Laser-induced Cavitation Bubble Research Articles

Interactions of a collapsing laser-induced cavitation bubble with a hemispherical droplet attached to a rigid boundary

We investigate experimentally and theoretically the interactions between a cavitation bubble and a hemispherical pendant oil droplet immersed in water. In experiments, the cavitation bubble is generated by a focused laser pulse right below the pendant droplet with well-controlled bubble–wall distances and bubble–droplet size ratios. By high-speed imaging, four typical interactions are observed, namely: oil droplet rupture; water droplet entrapment; oil droplet large deformation; and oil droplet mild deformation. The bubble jetting at the end of collapse and the migration of the bubble centroid are particularly different in each bubble–droplet interaction. We propose theoretical models based on the method of images for calculating the Kelvin impulse and the anisotropy parameter which quantitatively reflects the migration of the bubble centroid at the end of the collapse. Finally, we explain that a combination of the Weber number and the anisotropy parameter determines the regimes of the bubble–droplet interactions.

Experiment and Simulation Study of the Laser-Induced Cavitation Bubble Technique for Forming a Microgroove in Aluminum Foil.

The present work introduces a laser-induced cavitation bubble technique for forming an axisymmetric structure (i.e., microgroove) and the dynamics of a cavitation bubble from initial expansion to the collapse stages that were also simulated. Furthermore, the shock wave signals and dynamic properties of the cavitation bubble were recorded using a hydrophone and a high-speed camera. The experiments on microgrooves formed by laser-induced cavitation bubble stamping were carried out, and the effects of laser energy, the initial position of the bubble, and the number of impacts on the microformability of aluminum sheets are discussed. The depth of the microgroove was investigated using experiments, and it was found that the process can serve as a rapid technique for impressing microfeatures on thin-sheet metals. The experimental results showed that as the initial position of the bubble increased, the deformation depth decreased. As the laser energy and number of impacts increased, the deformation depth increased. The results of the response surface experiments showed that a laser energy of 27 mJ, 3 impacts, and a bubble position of 3 mm were optimal for the process. By using the optimal parameters, flat and smooth microgrooves with a forming depth of 102.54 µm were successfully fabricated. Furthermore, the maximum thickness thinning of the microgroove section occurred at the entrance areas, and this area had the greatest hardness. This also indicated that the greatest amount of plastic deformation of the material and grain refinement occurred in this area. On the other hand, the aluminum foil did not undergo oxidation during the plastic deformation process. These results demonstrated that laser-induced bubble stamping is an advanced micromachining method with promising applications.

The impacts of material acoustic impedance and thickness on single laser-induced bubble dynamics and determining factors in resulting pressure

The objective of this paper is to experimentally identify the primary sources of pressure when a laser-induced cavitation bubble is collapsing to a wall with specific emphases on the material acoustic impedance and thickness. Both high-speed videos and local wall pressure measurements were performed for various standoff ratios γ, bubble diameters, and wall materials. In the case of a rigid wall, in addition to the known high pressure for γ<0.6 where the bubble attaches and collapses on the wall (ring collapse), at γ≈1.12 where the jet is dominant, and low pressure obtained at γ≈0.913, where neither effect is significant, we further captured similar pressure profiles during the collapse after the first rebound at γ≈1.16 for the ring collapse, γ≈1.79 for the jet, and γ≈1.41 for the minimal, respectively. This indicates a strong jet is typically followed by a strong ring collapse. Generally, the pressure from the second collapse increases faster with the bubble size than that of the first collapse. For walls featuring smaller acoustic impedance or thickness, which cannot be approximated as rigid bodies or accessed by pressure sensor, our unique bubble edge analyzing tool shows that the ring collapse and jet effects are moved to smaller values of γ. The maximum pressure exerted on the wall in these cases is smaller than that on the rigid wall. Finally, we summarized the asymptotic evolution curves of each edge which bound the bubble dynamics at different standoff ratios.

