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Bubble puzzles: From fundamentals to applications

For centuries, bubbles have fascinated artists, engineers, and scientists alike. In spite of century-long research on them, new and often surprising bubble phenomena, features, and applications keep popping up. In this paper I sketch my personal scientific bubble journey, starting with single bubble sonoluminescence, continuing with sound emission and scattering of bubbles, cavitation, snapping shrimp, impact events, air entrainment, surface micro- and nanobubbles, and finally coming to effective force models for bubbles and dispersed bubbly two-phase flow. In particular, I also cover various applications of bubbles, namely in ultrasound diagnostics, drug and gene delivery, piezo-acoustic inkjet printing, immersion lithography, sonochemistry, electrolysis, catalysis, acoustic marine geophysical survey, and bubble drag reduction for naval vessels, and show how these applications crossed my way. I also try to show that good and interesting fundamental science and relevant applications are not a contradiction, but mutually stimulate each other in both directions.


I. INTRODUCTION
"How do you find the problems you work on?" And "What do you think is the difference between fundamental and applied research?" These are questions I am often asked. My short answer to the first question is: "Be curious!" And to the second one: "In principle, none". And what in particular holds for both fundamental and applied problems, both in finding and in solving them: "Watch, listen, and be open". The answer to both questions can be summarized as: "Work on problems you most enjoy. Strange things can happen on the way." (Walter Munk, UCSD). In the best case, the problem to work on is both relevant and outstanding, at the same time.
In this article I want to take the opportunity to give longer answers to these questions, in particular by giving examples from my own scientific biography and scientific journey. As the thread of the article I will choose "bubbles". I will report how I first incidentally bumped into the science of bubbles, what and how I learned about them, what wonderful science and great interactions and collaborations with colleagues this endeavor opened for me, and how I kept on bumping into very relevant applications of bubbles in technology. So bubbles have provided me both wonderful scientific problems and very relevant applied questions, to whose solution, I think, we have contributed over the last two and a half decades.
The length scales on which bubbles are relevant range from nanometers to at least tens of meters, and I will give examples for interesting and relevant bubble phenomena on all these scales. The richness of bubble fluid dynamics is reflected in the many dimensionless numbers that are relevant in the context of bubbles [1,2], namely the • Reynolds number Re = U R/ν, expressing the ratio of inertia forces to viscous forces.
Here U is the bubble velocity, R its radius, and ν the kinematic viscosity.
• Froude number F r = U/ √ gR, ratio of inertia to buoyancy, where g is the gravitational acceleration.
• Galileo number Ga = gR 3 /ν 2 , ratio of gravitational to viscous forces, • Weber number W e = ρU 2 R/σ, ratio of inertia to capillarity, where σ is the surface tension.
• Capillary number Ca = ηU/σ, ratio of viscous to capillary forces, • Ohnesorge number Oh = η/ √ ρσR = W e 1/2 /Re, ratio of time of viscous damping to time of the capillary oscillations.
• Eötvös number Eo = (ρ − ρ g )gR 2 /σ, also called Bond number Bo, ratio of buoyancy to capillarity, • Stokes number St, ratio of characteristic timescale of bubble to that of the flow, • Morton number M o = gη 4 (ρ−ρ g )/(ρ 2 σ 3 ), which is a material parameter for a bubble in a certain liquid, depending only on surface tension, density, density contrast, viscosity, and gravity.
• Lewis number Le = κ/D, ratio of thermal diffusivity κ to mass diffusivity D, and thus another material parameter, • Damköhler number Da, ratio between chemical reaction rate and (diffusive or convective) mass transport rate, • Jakob number Ja = ρc p (T − T sat )/(ρ v Λ), ratio of sensible heat to latent heat. Here c p is the liquid specific heat, Λ is the latent heat, ρ v the vapor density, T sat the saturation temperature of the liquid, and T the temperature of the surrounding liquid.
We will encounter most of these numbers in this article. The selection of which bubble problems I will report on is naturally subjective and, as said above, along my own scientific bubble journey. This will also be reflected in the citations, where I will restrict myself to papers that had a significant scientific impact on me and to references to our original work. Some of the given examples I briefly discussed before, in a short Proceedings [3] (without Web of Science Index). There are many more bubble problems and applications, which I cannot report here or which I am even not aware of, but I hope that the paper stimulates other scientists to look into the subject and be open towards both fundamental and applied bubble problems, because -as I hope to be able to show with this article -it is intellectually very rewarding, covers many areas of fluid dynamics, and extremely relevant in many applications.

II. SONOLUMINESCENCE -ILLUMINATED BUBBLE DYNAMICS
The first major scientific bubble problem I bumped into was sonoluminescence. In 1994, during my time as postdoc at the University of Chicago in the group of Leo Kadanoff, I attended a lecture by Brad Barber on his PhD thesis on single bubble sonoluminescence [4,5] (later summarized in [6]). This phenomenon had been discovered a few years earlier by Felipe Gaitan [7,8], then a PhD student in Mississippi, when he experimented with an air bubble trapped in a water-filled flask by piezo-acoustical forces (the so-call Bjerknes force [9,10], see figure 1 for a similar setup), which at the same time drive the bubble: When the pressure is low, the bubble expands, and once it is high, it is compressed. But what Gaitan observed was first not believed by anybody: Under certain conditions the bubble can emit light! How can this be? Typical acoustical energies are in the range of 10 −12 eV per molecule, typical light energies in the range of 1 eV. This means that there is an energy focusing factor of 10 12 ! Sound, radius, and light intensity as function of time are reproduced in figure 2a.
Directly after Barber's talk, I discussed this fascinating subject with my colleague Michael Brenner, then also a postdoc at the University of Chicago. We asked ourselves the two obvious questions: What is the light emitting process and under what conditions does this phenomenon happen, i.e., what is the phase space of single bubble sonoluminescence? We first focused on the second question, and started to read and learn about bubble dynamics.
We very soon found the seminal papers by Andrea Prosperetti on this subject, most visibly summarized in his review [12]. The core dynamical equation is the celebrated Rayleigh- Plesset (RP) equation for the bubble radius R(t), where p g (R(t)) is the pressure inside the gas bubble, P (t) = P a sin(ωt) the driving acoustical pressure with amplitude P a and frequency f = ω/(2π), P 0 the ambient pressure, η the dynamic viscosity, and σ the surface tension. A historical review of the development of this equation is given in ref. [12]. The typical RP bubble dynamics for increasing driving pressure P a is shown in figure 2b. The left-hand side of the RP equation (1) was already known to Lord Rayleigh who derived it in the context of an analysis of cavitation damage of ship propellors [1,13,14]. So already in Rayleigh's time applied and fundamental science came hand in hand. The solution to the inertial part RR + 3 2Ṙ 2 = 0 of the RP equation (1) is the power law with a diverging singularity in the bubble wall velocityṘ(t) ∝ (t s − t) −3/5 at time t s . This singularity reflects the violent bubble collapse which can occur for strong enough driving, see figure 2. In a nutshell, at collapse, the gas inside the bubble gets compressed, heats up, partly ionizes, and at recombination light is emitted [8,15]. The work of Andrea Prosperetti also led us to the conditions under which stable single bubble sonoluminescence can occur: An obvious necessary condition is the (spherical) shape stability of the bubble, which Eller and Crum [16] had experimentally and Prosperetti theoretically analysed [17]. We applied his results and determined under what conditions the collapsing bubble would be (spherical) shape stable so that it would on the one hand  2. (a) Acoustic driving pressure P (t) (in red), resulting bubble radius R(t) (in green), and light intensity I(t) (in blue), as measured by [11]. A negative driving pressure causes the bubble to expand; when the driving pressure changes sign, the bubble collapses, resulting in a short pulse of light (marked SL). The figure is taken from ref. [8]. (b) Solutions to the Rayleigh-Plesset equation (1) with a sinusoidal driving P (t) = P a sin(ωt) at forcing pressures P a = 1.0, 1.1, 1.2, 1.3 atm. The ambient bubble radius is R 0 = 2µm and the frequency f = ω/(2π) = 26.5 kHz. Material parameters are for water at room temperature.
