models the cavitation bubbles as individual particles driven by external forces. This enables the software to include fundamental physical mechanisms, like bubble-bubble interaction (secondary Bjerknes force), which are neglected or approximated otherwise. At lower bubble density, it is possible within a certain accuracy to reproduce bubble motion and bubble structure formation quantitatively (one-to-one) when compared to experimental observations. In other cases it is possible to give qualitative reproductions of experiments.
A main problem that still exists is the adequate modelling of bubble sources. This is rather a fundamental difficulty, because the physical mechanisms of bubble generation and the consequential spatial locations and bubble sizes are not known in detail. Further aspects that still have to be clarified or included refer to liquid streaming and objects submerged in the resonator bath.
The actual stage of the software is still experimental and not suited for a commercial distribution up to now. Such a state can be reached in the future if the fundamental difficulties above have been clarified. We like to mention that we are not aware of any commercial or experimental code that can solve cavitation structure problems at the moment with sufficient accuracy and completeness. This is still an active and ongoing field of research.

This result comprises a large collection of new optical recordings (high-speed images and movies) of cavitation bubbles under different experimental conditions. In typical acoustic resonator set-ups, many bubbles emerge and form localized patterns or structures. Some of these structures have now been classified into prototypes, and characterised with respect to their properties.
The processes involving cavitation bubbles are fast (typical excitation frequencies are larger than 20kHz) and small (down to micrometre scales). Therefore special observation techniques were used. One important method employed is a high-speed video recording (up to 2250 pictures per second), partly with magnifying optics and in stereoscopic view (two cameras). For other recordings, an ultra high-speed imaging device with eight subsequent frames and up to 100 million pictures per second has been used. Because such equipment is not widespread in the laboratories, the recordings, observations, and characterisations are mostly unique in the scientific community.
To be more specific, new insight has been gained about the following structures:
- Single bubbles moving towards a pressure antinode and the drag force acting on them;
- Single bubbles captured at a pressure antinode and the inversion of the primary Bjerknes force at lowered static pressure;
- Filamentary streamer structures ('dendritic trees'), the size and velocity distributions of the involved bubbles, and information about their detailed translational motion (tracks) in space and collisions;
- Details of the bubble interaction forces by comparison of the filamentary streamer recordings with particle model simulations;
- Double layer clusters ('jellyfish structures') which are important in free or weakly perturbed acoustic standing wave fields, and about their erosive and cleaning potential;
- Localised spot structures ('smokers') that appear on surfaces being eroded by cavitation.
The observations and characterizations are important for all applications of intense ultrasound in liquids, because acoustic cavitation is not only a wide spread phenomenon, but also mostly the active agent in these applications (like cleaning or
sonochemistry). Without detailed knowledge about the bubble distributions in space and time (i.e., the cavitation bubble structures) it is not likely to improve existing techniques significantly in the future.
The apparently simple question 'which structure appears under what conditions' naturally has to be answered before the next step, influencing the bubble distributions in a beneficial way, can be attacked. Our results represent a major distribution to the answer of this question, and can be viewed as part of a standard catalogue used by researchers and developers of acoustic cavitation devices and their applications.
The main innovation consist of the employment of high standard optical recording equipment for cavitation in technically relevant environment, namely in larger acoustic resonator baths. Additionally, advanced and novel evaluation tools have been developed and employed for automatic bubble recognition, three-dimensional reconstruction and tracking.
The recording devices are of high technical level, but nevertheless used at the limits of resolution in time and space. In particular the low spatial resolution of the high-speed video cameras puts restrictions on simultaneous bubble size measurements and tracking. A more fundamental problem is the proper characterisation of the driving sound field in the presence of cavitation, and the influence of liquid streaming on the observed effects.

The mathematical model of bubble cluster formed in a liquid exposed to acoustic field is proposed. The cluster size is assumed to be sufficiently small, so that liquid pressure inside the cluster can be considered as homogeneous. The motion of spherical cluster boundary is modelled by Releigh-Plesset equation with acoustic radiation. On the basis of this model, the dynamics of monodisperse and polydisperse clusters have been investigated.