Laser-induced cavitation bubble near boundaries

Laser-induced cavitation bubble near boundaries

Dynamics of laser-induced plasma and cavitation bubble at high pressures and the impacts on underwater LIBS signals

Characteristics of laser-induced plasma depends strongly on its ambient pressure, and a deeper understanding of the high pressure impact on the underwater plasma emission is essential for the application of LIBS in the deep-sea. In this work, we investigated the temporal evolution of underwater plasma and cavitation bubble at the high pressures up to 50 MPa, by using fast imaging and shadowgraph techniques. It showed that as the ambient pressure increases, the plasma expansion is much suppressed with a stronger emission intensity and a slower emission decay, but it quenches much earlier and ends with a sharp drop at the higher pressures. From the shadowgraph imaging results, it showed that the increase in pressure has a great impact on the bubble evolution. Both the bubble maximum radius and the bubble lifetime decrease dramatically with the pressure as a consequence of faster dynamics. By comparing between the plasma and bubble results, we demonstrated the critical role of bubble dynamics on the characterization of plasma emission at high pressures. At relatively low pressures, the bubble expansion time is long enough to finish the plasma radiation. While at high pressures, strong bubble-plasma interaction occurs that will guide the plasma evolution behavior at the late stage. The plasma can gain energy from the confinement effect caused by the bubble shrinking process which leads to an increase in the emission intensity, but it will quench immediately after the bubble collapse. Such effects caused by bubble dynamics at high pressures explain well the evolution behaviors of underwater LIBS signals of OH, Ca II, Ca I, and CaOH, considering the different lifetimes of these emitting species in the plasma. This work reveals the complex impacts of high pressure on underwater plasma emissions and underwater LIBS signals.

Plasma-based identification of gases in a laser-induced cavitation bubble

This study presents a general methodology and an experimental approach to identify the gas components within laser-induced cavitation bubbles. A needle electrode inside the cavitation bubble, which introduces low electric energy into the bubble, produces a homogenous plasma discharge inside the vapor cavity. The primary bubble dynamics remain identical while the rebound bubble becomes about twice as large when a discharge is applied. The effect of non-condensable gases and the electric charge on bubble dynamics is explored theoretically, and the role of the electric charge is found to be significant. Optical emission spectroscopy reveals the evolution of emission lines from gases inside bubbles. H lines and OH lines are persistently observed in all cases, providing a dominant presence of water vapor. The results also confirm that the gases, which are initially present in the water rather than transported from the water, contribute to the optical emission characteristics with different dissolved gases.

Dynamics of a Laser-Induced Cavitation Bubble near a Cone: An Experimental and Numerical Study

A bubble’s motion is strongly influenced by the boundaries of tip structures, which correspond to the bubble’s size. In the present study, the dynamic behaviors of a cavitation bubble near a conical tip structure are investigated experimentally and numerically. A series of experiments were carried out to analyze the bubble’s shape at different relative cone distances quantitatively. Due to the crucial influence of the phase change on the cavitation bubble’s dynamics over multiple cycles, a compressible two-phase model taking into account the phase change and heat transfer implemented in OpenFOAM was employed in this study. The simulation results regarding the bubble’s radius and shape were validated with corresponding experimental photos, and a good agreement was achieved. The bubble’s primary physical features (e.g., shock waves, liquid jets, high-pressure zones) were well reproduced, which helps us understand the underlying mechanisms. Meanwhile, the latent damage was quantified by the pressure load at the cone apex. The effects of the relative distance γ and cone angle θ on the maximum temperature, pressure peaks, and bubble position are discussed and summarized. The results show that the pressure peaks during the bubble’s collapse increase with the decrease in γ. For a larger γ, the first minimum bubble radius increases while the maximum temperature decreases as θ increases; the pressure peak at the second final collapse is first less than that at the first final collapse and then much greater than that one. For a smaller γ, the pressure peaks at different θ values do not vary very much.