survive the collapse, but at the same time would still collapse strongly enough so that the gas inside the bubble would be considerably heated [18]. Both a parametric instability and the Rayleigh-Taylor instability turned out to be relevant [15,18,19]. Another necessary condition for stable single bubble sonoluminescence is the diffusive stability of the bubble. Also the diffusive bubble stability had been analysed before, namely in the seminal work by Epstein and Plesset [20], later extended by Fyrillas and Szeri [21] to oscillating bubbles, for which rectified diffusion [22][23][24] can occur: For large pressure, the bubble loses gas to the outside liquid, but during the low pressure period, it can gain gas from outside. For very strong driving the growth can win, mainly due to the thin boundary layer during that time but also due to the much larger bubble size. Applying these ideas to the regime of sonoluminescing bubbles could account for the experimentally observed diffusively stable single bubble sonoluminescence [25], and combining the conditions of shape stability, diffusive stability, and energy focusing led to the phase diagram of sonoluminescing bubbles [19], which for pure argon bubbles was in good agreement with the experimental observations. However, air bubbles were found to be stable for 100 times larger gas saturation than pure argon bubbles, see figure 3. The reason for this turned out to be the chemical stability of the gas inside the bubble [26]: The bubble is collapsing so strongly that the gas inside is nearly adiabatically compressed. This means that the collapse of the bubble is so violent that no thermal equilibrium with the surrounding water can be established: The bubble is heating up, to about 15000 K, as we now know both theoretically [27] and experimentally [28]. For molecular gases such as O 2 or N 2 this is much too hot and they dissociate: The resulting radicals react with each other and with the dissociation products of water vapor. NO, NH, etc. are formed, which dissolve in water. Therefore it is mainly argon that remains in the bubble, which is contained in air with a concentration of about 1%, explaining the factor of 100 higher gas concentration required for stable single bubble sonoluminescene with air as compared to the pure argon (or any other inert gas) case. This theory later got confirmed through various experimental results, see e.g. [29,30]. Another advantage of argon for achieving strong SBSL is that in contrast to O 2 or N 2 , argon has no internal degrees of freedom. Thus the focused energy of the bubble collapse can directly be transferred into heat. The 15000 K which are reached in this way are sufficient to partly ionize the gas. Recombination of ions and electrons leads to light emission through thermal bremsstrahlung [31,32]. Later we extended the ideas of chemical stability of ref. [26] to include various other chemical reactions of air with water [27] to find very good and quantitative agreement between experimental and theoretical phase diagrams, see figure 3b. (a) Phase space of single bubble sonoluminescence in the argon concentration P ∞ /P 0 vs. driving pressure P a /P 0 phase plane for pure argon bubbles, showing the three phases "no SL", "stable SL" (only possible for very small argon concentrations), and "unstable SL", where the bubbles grow by rectified diffusion while emitting light and finally run into a shape instability [19]. Adopted from [19]. (b) Phase space of single bubble sonoluminescence in the bubble radius vs. driving pressure phase plane for air bubbles, taken from ref. [27]: Bubble equilibrium radius R 0 versus driving pressure P a for three different air concentrations (10%, 20%, and 40% of saturation; the driving frequency is 33.4 kHz). The curves, signalling stable bubbles, follow from our parameterfree theory [27]; the data points had been measured by Ketterling and Apfel [30]. The bubble emits light only on the right curves where argon has accumulated in the bubble (filled diamonds). On the left curves, the losses by chemical reactions and the growth by rectified diffusion balance.
In a sense, single bubble sonoluminescence can be viewed as illuminated bubble dynamics, with the RP dynamics as backbone. A combination of concepts from hydrodynamics (both shape stability and diffusive stability), chemistry, plasma physics, applied mathematics, thermodynamics, and acoustics led to the phase diagrams, which are in good agreement with the experiments [15,27]. In its conceptual simplicity -an isolated, fixed, non-interacting single bubble in a flask -it can also be seen as "hydrogen atom of bubble fluid dynamics", on which we learned a lot.
Our work on sonoluminescence started off as pure fundamental research, driven by curiosity. We had not asked ourselves whether there would be any applications. Single bubble sonoluminescence simply was a fascinating and outstanding problem, with major open questions. In answering them, we learned tremendously, including on 1. Acoustic and other forces on bubbles 2. Bubble dynamics and bubble shape stability 3. Diffusive bubble stability 4. Chemical bubble stability 5. Bubble nucleation 6. Bubble collapse and cavitation 7. Plasma formation and thermal bremsstrahlung As we will see, all these items turned out to be very relevant in connection with various applications -sometimes very unexpectedly. The insight obtained from single bubble sonoluminescence therefore helped us enormously to discover and identify applied problems and to help to solve them or at least make some progress on them. In the next chapters I will report how this came about.
Namely, I will report on: • How our understanding of the bubble dynamics and of the bubble shape stability (subject 2 in above list) contributed to ultrasound diagnostics and to improve ultrasound contrast agents. I will also report on other applications of bubble dynamics in the medical context (section III).
• How we found out that the sound of snapping shrimp originates from a cavitating bubble (section IV, originating from subjects 5 and 6 in the above list).
• How the collapse of a bubble or a void close to a surface focuses the energy, leading to a major jet (section V, originating from subject 6 in the above list).
• How an entrained bubble in a piezo-acoustic inkjet channel can cause major trouble due to rectified diffusion (subject 3 in the above list) and how to solve this problem (section VI).
• How bubbles can nucleate on a microstructured surface and, when acoustically driven, collapse in a controlled way, enhancing the efficiency of ultrasonic cleaning and chemical reactions (section VII, originating from subjects 4, 5 and 6 in the above list).
• How our understanding of diffusive bubble stability (subject 3 in the above list) was instrumental to figure out why surface nanobubbles and surface nanodroplets are stable, with various applications in electrolysis, catalysis, diagnostics, and the food and remediation industry (section VIII).
• Finally, how our understanding on bubble forces (subject 1 in above list) brought us to bubbly two-phase flow, including studying drag reduction in turbulent bubbly flow, for which bubble deformability (subject 2) is crucial (section IX).
The paper closes with conclusions and with a short outlook (section X). In particular, I will motivate why from my point of view we live in the golden age of fluid dynamics. Both wonderful bubble science and very relevant bubble applications are ahead of us.

III. ULTRASOUND DIAGNOSTICS, ULTRASOUND CONTRAST AGENTS, AND OTHER APPLICATIONS OF BUBBLES IN DIAGNOSTICS, THERAPY, AND MEDICINE
In an outreach effort, in order to popularize physics, fluid dynamics, and bubbles, in 1995 I had written an article on single bubble sonoluminescence in "Physikalische Blätter", which was the German analog of Physics Today. Based on this, I got contacted by a physicist working for a pharmaceutical company on ultrasound contrast agents (UCA), which are introduced into the blood to enhance the acoustic scattering and which contain small, encapsulated microbubbles. These very effectively scatter ultrasound, see fig. 4a. In this way, it is e.g. possible to visualize the perfusion of tissue, like the heart muscle. The images are meanwhile used to obtain diagnostic information from the volume, shape and movement of the heart ventricles, in studying the blood flow in small blood vessels, in blood perfusion measurements, and in targeted molecular imaging, among others [33][34][35].
(a) Ultrasound image of a heart without (left) and with (right) previous injection of micro-bubbles. In the second case the structures become clearer as the bubbles act as ultrasound contrast enhancers. (b) Electron micrograph of a microbubble coated with a protein layer, taken from ref. [36]. (c) Eigen frequency f 0 of a sound-driven coated with DPPC monolayers as function of the ambient bubble radius (blue data points, from [37]). The solid blue line shows the fit to the Marmottant model [38], whereas the dashed line shows the Minnaert frequency (3). Figure adopted from ref. [39]. (d) With the coating the sound-driven bubbles show the typical "compression only" behavior, with the compression being much more pronounced than the expansion. Figure taken from ref. [38].
One of the nagging questions in ultrasound diagnostics in the mid 1990s was: How does 8 8 Appeared in Phys. Rev. Fluids 3, 110504 (2018).
one increase the signal-to-noise ratio? Namely, when detecting the emitted sound from the bubble at the driving frequency, the signal is obscured by the reflections from tissue. To improve the signal quality, it had been proposed [40] to detect higher harmonics of the driving frequency in the sound emission spectrum of the bubble. Vice-versa, also subharmonics had been suggested for better contrast. The immediate question was: What bubble properties are optimal for these purposes, in particular what bubble size, for given driving frequency? This indeed was the question with which the pharmaceutical industry approached us, and thanks to our work on single bubble sonoluminescence, we could straightforwardly provide an answer, namely by simply solving the RP equation with the relevant parameter for medical ultrasound imaging: Here, rather than in the range of 20 kHz -30 kHz as common for single bubble sonoluminescence, the typical driving frequencies are between 2 MHz and 10 MHz and typical bubble radii are a few micrometers. From linearizing the RP equation (1) it follows that the eigenfrequency of the volume oscillations of an acoustically driven bubble with ambient radius R 0 and under isothermal conditions approximately is [1,12,41] This eigenfrequency is called Minnaert frequency. With the material parameters for water under ambient conditions this gives the well-known rule-of-thumb [41] f 0 R 0 ≈ 3 MHz µm = 3 kHz mm = 3 Hz m (4) for the resonance frequency f 0 = ω 0 /(2π). For the frequencies of medical ultrasound imaging, the resonance radii are thus in the micron range. Also in the fully nonlinear case, thanks to the full RP equation we could calculate optimal parameter values for maximal sound emission in the second harmonic and in subharmonics, and could make statements on the expected bubble shape stability in those regimes [42]. One issue we had first ignored was that the ultrasound contrast agent bubbles are not "naked", but coated with lipids and polymers (figure 4b), to avoid the obviously undesirable bubble clustering in the body and to increase their lifetime. The coating however modifies the oscillation behavior of the bubble in an a priori unknown way. Therefore, a few years later, to take the effect of the coating on the bubble dynamics into consideration, we developed a model to quantitatively describe this modification of the RP dynamics (1), which is now known as "Marmottant model" [38]. The key idea behind this model is to introduce an instantaneous bubble-size dependent surface tension, reflecting the buckling behavior of the bubble coating when compressing the bubble, an elastic regime, and a shell-ruptured regime. An excellent recent review of such modified bubble dynamics can be found in ref. [39].