The diffusion of gas dissolved in liquid and the stability of bubbles spherical form have been studied. It is shown that in polydisperse cluster (with bubbles of different initial radii) the synchronisation phenomenon of collapse phases occurs. Such synchronisation can lead to a more violent collapse of bubbles in a cluster. In monodisperse cluster two different diffusion stable bubble radii exist at which mean bubble size doesn't change over the oscillation period. In the case of two-fraction cluster the tendency to become monodisperse is observed so that bubbles of one size dominate in the cluster in the course of time. The investigation of stability of spherical bubble shape shows that bubbles in cluster are more stable at high-pressure amplitudes than a single bubble.
The phenomenon of bubble growth due to rectified diffusion of liquid-dissolved gas is a very important mechanism for wastewater cleaning. The investigations of diffusion processes have shown that the diffusion growth in monodisperse cluster differs from diffusion growth in a single bubble: an equilibrium radius in a cluster may be observed at higher values of dissolved gas concentration than for a single bubble. Thus, in a liquid that has not been degassed the cluster formation is more probable than the single bubble formation. Investigation of the rectified diffusion in polydisperse cluster shows that the cluster has a tendency to become monodisperse. Due to rectified diffusion, cavitation bubbles absorb gas dissolved in the water, collide, float up or concentrate locally, thus degassing and cleaning wastewater.
The main innovation consists of creation of mathematical model based upon multiphase mechanics principles and the effective (fast) numerical algorithm to calculate the collective violent oscillations of groups of bubbles in acoustic field. There are two main problems, which stay unclear today. First, the present mathematical model of bubble cluster dynamics uses assumption of spherically-symmetric type of bubble motion, which is not true in common case. At the same time the model, which accounts for non-sphericity of bubble surface seems to be too complex to be successfully implement and resolved numerically now. Second problem is to combine single bubble dynamics into multibubble systems with minimum corrections in both models. The next step in a bubble cluster dynamics modelling is connected to the mentioned problems, namely, we need to account two- and three- dimensional effects in single bubble dynamics. This will enable to model interactions between few bubbles, including effects of their coalescence and breaking.

The objective of the work was to extend the theory of acoustic radiation forces, which are also known as Bjerknes forces. These forces are experienced by gas bubbles (and other particles) in acoustic fields and cause them to migrate, cluster in certain space areas, interact with each other, etc. The pre-existing theory was based on a large number of simplifying assumptions, which restricted its accuracy and applicability and did not account for many experimental observations. Owing to the new extensive studies the old theory was considerably advanced. This was achieved by solving the following particular tasks:
- An analytical expression was derived for the time-averaged radiation force induced by an acoustic field between arbitrary compressible particles suspended in an ideal compressible fluid. This expression takes into account multiple re-scattering of sound between the particles and shape modes of all orders and does not impose any restrictions on the size of the particles, separations between them, and their number. It can be applied to modelling the dynamics of individual bubbles and multi-bubble clusters and also to compound systems including drops. For bubble-bubble interactions, the obtained results show that the classical Bjerknes theory is only valid for intermediate separation distances, while it fails for small and large separations.
- The influence of the second harmonic of volume oscillations of two interacting bubbles on their mutual interaction force was investigated. It was found that under certain conditions the interaction force could change from attractive to repulsive, due to the growth of the second harmonic, thus preventing the bubbles from coalescing.