To experimentally test such models for the relevant frequencies in the MHz regime, one unavoidably needs ultra high-speed imaging, with frame rates 1 MHz. In an effort led by Nico de Jong and Michel Versluis, we therefore developed [43,44] an ultrafast camera, which allows imaging 128 digital frames with a frame rate of up to 25 MHz. We called it "Brandaris 128", as it is based on a rotating mirror, just as the famous Dutch lighthouse Brandaris on Terschelling. This camera allowed us to gain insight into the volume and shape oscillations of ultrasound contrast agent bubbles [45][46][47], in particular when combining it with measurements of the acoustic emission of such coated bubbles [45]. This procedure allowed us to adjust the model parameters of the Marmottant model [38] to the experimental data, very nicely reflecting the observed so-called "compression-only" behavior of the UCA bubbles [38,45,48,49] (see figures 4c,d), which is nothing else than -thanks to the bubble coating -modified RP dynamics, thus giving the ultrasound contrast agent community a very relevant tool.
Another necessity which arose out of the applications of bubbles as ultrasound contrast agents was to produce large numbers of relatively monodisperse and coated microbubbles. The monodispersity is desirable to enhance the scattering property of the bubbles, which is optimal close to the bubble resonance size given by eq. (4). We achieved this with a so-called co-flow device originally developed by Howard Stone, Dave Weitz and coworkers [50][51][52] and in the case of bubbles by Gordillo and coworkers [53], but now operated in a regime in which we could produce particularly small monodisperse bubbles in large quantities [54], see figure  5. Meanwhile this method has been commercialized within a start-up company, as spin-off from our Physics of Fluids group.  [54] to produce monodisperse microbubbles. The control parameters are the gas flow rate Q g , the liquid flow rate Q , and the geometric parameters, including the thickness (not shown) of this quasi-2D device. The scale bar in the lower right is 50 µm. Figure taken from ref. [54]. (b) Monodispersed ultrasound contrast agents produced with such a co-flow device, employing the principles of ref. [54]. The scale bar in the lower right corner is 50 µm. Figure  The development of the Brandaris 128 ultra-high-speed imaging facility also allowed us to study the interaction of ultrasonically driven bubbles with cells. This interaction is sometimes spectacular, as seen in figure 6 [55], which shows a HeLa cell culture (a commonly used human cell line) grown on a glass plate, just after a bubble has collapsed close to it. The collapsing bubble exerts such strong shear forces on the cell that they detach from the glass plate, or, if they are more remote, holes in the cell membrane are induced. These holes, which can close again after some time, allow drugs or genes to invade the cell. Therefore ultrasonically driven bubbles can be used for local application of genes or drugs. This includes employing emulsions of droplets composed of liquid perfluorocarbons, which are acoustically activated to undergo a phase change into a bubbly dispersion, a procedure termed acoustic droplet vaporization [56].
I take the opportunity to stress the fundamental differences between vapor and gas bubbles, which in detail are elaborated and explained by Prosperetti in his recent review on vapor bubbles [58]. While for gas bubbles it makes sense to ascribe them an ambient radius as the gas exchange processes with the environment are slow due to the slow gas diffusion, it does not make sense for vapor bubbles, which are controlled by the much faster heat diffusion and condensation and evaporation. Also the resonance frequency of vapor bubbles does not scale like the inverse radius as for gas bubbles (equation (4)), but as ∼ 1/R 2/3 [58]. Note that an expanding vapor bubble is not only invaded by evaporating liquid, but also by gas dissolved in the liquid, which in the long term crucially determines its dynamics and lifetime, as we showed for vapor bubbles generated with water-immersed plasmonic nanoparticles [59].
Such so-called plasmonic microbubbles [60,61] indeed also have potential biomedical applications [62][63][64][65][66], again both in diagnosis and therapy (next to other potential applications in micro-and nano-manipulation, catalysis, and solar energy harvesting [67]), and understanding and controlling the dynamics of these microbubbles is key to successfully exploit them -and to recognize potential risks. In figure 7a we show the life-cycle of such a plasmonic nanobubble, nucleating in air-saturated water thanks to laser-illumination of plasmonic gold nanoparticles, each with a diameter of about 100 nm. Note the very different timescales in between the four snapshots. After some delay time τ d after the beginning of the illumination, the bubble explosively grows to giant size (as compared to the size of the nanoparticle), up to a maximum radius of 80 µm, and collapses again within ≈ 10µs (bubble life phase 1, which we time-resolved with ultra high-speed imaging in figure 7b). The maximum bubble volume V max remarkably increases with decreasing laser power P , see figure 7c, and, also remarkably, decreases with increasing gas saturation of the water.
We could explain [57] these remarkable features, based on the phase diagram of water (see figure 7d for a sketch) and in particular the lines of attainable superheat therein, which are in between the line of liquid-vapor coexistence and the liquid spinodal line. We first measured the delay time τ d from the beginning of the illumination up to nucleation, which drastically increases with decreasing laser power, leading to less total dumped energy E = P τ d . This   dumped energy E shows a universal linear scaling relation with V max , irrespectively of the gas concentration of the surrounding water (figure 7e). This finding supports the interpretation that the initial giant bubble is a pure vapor bubble. In contrast, the delay time does depend on the gas concentration of the water, as gas pockets in the water facilitate an earlier vapor bubble nucleation, which leads to smaller delay times and lower bubble nucleation temperatures (see again the phase diagram of water, figure 7d). After the collapse of the initial giant bubbles, first much smaller oscillating bubbles form out of the remaining gas nuclei (bubble life phase 2, up to typically 10 ms, see fig. 7a). Subsequently, a vaporizationdominated growth phase takes over and the bubble stabilizes (life phase 3). In the final life phase 4, the bubble slowly grows by gas being expelled due to heating of the surrounding.

IV. SNAPPING SHRIMP AND THE UNDERWATER SOUND OF BUBBLES
In another outreach effort, in 1999 I gave a colloquium talk on single bubble sonoluminescene at the Technical University of Munich (TUM), also addressing the issue of sound emission from the collapsing bubbles. After the talk, I met with a TUM zoologist who showed me a signal of the sound emission of so-called snapping shrimp, and the same evening I met these animals in the lab. They are about 5 cm long and live in the tropical ocean. With the help of a huge claw they can make considerable noise. This animal is very unpopular with the navy: First, it disturbs underwater communication between submarines. Second, even worse, hostile submarines use shrimp colonies to "acoustically hide" themselves.
The first obvious question to ask is: How does the shrimp make such noise? Zoologists thought that the sound pulse is caused by mechanical vibration on claw closure. Knowing the sound emission of a collapsing bubble from the work on sonoluminescence, I had my doubts on this hypothesis. Moreover, I knew the beautiful paper of Prosperetti, Crum, and coworkers on the underwater noise of rain [69], in which it is shown, by correlating highspeed imaging and sound detection with a hydrophone, that the noise arising when rain drops fall on a water surface does not originate from the impact, but from the oscillations of an entrained bubble. Following their example, we made high-speed movies of the snapping event (the shrimp had to be tickled) and correlated them with the corresponding sound track [68]. What we saw was that the shrimp closes its claw so quickly that a fast water jet develops. High velocities imply low pressure. Just as in single bubble sonoluminescence this leads to growth of bubbles. Once the pressure has equilibrated, the bubble collapses, leading to sound emission (figure 8) at bubble collapse, Indeed, the singularity (2) in the RP bubble dynamics is reflected in a singularity P s (t) ∼ (t s − t) −6/5 in the sound emission. The second obvious question to ask is: Why is the snapping shrimp doing all this? The answer is simple: It wants to eat! The emitted sound pulse of the collapsing bubble is so strong that little fish or shrimp get stunned or even killed by it and are then eaten up by the snapping shrimp. Our explanation immediately solved another paradox: Why aren't there any snapping shrimp in the deep ocean, but only in water up to about 50 m depth [70]? The reason is that the hydrostatic pressure increases with increasing depth so that eventually the shrimp can no longer generate a cavitating bubble. Thus it would starve in deeper water.
With our background in single bubble sonoluminescence, we could not resist looking into possible light emission from the shrimp-produced collapsing bubbles: Indeed, there was a faint light emission, a phenomenon we called shrimpoluminescence [71]. Sound emission from oscillating or collapsing bubbles is not only relevant for snapping shrimp, but also on a much larger length scale. Eq. (4) is not only valid in the micrometer or millimeter range, but can even be employed as an estimate in the meter range, implying that the resonance frequency of a bubble with an ambient radius R 0 = 1 m is 3 Hz. Such extremely low frequencies are of great interest, for example, in the oil industry for acoustic marine geophysical exploration. The giant bubbles are generated with so-called airgun acoustic sources. Their operating principle is as follows: During the charging process, air is put under very high pressure in a cavity in the airgun. This air is laterally released during the discharging process, leading to a giant air bubble with typical diameters of one meter and beyond. The oscillations of this air bubble leads to sound generation and emission, according to eq. (5). The sound gets reflected at the ocean-bottom and is detected both with hydrophone streamers and with ocean bottom seismometers and hydrophones, see figure 9. To extract the relevant information from these data, low frequencies around 1 Hz are crucial, whereas high frequencies beyond 120 Hz are attenuated in the earth. This is the reason why it is essential to generate low frequencies with the airgun (or airgun clusters) and thus employ big bubbles.
Bubbles of this size will no longer be strictly spherical during their whole period of life, and this will affect their sound emission behavior, both with respect to intensity and direction. The way to calculate the dynamics of such large bubbles or voids from the Navier-Stokes equation (or, to be precise, from its potential flow approximation) are boundary integral methods, in the context of fluid dynamics again pioneered by Andrea Prosperetti [72]. With such methods one can optimize the sound emission in the low frequency range. This is not restricted to single bubbles, but one can also analyse bubble clusters and their collective sound emission, either in 2D by employing some symmetry, or fully 3D. Presently, together with the oil company Shell, who contacted us on this problem, we are pursuing such calculations.

V. IMPACT ON LIQUIDS AND ON "SOFT" SAND
We had started to work with boundary integral methods in the context of impact events. What had triggered us was again the above mentioned pioneering paper by Prosperetti, Crum and coworkers on rain drops falling on the ocean [69,72], leading to bubble entrainment and sound emission. Apart from the question on sound emission, another important question to ask is: How much air ends up in the water? The answer is relevant, for example, in climate models, in which models for the atmosphere are coupled to those of the ocean. To answer this question, Prosperetti and coworkers had employed boundary integral methods [72][73][74], finding very good agreement with the experimental results.
In the late 1990s and in the first two decades of this century, the development of digital high-speed cameras has boomed, ever increasing in frame-rate, resolution, and storage, and lowering in price considerably. That gave us the opportunity to look into the impact events and the subsequent void collapse in more detail. This line of research was also triggered by single bubble sonoluminescence, namely to analyze in detail the hydrodynamic singularity at collapse; here not the spherically symmetry bubble collapse, but the axially symmetric void collapse. In fact, mathematically the collapse of the void formed after impact can approximately be described by a two-dimensional Rayleigh equation [75][76][77][78][79] analog to the 3D Rayleigh equation (1), which has been so successful in describing the collapse of the sonoluminescing bubble. The 2D version of the inertial part of the Rayleigh equation reads RR +Ṙ 2 = 0, with the singularity solution R(t) ∝ (t s − t) 1/2 . However, the collapse of a void emerging at impact is not purely 2D and correction terms emerge. Indeed, experimental studies have found that the exponent of the power law is higher than 1/2 (typical values found are 0.54 -0.60) [77][78][79][80][81][82] and theoretical studies have shown that the exponent indeed has a weak dependence on the logarithm of the remaining collapse time, approximating to 1/2 only asymptotically at the end [83][84][85].
To analyze these questions in a controlled way, rather than letting droplets or spheres fall on a water surface, we pulled a disk with controlled velocity V (defining the Froude number F r = V 2 /(gR disk ) as dimensionless control parameter) through the air-water interface and performed high-speed imaging of the void collapse and jet formation [79,82,85]. Figure 10a shows how the cavity develops and then collapses due to the hydrostatic pressure from the side. At singularity two jets emerge (figure 11): one upwards straight into the air, the other downwards into the developing bubble. Just as in 3D the focusing power of the collapsing (sonoluminescing) bubble is converted into sound and light emission (and of course heat), in 2D this focusing power is converted into the jet formation. As one can see from figure 10a, excellent agreement between experiments and the boundary integral simulations can be achieved. Even the effect of the air flow can be included [87], which reverses during the collapse from downwards during void formation to upwards during void collapse, leading to a supersonic air flow out of the closing void. 10. (a) A disk (diameter 6 cm) is pulled through an air-water interface with constant velocity of 1m/s. The emerging void is collapsing due to the hydrostatic pressure. The photos are taken 88ms, 115ms, and 131ms after the impact of the disk, the solid line results from a boundary integral calculation without any free parameter and shows excellent agreement with the data. The pictures are taken from ref. [79]. (b) Here the pulled impact disk (diameter 4 cm) has a small azimuthal asymmetry with a mode m=20 and an amplitude of 4%. Again, the impact velocity is 1 m/s. The three snapshots show the evolution of the shape distortions. Taken from ref. [86].
FIG. 11. A comparable process as in figure 10 at a later stage: Two jets have developed at the singularity: One upwards and one downwards into the bubble. This photo is taken from ref. [14].
The analogy between the 3D bubble collapse and the 2D void collapse goes so far that even the shape stability can be analysed in one-to-one analogy, as again first done by Prosperetti [88]. Later, we extended this analysis both theoretically and experimentally to find the shape of a collapsing non-axisymmetric impact-created air cavity in water [86], see figure  10b.
To proceed to even larger scale: How comparable is the impact of a ball on a water surface with that of an asteriod on the surface of a planet? We downscaled such an astroid impact to lab-scale by fluidizing very fine sand. Before the impact event, the air-flow is turned off. The first reason for the fluidization is to create reproducible conditions. Second, the fluidization implies that the energy stored in the ground decreases by orders of magnitude, corresponding to the much smaller kinetic energy of the falling ball in the lab as compared (a-f ) Impact (at t=0 s) of a steel ball (diameter 2.5 cm) on soft, decompactified sand (grain diameter typically 40 µm). The splash and the jet emerge, just as in water. The grains in the jet cluster due to their inelastic collisions. The last frame shows a granular eruption caused by the rising air bubble. (g) A cross section of the void collapse following from the axially symmetric Rayleigh-type model of ref. [76]: The void is pressed together by the hydrostatic pressure from the side, leading to a singularity and an upward and downward jet. Both series of images are taken from ref. [76].
to that of an impacting astroid. The idea is to achieve similar Froude numbers and Newton numbers (the ratio between yield stress of the surface and kinetic energy of the intruder) as in the geophysical event, hoping for similar dynamical behavior.
Indeed, the phenomena of the impact of a ball on such prepared sand turned out to be very comparable to those of the impact on water [76]: First a splash is formed and then a jet develops (figure 12), just as in the water case ( figure 11). Even the bubble, which forms in water, again develops and slowly rises, finally, when hitting the sand-air interface, causing a granular eruption. It however also turned out that the air in between the sand grains has a major role in the emergence and intensity of the jet [89][90][91], namely, when prior to the impact event (partially) evacuating the air from the container with the sand, the jet is much less pronounced as the impacting object can intrude less deep, leading to a weaker "hydrostatic" collapse. An excellent review on impact of objects on granular beds can be found in ref. [92]. Droplet distortion (droplets 28-29) caused by a dirt particle around the jetting nozzle. Such a dirt particle can lead to bubble entrainment. In the shown case here the normal droplet formation process recovers and becomes regular again. Figure taken from ref. [93]. (c) Upper: Infrared bottom view through the silicon around the nozzle into the ink channel. In the center the nozzle is seen. In three of the four corners bubbles got entrained which affect the printing process. The nine smaller images show the diffusive dynamics of the bubbles, clearly revealing bubble growth (by rectified diffusion) and Ostwald ripening. The times are given in seconds. Images taken by Arjan Fraters, Physics of Fluids group, Twente, in collaboration with Océ.
developing piezo-acoustic ink-jet printers and obviously various great fluid dynamics challenges come with this, starting from the flow in the nozzle, to the jetting and droplet process, down to drop impact, drop spreading and drop-paper interaction, and finally (partial) drop evaporation or solidification. One of the most burning questions Océ had in those days was on the fluid dynamics in the print-head.
A schematic cross-section through a typical modern MEMS-based print-head is seen in figure 13a. The piezo-actuator -similar to the piezo used to generate single bubble sonoluminescence -gives short pulses with -depending on the printer -a frequency of around 20 kHz to 100 kHz. Each pressure pulse drives out a little droplet. Altogether, typically there are presently hundreds of nozzles within one silicon chip and several chips are integrated in one print-head. A modern inkjet printer has tens of such print-heads. Hans Reinten then told me that unfortunately this very fast and precise printing facility can break down from time to time: After billions of cycles a distortion of the droplet formation in a channel can develop. This distortion either vanishes after a short time (figure 13b), or the jetting process of that channel eventually completely breaks down. The only solution then is to turn off the piezo and wait for a minute or so, which of course is extremely annoying for a high-speed printer (though other nozzles might take over during that time). The suspected culprit for the problem was a bubble within the inkjet channel. But how does the bubble get there and what is its dynamics? And how to avoid this trouble?
To solve this problem, we employed the same method as for the snapping shrimp [68] or the entrained bubble at water droplet impact [69]: watch and listen. I.e., we measured the acoustic response of the channel and combined it with high-speed imaging. Indeed, we found that the distortion of the droplet is correlated with a modification of the acoustic response of the channel. This result indeed suggested that the distortion originates from a bubble, because bubbles modify the acoustical behavior of the channel [94,95]. In fact, we were even able to "hear" the size of the bubble, see figure 14. But how does the bubble get there? Is it nucleated or entrained at the nozzle?