- A refined expression was derived for the secondary Bjerknes force of two spherical bubbles in a viscous incompressible liquid, assuming that the spacing between the bubbles is much larger than their radii. This result is a substantial extension of the Bjerknes theory as it accounts for the translational oscillations of the bubbles, the linear vortical motion of the liquid and acoustic streaming around the bubbles. Two types of the boundary conditions on the gas-liquid interface were examined, namely, the no-slip condition, which corresponds to bubbles with surface impurity, and that allowing slip, which corresponds to clean gas bubbles. It was found that in both cases, the inter-bubble force could considerably differ from the force given by the Bjerknes theory. There is also an essential difference between the two cases themselves. In particular, while the interaction force between two small (driven well below resonance) bubbles with coating (no slip) is always attractive, only decreasing in magnitude (as compared with the Bjerknes theory) in the limit of high viscosity, the force between similar clean bubbles (slip allowed) can be repulsive within a relatively wide parameter range.
- The influence of neighbouring bubbles on the primary Bjerknes force experienced by a small cavitation bubble in a strong acoustic field was investigated. It was shown that bubbles very substantially affect each other's primary forces even if separation distances between them are large compared with their size. As a result, the peculiar features of the primary forces in strong fields, such as the change of sign with increasing driving pressure, manifest themselves much earlier and more vigorously.
The results obtained provide solutions to a number of important theoretical problems in the field of bubble dynamics. Altogether they make a major contribution to the theoretical basis that is needed for an adequate simulation of bubble motions in acoustic fields. All of the above listed results were published in well reputed scientific journals and are thus available for wide use in many scientific and industrial ultrasound applications, such as wastewater treatment by ultrasound, biomedical ultrasonics, acoustic levitation, containerless processing of materials, ultrasonic coagulation and precipitation of aerosols, acoustic purification of liquid solutions and melts, etc.

The objective of the work was to extend the theoretical basis that is needed for an adequate simulation of the translational motion of individual bubbles and bubble pairs in strong acoustic fields. In this context, a number of important theoretical problems were formulated and solved and extensive numerical studies were conducted for various values of the system parameters. The following main results were obtained:
- Using the Lagrangian formalism, coupled equations of radial and translational motions of a spherical gas bubble in a strong acoustic field were derived. The equations were then solved numerically for the purpose of studying the translational motion of a bubble in a plane standing wave of high intensity. It was shown that, if the acoustic pressure amplitude is high enough, a bubble driven below resonance, instead of moving to the pressure antinode as it does in a weak field, reciprocates about the pressure node plane. This result is of interest for an understanding of irregular bubble motions observed experimentally in strong acoustic fields.
- A theoretical model was proposed that makes it possible to calculate radial and translational motions of two interacting spherical bubbles in a strong acoustic field. The model allows for the radiation coupling between the bubbles up to terms of third order in the inverse separation distance, viscous forces on the bubbles, and effects of liquid compressibility on their volume pulsations. Using this model, a numerical investigation of the translational motion of two small, driven well below resonance, bubbles in sound fields with pressure amplitudes exceeding 1bar was made. It was shown that for most combinations of bubble radii characteristic of acoustic streamers, a mutual approach results in a dynamic steady state in which the distance between the bubbles remains, on average, constant. This result is of interest for understanding and modelling collective bubble phenomena in strong fields, such as acoustic streamer formation.
- Nonlinear coupling between the volume pulsation, translational motion and shape modes of an oscillating bubble was studied in the context of translational instability, known as "dancing motion", that is demonstrated by bubbles in acoustic standing waves of high intensity. A set of coupled equations was derived that describes volume pulsations of a bubble, its translation and shape modes evolving on the bubble surface. Unlike earlier work on this subject, where only two adjacent shape modes with given natural frequencies are taken into account, shape modes of all orders were allowed for and no limitations were imposed on their natural frequencies. As a result, additional important features were uncovered, which are inherent in the mutual interactions of the bubble shape modes as well as the shape modes and the translational motion.
The above listed results make a major contribution to the theoretical basis that is used for simulation of bubble dynamics in strong acoustic fields. Most of them already appeared in well-reputed scientific journals and are thus available for wide use in scientific and industrial ultrasound applications. The results pointed out in the last item were submitted for publication in the Journal of Fluid Mechanics, one of the leading journals in the field of bubble dynamics, and are now under reviewing.