By combining high-speed imaging for the inkjet and infrared imaging for the interior of the ink-channel, we succeeded to visualize how a bubble is entrained at the nozzle [93,96] and what are its dynamics inside the channel (figure 13c). Here small dirt particleseither on the nozzle plate or in the ink channel -play a crucial role. Once a tiny bubble is entrained at the nozzle, the acoustical forces pull it into the channel. Just as in single bubble sonoluminescence, the oscillating bubble then grows in the acoustic field by rectified diffusion. So the knowledge which we had acquired from single bubble sonoluminesence -namely on acoustical forces on a bubble and on rectified diffusion -was essential in solving this problem.
Once the bubble has grown by rectified diffusion to considerable size, the actuation pressure pulse to jet a droplet simply leads to bubble compression, and not to a pressure increase at the nozzle. Therefore, no droplets can be jetted any longer. Only after the acoustic pressure has been turned off, can the bubble dissolve by diffusion so that printing becomes possible again.
The final goal of course must be to avoid the entrainment of the bubble or to immediately get rid of it again, e.g., by applying an acoustical pulse immediately after the bubble has been detected. For the next step -the development of even faster printers with even smaller droplets -further fundamental work on the meniscus instability leading to the bubble entrainment remains essential. Bubble entrainment also turned out to be crucial for so-called immersion lithography, pioneered by ASML, the world's leading supplier of lithography systems for the semiconductor industry. Once the photolithography of wafers had been at the edge of optical resolution, even smaller structures were realized by introducing lithography machines using immersion technology, i.e., lithography under a film of water (with a refractive index of 1.33, rather than 1.00 for air) between wafer and optical lens, see figure 15a, allowing for smaller structures on the wafer than standard lithography in air. However, the development of immersion lithography machines has been hindered by two fundamental fluid dynamical problems, namely, the entrainment of bubbles into the water and the loss of water from the film, see fig. 15b. Both of these contact line instabilities have challenged the further development of immersion lithography, as obviously the entrainment of light-scattering bubbles into the film or the partial loss of the film is unacceptable for the lithography process. These hydrodynamic instabilities set in at a certain velocity with which the wafers under the lens are pulled away. It is thus this hydrodynamic instability which sets the production rate of the wafers and therefore the price of the lithographic system.
In the context of this problem, together with ASML we looked at the entrainment of bubbles under droplets impacting on a solid substrate, following pioneering work of the Chicago, Harvard, Kaust and other groups [98][99][100][101][102][103] (for recent reviews, see refs. [104,105]). By combining high-speed color interferometry [106], scaling arguments, and numerical simulations with the boundary integral method for the droplet coupled to a viscous lubrication approximation for the gas flow in the thin and narrowing gap between impacting droplet and substrate, we found that there is an optimal velocity for maximal bubble entrainment [97]: For lower velocities the impacting droplet remains more spherical (capillary regime) and for higher velocity (inertial regime) the drop is smashed against the surface so much that not much air can be entrained, either, see fig. 15c. Obviously, this work is also of interest for the inkjet industry and the coating industry.

VII. FROM SURFACE BUBBLES TO SONOCHEMISTRY & ULTRASONIC CLEANING
As stated above, single bubble sonoluminescence can be seen as the hydrogen atom of bubble fluid dynamics. But atomic physics did not stop with the understanding of the hydrogen atom but moved ahead to more complicated and interacting atoms, molecules, and condensed matter. So it was also our desire to better understand interacting bubbles, but in the most controlled way, with fixed distance. To achieve this acoustically is difficult, as acoustically driven microbubbles either repel or attract each other through the so-called Bjerknes forces of the second kind [1,109], depending on their size and the driving pressure. If trapped by the primary Bjerknes forces in different pressure antinodes of the acoustic field, they are too far away (7.5 cm corresponding to the acoustic wavelength for f = 20 kHz in water) from each other to considerably interact. Therefore we came up with the idea to trap the bubbles through pinning forces to hydrophobic micro-machined microcavities which act as gas traps when the substrate is immersed in water [107,110], see figure  16a,b. The patterning of the substrate allows us to control the number of bubbles and the distance between them. Each hemispherical bubble experiences the effect of its mirror image. Correspondingly, an isolated hemispherical bubble together with its mirror image behaves like a free spherical bubble, i.e., its dynamics is well described by the Rayleigh-Plesset equation (1). By putting the micro-cavities close to each other, we could study interacting microbubbles, either in a row (i.e., in 1D) as in figure 16b, where we could compare the results with boundary integral simulations (i.e., under the potential flow approximation) with axial symmetry, or in 2D with bubbles arranged on a surface in any order.
In fact, by varying the diameter of the microhole down to 100 nm and less, we could quantitatively test [108] the crevice model of bubble nucleation [111][112][113][114]. Figure 16c compares the nucleation threshold calculated from the crevice model [114] with the experimental data, finding good agreement. We also used such hydrophobic micro-machined pits as artificial crevices for bubble nucleation to achieve higher sonochemical yields at ultrasound powers that would otherwise not produce a significant chemical effect [115] and for ultrasonic cleaning purposes [116], in both cases making use of the energy focusing power of the collapsing bubbles. Out of this activity another spin-off company emerged from our group. Finally, with such pits we enhanced the heat flux in thermal convection by vapor-bubble nucleation [117].

VIII. FROM SURFACE NANOBUBBLES TO CATALYSIS AND ELECTROLYSIS
Being interested in tiny surface bubbles, the so-called surface nanobubbles caught my attention, which from about 2000 on have been found in atomic force microscopy (AFM) images of water-immersed, preferentially hydrophobic substrates [118][119][120], see figure 17a. One of course immediately wonders why such surface nanobubbles are stable. Because of the diverging Laplace pressure p Laplace = 2σ/R, where σ is the surface tension and R the radius of curvature, tiny bubbles should dissolve immediately: In water the gas pressure inside a bubble with R = 10 nm is p gas = P 0 + p Laplace ≈ 145 atm. With Henry's law this translates to a gas concentration at the edge of the bubble c R = p gas c s /P 0 which is 145 times larger than the saturation concentration c s and therefore to a large concentration gradient away from the bubble, leading to very fast dissolution. This even holds in the case of (slight) gas oversaturation, i.e., as long as the gas oversaturation is not too large. In the above mentioned classical paper by Epstein and Plesset [20], known to us from our work on single bubble sonoluminescence, the dissolution time of such a bubble had been calculated from the diffusion equation and the corresponding boundary conditions. The result for the typical dissolution timescale is τ EP = R 2 0 ρ g /(2c s D), where D is the diffusion constant and ρ g the gas density. For the bubble with R = 10 nm and the material constants for water, one indeed gets τ EP ≈ 3 µs, i.e., such nanobubbles should dissolve basically immediately. The surface nanobubbles in the AFM image figure 17a were generated with the so-called solvent exchange process, which had been pioneered by Xuehua Zhang [122,123]. Here, a gas-saturated liquid having high gas solubility (e.g., ethanol) is replaced by another liquid with lower gas solubility, e.g. with water. This leads to a local gas supersaturation ζ > 0 and therefore to the nucleation of bubbles. Macroscopically, we know this effect of everyday life: When we fill a water glass with cold tap water and leave it for a while in a warm room, small air bubbles form on the inside of the glass. The reason is that in general tap water is oversaturated with air and gases in cold water dissolve much better than in warm water. If the tap water in the glass slowly warms up to room temperature, the gas solubility is reduced and bubbles nucleate on the edge of the glass. Depending on the size of the glass, the bubbles last about four days, as anyone can easily try.
First it was speculated that the stability of surface nanobubbles is due to surfactants [124], which however for various reasons (explained in ref. [121]) could be ruled out. Moreover, with the help of fluorescence lifetime imaging microscopy it could be shown [125], that the objects observed with AFM are indeed air bubbles, and not nanodroplets of a contaminating liquid. In addition to the remarkable stability of the nanobubbles, another paradox was that their contact angle (measured on the bubble side) was not Young's angle, as known from macroscopic measurements, but much smaller [126,127].
When there is such a large gap between experiment and theory, numerical simulations often help. We therefore performed MD (molecular-dynamics) simulations of surface nanobubbles [128], and at least those adhered to the theoretical expectation: they dissolved in microseconds.
The key to solving this paradox came from the experiments of Xuehua Zhang and coworkers [129] who observed that surface nanobubbles exposed to gas-undersaturated water (ζ < 0) dissolve slowly, but (initially) not by reducing their lateral size, but by reducing their contact angle (on the gas side), see Fig. 18a: The three-phase contact line remains pinned. This dissolution mode is called "CR-mode", standing for constant contact radius, in contrast to the so-called "CA-mode", standing for constant contact angle [130,131]. Pinning dramatically changes the dissolution scenario: the Laplace pressure p Laplace = 2σ/R now no longer diverges, but approaches zero, see Figure 18b. Thus, when the bubble dissolves, no large internal pressure can build up and thus no concentration gradient from the outside of the bubble to the predetermined concentration level c ∞ far away from the bubble: The bubble becomes stable.
The reason for the pinning lies in the unavoidable surface inhomogeneities of geometric and/or chemical nature. These are also relevant in the above mentioned daily life phenomenon of bubble formation in a glass with cold tap water which warms up or when we pour soda water into a glass: In both cases bubbles nucleate out of oversaturated water on such inhomogeneities. Macroscopically, the surface inhomogeneities lead to contact line hysteresis, as studied extensively by de Gennes and co-workers in the 1990s [132,133].
We could generalize the classical Epstein-Plesset calculation [20] to calculate the diffusive dynamics of pinned surface bubbles [134]. Here the key idea was to adopt the quasistatic calculation of Popov [135] for the so-called "coffee stain problem" [136,137], which is on the evaporation of a liquid drop on a plain substrate. This is not surprising, because both processes are controlled by diffusion outside of the drop / bubble: in the evaporating droplet case, of diffusion of water vapor in air, and in the surface nanobubble case, of air into water. For bubbles with constant contact diameter L (i.e., in the CR-mode), the result of this adopted (quasistatic) calculation reads [134] with a positive definite f (θ) given in ref. [135] and a critical lateral extension L c = 4σ/P 0 ≈ 2.84µm for air bubbles in water with 1 atm ambient pressure. From eq. (7) it immediately follows that for gas undersaturation −1 ≤ ζ < 0 no stable surface nanobubbles can exist, as then the right-hand side of eq. (7) is always negative: the bubble dissolves down to θ = 0. For gas oversaturation ζ > 0, however, a stable equilibrium with the equilibrium contact angle [134] sin θ e = ζ L L c (8) unpinned: "CA-mode" pinned: "CR-mode" 18. (a) The nanobubbles can dissolve in two different "modes" (the time direction is indicated by the arrow): In the CA mode (constant contact angle, above), the contact angle is constant. As a result, the radius of curvature becomes smaller and smaller, which leads to the divergence in the Laplace pressure, as can be seen in (b) (red curve), where we show ∆p = p Laplace as a function of the bubble volume V. In the CR mode (constant contact radius, bottom), the contact radius is constant and the contact angle becomes smaller. As a result, the radius of curvature increases in the course of the dissolution process and the divergence in Laplace pressure does not occur (blue curve in (b)). In (c), we show a sketch of the phase space for the stable equilibrium, which results in "pinning" (CR-mode, bottom), with the equilibrium contact angle θ e , given by equation (8)), and a sketch of the phase space for unstable equilibrium without pinning (CA-mode), in which the surface bubble either shrinks or grows. In (d) the used notation is introduced: L is the lateral extent of the bubble at the substrate (contact diameter), H is the bubble height, θ is the contact angle at the gas side, and R the radius of curvature.
can exist. That the equilibrium indeed is stable is seen from the phase space figure 18c, bottom. In this stable equilibrium, Laplace pressure (causing gas flux out of the bubble) and gas overpressure (causing gas influx) are in balance.
From equations (7) and (8) we also see that for too large oversaturation ζ > L c /L there is no stable equilibrium and the surface bubble keeps on growing so that it will finally detach. The condition L < L c /ζ is the reason that there can be stable surface bubbles only on a microscopic scale. The stabilization mechanism does not work on a macroscopic scale, because then the Laplace pressure is too weak and cannot compensate for the gas overpressure from the outside. We also see that on the microscopic scale, the radius of curvature of a surface bubble is not given by Young's equation but by the relationship (8). We note that eq. (8) and the stability of the equilibrium have also been confirmed both by numerical simulations of the full diffusion equation [138], employing immersed boundary methods for the growing or shrinking bubble, see figure 19, and in addition also by MD simulations -but now with built-in pinning [139], see figure 20. As we see, the size L of the pinning site and the oversaturation ζ > 0 determine the stability of the surface nanobubbles. But how is the oversaturation ζ determined? When the liquid container with the surface nanobubbles on some substrate is closed and in equilibrium, ζ remains constant and hence the equilibrium contact angle θ e , Eq. (8). In an open vessel, on the other hand, an initial gas oversaturation ζ > 0 will not last long due to diffusive processes with the outside world [140]. If the distance to the outside world is , the typical diffusive time scale is τ outer ∼ 2 /D. For = 1 cm we get τ outer ≈ 14 hours and for = 3 cm τ outer ≈ 5 days: This is exactly the time scale that we also observe for the dissolution process of air bubbles which nucleate at the edge of the above mentioned glass filled with cold, gas-supersaturated tap water when put into a warmer room. Just as single bubble sonoluminescence can be seen as the hydrogen atom of inertial bubble dynamics [15], a single surface nanobubble can be seen as the hydrogen atom of diffusive bubble dynamics. Its properties -in particular its on first sight surprising stability and its small contact angle -are meanwhile reasonably well understood, see our review article ref. [121]. As a next step, we have moved towards diffusively interacting surface nanobubbles. Just as for inertial bubbles (section VII and figure 16b), one can best study also the diffusive behavior of surface bubbles by fixing their distance by offering "weak spots" on a hydrophobic surface, namely by micro-machining the surface: During the solvent exchange, the bubbles will nucleate in the cavities and grow [141]. In fig. 21a we show snapshots of this growth process over a period of three minutes. Note the very different timescale as compared to the The color scale corresponds to a height of 0µm (blue) to 0.9µm (red) of the droplets. The dissolution process was stopped by polymerisation of the HDODA by UV-radiation, so that a visualization of the droplets by AFM became possible, a trick introduced in ref. [146]. Figure taken from [147].
inertial bubble dynamics of figure 16b, where the whole series of snapshots is less than 25 µs. For interacting surface nanobubbles, according to above sketched theory [134] one would expect that in equilibrium all surface nanobubbles would have the same radius of curvature R e = L c /(2ζ). On first sight, one may expect that this equilibrium is unstable due to Ostwald ripening of the bubbles [142]: Small bubbles shrink due to their larger Laplace pressure and neighboring larger ones grow. However, it turns out that again it is pinning which stabilizes these neighboring bubbles against Ostwald ripening [143,144].
As stated above, a central idea of solving the surface nanobubble paradox originated from the analogy to the pinned coffee stain [136]: The first problem is controlled by diffusion of air in liquid, the latter one by diffusion of vapor in air. In between these two cases is the one of a liquid surface nanodroplet (see figure 17b) in a sparsely miscible host liquid, for which the stability (with pinning, or lack thereof, without pinning) is given by the same equations and mechanisms [121].
Not only the dissolution processes of such surface nanodroplets (see figure 21b) are completely analogous to those of surface nanobubbles, but also the nucleation and growth processes. In particular, it is again the solvent-exchange process that allows surface nanodroplets to be generated in a controlled manner [146,148]: Now a high solubility liquid saturated with a certain substance is replaced by another with less solubility. The emerging oversaturation leads to droplet nucleation.
As an example, consider the Greek liquor "Ouzo" (alternatively Pastis (France) or Sam-buca (Italy) or Raki (Turkey)): Ouzo consists of alcohol, water and anise oil, which is very soluble in alcohol. When served, the ouzo is diluted with water. As a result, the alcohol concentration and thus the solubility of the oil decreases: Oil droplets then nucleate and grow in the supersaturated solution. This leads to the known milky appearance of the ouzo-water mixture. In the presence of a hydrophobic surface, nucleation naturally takes place at this surface and surface nanodroplets are formed in complete analogy to the surface nanobubbles. With the help of the control techniques of microfluidics, one can study very precisely the dependencies of the nucleation process on the control parameters such as flow velocity, channel thickness, oil concentration, size and shape of the pinning sites etc. and understand them theoretically and quantitatively [148][149][150].
The technological relevance of surface bubbles and droplets and the solvent-exchange process can hardly be underestimated. In our review article [121] we have dedicated a whole chapter to it; here I restrict myself to some particularly relevant and beautiful examples.
Liquid-liquid extraction -the transfer of a substance from one solvent to another -is one of the key processes of chemical technology and analysis. Since the work of Nobel Prize winner Fritz Pregl [151], efforts have been made to further miniaturize processes for chemical analysis and chromotography. The need to do this has increased in recent years [152]. First, it is important to be able to demonstrate ever smaller traces of chemical substances, be it in a medical context or for environmental reasons. Moreover, this proof must be fast and small samples should be sufficient, also to come to a "greener" chemical technology. One way to achieve this is single drop microextraction, where a substance A dissolved in water slowly dissolves in a drop of an organic solvent with a much higher solubility for A (e.g. carbon tetrachloride) held in the water. The drop is then extracted and analyzed. However, this process generally takes a long time and the efficiency is low.
These limitations were elegantly overcome with the so-called disperse liquid-liquid microextraction (DLLME) technique [153][154][155] invented just 12 years ago. The technique involves a third liquid that is miscible with water and with carbon tetrachloride, e.g., ethanol. If a mixture of carbon tetrachloride and ethanol is now added to the water containing the substance A to be extracted, instantaneously many nano-and microdroplets of carbon tetrachloride nucleate, due to the enhanced water concentrations these molecules experience and the resulting lower solubility. With their very large total surface area, these nano-and microdroplets are very efficient to extract the substance A. As a final step, the carbon tetrachloride microdroplets are centrifuged off. The highly enriched substance A can then easily be detected by conventional methods. So far, DLLME is mostly optimized by trial-anderror; however, to achieve further control and progress, from my point of view it is very promising to apply the well-controlled methods of microfluidics and fluid dynamics to this process [156].
A good understanding and control of the diffusive processes in and around microdroplets and microbubbles may even save lives: in open-heart surgery, the blood stream is decoupled from the heart and maintained by a heart-lung machine [157] and at the same time the blood is cooled down. In the cold blood more air dissolves than when it is warm. However, warming up the patient after surgery can be seen as a solvent-exchange process that can result in the nucleation of (surface) nano-and microbubbles that may significantly affect the blood supply locally. Here, too, it is essential and possibly life-saving to well understand bubble nucleation in diffusive processes.
Further examples are electrolysis and catalysis: In both processes, the forming surface nano-and microbubbles considerably impair the efficiency, as they cover the electrode or catalyst. How does one control the growth of the bubbles and how to get rid of them?
We have taken some first steps to answer these questions [159][160][161][162]. Figure 22a shows catalytically grown oxygen bubbles on a platinum surface immersed in a hydrogen peroxide (H 2 O 2 ) solution. The platinum acts as a catalyst which triggers the decay of the peroxide to oxygen (evolving in bubbles) and water. For such bubbles evolving out of chemical reactions the relevant dimensionless number coming into play is the Damköhler number Da, which is the ratio between the chemical reaction rate and the (diffusive or convective) mass transport rate. If Da is very large, the chemical reactions is controlled by the latter.
An example of a carbon dioxide bubble which is purely growing by diffusion is shown in figure 22b. As discussed above, according to the Epstein-Plesset theory [20], for such bubbles the radius should grow in time with a square-root behavior. As we found in ref. [158], in the beginning it indeed does, but later on convective effects take over, as the dissolved carbon dioxide affects the density of water. We also found that in the long term gas depletion effects become very relevant and bubbles nucleating later at the same location grow much slower than the earlier ones [163]. After growth, the bubbles finally detach. This happens because of buoyancy, either directly once they are so large that the buoyant forces overwhelm the capillary forces (the so-called Fritz radius [75,164]), or when they coalesce and then detach, as shown in figure 23. In the context of catalysis and electrolysis, the bubble detachment is of course advantageous, as pointed out above.
The area of electrolysis and catalysis seems to me to be a very important and promising future field for physics of fluids, building a bridge from fluid dynamics to process technology and (colloid) chemistry.
FIG. 23. Snapshots of the coalescence of two diffusively grown gas bubbles pinned to 50 µm radius pits. When they touch and coalesce, both bubbles have a diameter of 600 µm. Three main stages can be identified: neck formation between the two bubbles, images (a-d); propagation of the capillary waves along the upper and lower half-surface of the coalescing bubble and their final convergence at its side, images (e-l); and final detachment, upwards jumping, rising and oscillation with the Minnaert frequency, images (m-o). Adopted from ref. [165].

IX. EFFECTIVE BUBBLE FORCE MODELS, DISPERSED BUBBLY FLOW, AND BUBBLE DRAG REDUCTION
I now come back from the predominantly diffusive bubble dynamics to further inertial bubble phenomena. As explained in section II, the acoustic forces ("Bjerknes forces") are the reason why the bubble is pushed towards the pressure antinode where it can oscillate in the strongest way, possibly even leading to single bubble sonoluminescence [15]. In general, these acoustic forces will compete with inertial forces such as the added mass force, drag, and lift, and of course with buoyancy. In 2001, we aimed to understand this competition in more detail, in order to gain more insight into the hydrodynamic forces on bubbles such as drag and lift (which had very nicely been reviewed by Magnaudet and Eames [166]) and in order to better control the motion of bubbles in a shear flow with the help of an external acoustic field. We thus built an experimental setup with an acoustic spherical glass resonator through which we pumped some liquid in a controlled way, thus generating controlled shear flow [167]. Remarkably, for certain parameters, the entrapped bubbles performed a spiralling motion, out of which we could extract various pieces of information on the drag and in particular on the lift. The work -and follow up work in similarly controlled geometries such as a rotating horizontal cylinder [168,169] -also confirmed the usefulness of the concept of effective bubble forces, even for deformed or oscillating bubbles. In figure 24, for a bubble spiralling in the rotating horizontal cylinder [169], we show the forces acting on the bubble (a), the spiralling trajectories for certain parameters (b), and the extracted rotational lift coefficients C L , which remarkably can take both negative and positive values (c) [169]. In a particularly counter-intuitive example, employing the Basset history force as a non-local-in-time effective force [170], we could explain how viscosity can destabilize the path dynamics of sonoluminescing bubbles [171], a phenomenon discovered by Suslick and coworkers for sonoluminescing bubbles in glycol and called moving-SBL [172]. The unstable, chaotic path-trajectory of such a bubble is shown in figure 24d.  [169], to which we refer for more details. (d) y-position and three-dimensional trajectory of a moving SBSL-bubble in N-methylformamide [172] from a dynamical model based on effective forces (including the history force), for the same parameters as in experiment [172], where similar trajectories are observed. Taken from ref. [171].
Encouraged by these successes, we [175,176] dared to transfer Maxey and Riley's seminal idea of using effective forces on small particles to describe their (collective) dynamics in dispersed multiphase flow with point-like particles [173] to turbulent dispersed multiphase flow [177][178][179] with point-like bubbles, following refs. [180][181][182][183][184][185][186][187][188][189][190]. The motion of the small and thus non-deformable bubbles of radius a, volume V b = 4πa 3 /3, velocity v(x, t) and embedded in a velocity field u(x, t) was modelled by (see e.g. [166,180,187,188]) and each bubble at position y(t) exerted a δ-type force on the liquid flow [176], Here, ρ f is the liquid density and ρ g ρ f is the (low) gas density. The terms on the right-hand side of eq. (9) represent buoyancy, drag, added mass plus fluid acceleration, and lift, where C D , C M and C L are the corresponding coefficients, which are modelled. Though in such complicated turbulent flow situation in particular the lift coefficient is not exactly known, as an approximation -and in order to explore the effect of the lift force on the overall flow dynamics -we took the standard value C L = 1/2 [166] for small spherical bubbles in shear flow. Following this approach, we studied the effect of the bubbles on the turbulent energy spectra [176], the Lagrangian statistics of bubbles in turbulence [191] finding very pronounced intermittency of the bubble acceleration, and the clustering of bubbles in the vortices [174,192], see figure 25, in which we compare the distribution of bubbles with that of particles in dispersed multiphase flow. We also extended the effective bubble force idea to vapor bubbles in turbulent flow which can shrink and grow thanks to evaporation and condensation [193][194][195]. In this way we could reveal the physics of the considerable enhanced heat transfer in boiling thermal convection as compared to normal thermal convection. Two enhancing effects compete: Enhancement due to additional mixing by the rising vapor bubbles, and enhancement due to the bubbles as directed carriers of (latent) heat from the boiling bottom plate to the cold top plate. Due to the many modelling assumptions a quantitative comparison with experimental data [117] however remains difficult.
One of the most remarkable features of dispersed bubbly flow is that a bubble addition of only a few percent leads to remarkable drag reduction of up to 80% [197,198], with great potential for naval applications [199] (see figure 26a), as it can lead to significant reduction of  the fuel consumed by ships without adding substances into water. As far as I can remember, I myself first heard of this phenomenon at an APS DFD meeting in a talk of Kazu Sugiyama, then a postdoc in the group of Yoichiro Matsumoto and Shu Takagi in Tokyo/Japan. For excellent recent reviews on bubble drag reduction I refer to refs. [200][201][202]. Despite all efforts done to understand the fundamental mechanisms behind this effect, a solid understanding of the drag reduction mechanisms occurring in bubbly flows has still been missing. To achieve such an understanding, we wanted to study bubble drag reduction under the well-defined conditions of turbulent Taylor-Couette (TC) flow, the flow between two coaxial co-or counter-rotating cylinders. For recent reviews on low-Reynolds number and high-Reynolds number TC flow, we refer to ref. [203] and ref. [204], respectively. The advantage of this canonical flow of physics of fluids is that it takes place in a closed system and that both global measurements (the overall torque required to drive the system, i.e., the drag) and local measurements (by PIV or LDV) are possible. In a series of papers on turbulent bubble drag reduction [196,205,206], we could indeed measure major bubble drag reduction in this TC system, namely in the strongly turbulent case up to 40% with only 4% volume fraction of the bubbles, see figure 26b, and in addition characterize bubble concentration and velocity profiles. The measurements were consistent with the conjecture that it is the bubble deformability that is responsible for the strong drag reduction, i.e, we only found substantial drag reduction for bubble Weber numbers We larger than 1 [196]. These findings were in line with the results from numerical simulations by Gretar Tryggvason's group [207], employing the front-tracking technique. The modification of the lift force on the bubbles due to their deformability (for We > 1) seems to play a central role in the mechanism and the bubbles arrange themselves in between the turbulent bulk and the boundary layers, blocking the momentum transfer and therefore reducing the overall drag.
How to further validate this hypothesis? If the hypothesis were correct, one should change the bubble deformability during the experiment and then the drag should change on the spot. But how to achieve this? In 2010 I had visited the group of Yoichiro Matsumoto and Shu Takagi in Tokyo and saw their impressive huge bubble column experiment, in which the addition of a few drops of surfactants (Triton X) immediately changed the flow characteristics and bubble size, see ref. [208]; for a recent review of this phenomenon, see [209]. The main reason is that the Triton X surfactant prevents bubble coalescence, and that therefore in strongly turbulent flow in which bubble splitting is omnipresent, the average bubble size will dramatically decrease after the addition of Triton X.
We decided to apply the same trick in turbulent bubbly TC flow, and indeed found that also here the addition of a few drops of Triton X (to 111 liter of water in our TC setup [210]) dramatically changed the flow [211], see figure 27, in spite of the unchanged bubble volume fraction: Optically ( fig. 27a-f), because the flow became very opaque, just as in ref. [208], but in particular also mechanically ( fig. 27g-h), as the effect of bubble drag reduction nearly vanished. The combination of these local and global findings strongly supports the hypothesis that it is the bubble deformability which is relevant for strong turbulent bubble drag reduction.
How to numerically model turbulent bubble drag reduction? Given that the mechanism is bubble deformability, which requires to resolve the gas-liquid interfaces of the bubbles and given the large Reynolds numbers, this is extremely challenging. With the front-tracking technique, Tryggvason and coworkers could simulate a few large bubbles in a "minimum turbulent channel", indeed qualitatively finding bubble drag reduction due to deformability  [214], the numerical simulations of Sugiyama et al. [215], and the numerical simulations by Spandan et al. [216]. One clearly sees that for these low Reynolds numbers the drag reduction is decreasing with increasing Reynolds numbers. (b-d) Corresponding averaged azimuthal velocity contours with point-bubbles included. Re i = 2500 and single phase flow is compared with two different Froude numbers F r = 0.16 (strong buoyancy, weakening the Taylor rolls considerable) and F r = 1.28 (weak buoyancy). In the latter case, the Taylor rolls are hardly affected by the bubbles which get trapped in them. Figures taken from ref. [216]. [207]. Also his later work with more bubbles confirmed this interpretation [212,213].
For the numerical simulations of bubble drag reduction in TC flow, we started off moderately, namely for relatively small Reynolds numbers (based on the inner cylinder rotation, the outer cylinder was at rest) Re i < 5000, for which the flow is not yet turbulent. Note that these Reynolds numbers are more than two orders of magnitude smaller than what I discussed in figures 26 and 27. Interestingly enough, also in that low Re-regime Murai et al. [214] had experimentally found bubble drag reduction, namely by microbubble injection, see figure 28a. But due to the lack of bubble deformability (We < 1 for the injected microbubbles), the physical mechanism in that regime must be very different. The good news is that this regime is accessible to numerical simulations, and even with effective force models, as the microbubbles can be considered as point-like, with W e < 1 and a diameter d b < 10η, where η is the (global) Kolmogorov length of the flow.
Our numerical simulations of bubbly TC flow at these low Re i < 5000 and with point-like bubbles [215,216] indeed resulted in bubble drag reduction in quantitative agreement with Murai's data [214], see figure 28a. In contrast to the bubble drag reduction in the large Reynolds number regime, which increases with increasing Reynolds number (figure 26b),  Reynolds numbers Re i from a few hundreds up to more than 10 6 . The left block of data are experiments and numerical simulations for microbubble drag reduction from refs. [214,215,222,223], and the right block of data are our large Re experiments with deformable bubbles [196,205,206].
in the low Reynolds number regime the effect of bubble drag reduction gets smaller with increasing Reynolds number (figure 28a).
As the relevant mechanism for bubble drag reduction in TC flow in the low Reynolds number regime, we could identify the considerable weakening of the Taylor rolls through the rising bubbles, see figures 28b-d. The crucial parameter in this low Reynolds number regime is therefore the Froude number (and not the Weber number as in the large Reynolds number regime), here defined as square root of the ratio between centrifugal force and buoyancy, F r = ω i r i /g, where r i is the bubble radius, ω i the angular velocity of the inner cylinder and g gravity. For small F r < 1 (figure 28c), buoyancy wins, and the bubbles rise through the Taylor rolls, thereby weakening them. This reduces the angular momentum transfer from inner to outer cylinder and thus the drag. For large F r > 1 (figure 28d) the bubbles simply get trapped in the Taylor rolls, without affecting them much, and the overall drag remains nearly unchanged [217]. For increasing Reynolds numbers the Taylor rolls more and more lose their flow dominance which explains why in this low Reynolds number regime the bubble drag reduction decreases with increasing Reynolds number (figure 28a).
The challenge for the next years will be to close the gap between the microbubble drag reduction experiments at lower Re of Murai et al. [214] and our large Re experiments with deformable bubbles [196], see figure 29. Our first step on this way was to include bubble deformability in point-particles by picking up an idea of refs. [218,219] and couple the deformation dynamics of ellipsoidal bubbles to the flow. Indeed, we found that bubble deformation helped the overall drag reduction [217]. In a second step [220], we coupled the well-resolved bubbles with an immersed boundary method to the Navier-Stokes equation, again finding that an increase in the bubble deformability (i.e., its Weber number) indeed implies larger drag reduction, see figure 30a. Profiles of the local angular velocity flux show that, in the presence of bubbles, turbulence is enhanced near the inner cylinder while attenuated in the bulk and near the outer cylinder. We connect the increase in drag reduction to the decrease in dissipation in the wake of highly deformed bubbles near the inner cylinder [220], as visualized in figure 30b,c. As in the closed TC system the total dissipation is proportional to the total torque [204], the decrease in dissipation implies a decrease in the overall drag. Bringing all these fundamental insights into bubble drag reduction towards applications in naval industry however still is a very long way to go and other concepts like air cavities may be more efficient [221]. We are presently collaborating with the Dutch Maritime Research Institute (MARIN) in Wageningen on this subject.

X. CONCLUSIONS
I hope to have demonstrated -along my own bubble trajectory -how fascinating and broad the fluid dynamics of a bubble is, from the nanometer scale to geophysical scales, with both outstanding fundamental questions and relevant applications. Clearly, the close interaction between experiment, theory, and numerics is crucial in physics of fluids and in particular in physics of bubbles, and makes this subject so particularly nice. Only a combination of all three methods gives real insight into the phenomena.
I also hope to have demonstrated with the chosen examples how crucial and essential for innovation of any kind fundamental research is. In what direction creative fundamental research will go is often unpredictable. If one ends up exactly where one had expected, the chance is large that the research was standard. What remains crucial is to be open, watch, and listen.
I want to close with some more general comments on fluid dynamics: From my point of view we presently live in the golden age of fluid dynamics: The reasons are that (i) Moore's law is kept on still being followed for the computational power, now making simulations possible of which even ten years ago we did not dare to dream of, and (ii) a similar revolution (for the same reason) in digital high-speed imaging, now being able to routinely resolve the millisecond time scale and even smaller time scales [224], revealing new physics on these scales which up to now was inaccessible and producing a huge amount of data on the flow. Also other advanced equipment like confocal microscopy, digital holographic microscopy and atomic force microscopy get more and more used in fluid dynamics. With all of these advances together, the gap between what can be measured and what can be ab-initio simulated is more quickly closing than we had anticipated at the end of the last century.
Also other gaps are closing: Fluid dynamics and in particular that of bubble fluid dynamics is bridging out to various neighboring disciplines such as chemistry and in particular colloidal science, catalysis, electrolysis, medicine, biology, computational science, and many others. Here the techniques, approaches and traditions from fluid dynamics can offer a lot to help to solve outstanding problems -vice versa, these fields can offer wonderful questions to fluid dynamics. Academic (bubble) fluid dynamics is also bridging out not only to traditional applications on large scales such as in chemical engineering, in the food industry, or in geophysics, but also to various new high-tech applications, be it in inkjet printing, immersion and XUV lithography, chemical diagnostics, and lab-on-a-chip microfluidics.

ACKNOWLEDGEMENT
Science is a collective activity. The scientific insight in this paper is gained during my bubble journey over more than two decades together with my colleagues, postdocs, PhD students, and students and I would like to thank them for all their work and contributions, and for the stimulation and intellectual pleasure we have enjoyed when doing physics. But first of all, I would like to express my sincere gratitude to my scientific teacher, Professor Siegfried Grossmann, for all what I learnt from him, which goes way beyond physics. Then, in particular, on this subject of bubbles, I would like to thank Michael Brenner, Sascha Hilgenfeldt and Michel Versluis, with whom my bubbly journey started, and the "part-time professors" of our Twente Physics of Fluids group, namely my colleagues and friends Andrea Prosperetti (who also inspired us on the beauty of bubbles in art [225]!), Roberto Verzicco, Chao Sun, and Xuehua Zhang, whose intellectual contribution to our group is way beyond "part-time", from whom I learned tremendously, and with whom it has been a great pleasure and privilege to work.
I also gratefully acknowledge generous financial support over the years, mainly from NWO via various programs and from the European Union via ERC Advanced Grants.