Research on Dynamic Process and Droplet Splash of Laser-Induced Cavitation Bubble Collapse within a Droplet

The cavitation bubble within a droplet is one of the frontier topics in bubble dynamics, with applications in many industrial fields. In the present paper, the dynamics of the cavitation bubble wall and the droplet surface, with different radius ratios, are compared and analyzed. The relationship between cavitation bubble collapse and droplet splash is disclosed. Research shows that, firstly, under the same type of splash, there is a positive correlation between the radius ratio with the displacement of the feature point and the distance between the two ends of the X-axis. It shows that the splash becomes more prominent with the radius ratio increase. Secondly, under different splash cases, the radius ratio also shows a significant impact on the trend of interface displacement and the splash dynamics. In addition, as the radius ratio increases the modification coefficient of the collapse time being smaller.

Collapsing and Splashing Dynamics of Single Laser-Induced Cavitation Bubbles within Droplets

In the present paper, the cavitation bubble dynamics model for a single bubble oscillating within a droplet is improved based on the classical Rayleigh–Plesset bubble dynamics equation and the effects of liquid surface tension and viscosity are both considered. In the aspect of the experiment, the collapsing dynamic process of a bubble within a droplet is carried out by building a high-speed photography experimental platform. In addition, the numerical solution of the dynamic equation for the collapse time of the bubble within the droplet is also carried out. The findings are given as follows: (1) The bubble dynamic equation considering liquid surface tension and viscosity of bubble within droplet is proposed. (2) The surface of liquid droplets induced by the bubble motion could be divided into three modes: no splashing, scattered splashing, composite splash consisting of scattered and flaky splash. (3) The bubble interface during the first collapsing stage could be divided into three types: spherical, conical, and fungiform. (4) The numerical solution shows an accurate prediction of the bubble collapse time within the droplet especially under the condition of medium radius ratio.

Early dynamics of laser-induced plasma and cavitation bubble in water

The knowledge of early laser-induced plasma and cavitation bubble dynamics are of great importance for a better understanding of underwater laser-induced breakdown spectroscopy. In this work, we investigated the temporal evolution of the plasma and bubble at an early stage before 2 μs, by using fast imaging and shadowgraph techniques. It showed that the initial plasma forms at the laser focal point and then grows up toward the incident laser. As time evolves, the plasma expands simultaneously forward and backward to its peripheral region and a sub-plasma structure can be observed at low laser energies. The plasma has a good pulse-to-pulse repeatability during the laser pulse, while soon after the laser pulse (from 20 to 50 ns), it undergoes a severe pulse-to-pulse fluctuation with an inhomogeneous emission distribution. At longer times, the plasma becomes much stable again. To clarify the interrelationship between the bubble dynamics and the plasma characteristics, shadowgraph images of the bubble were shown and compared with the plasma images. It showed that the morphology of the bubble and its evolution have a good consistency with the morphology of plasma. The bubble radius as a function of time can be well expressed by the equation R=at0.4, and a is a constant related to the laser energy. By inspecting the evolution curve of the bubble wall and the bubble pressure calculated based on the Gilmore model, a transition phase from 20 to 50 ns can be identified between the moving breakdown phase and the thermal expansion phase. In this transition phase, both the bubble expansion speed and the bubble pressure drop drastically that may account for the severe plasma fluctuations as observed in this period. These results of early bubble dynamics provide insights into the critical role of bubble-plasma interaction on the characterization of laser-induced plasma in water.

MP35-10 MECHANISMS OF FIBER-TIP DEGRADATION DURING LASER LITHOTRIPSY

You have accessJournal of UrologyCME1 Apr 2023MP35-10 MECHANISMS OF FIBER-TIP DEGRADATION DURING LASER LITHOTRIPSY Yuri Pishchalnikov, William Behnke-Parks, and Marshall Stoller Yuri PishchalnikovYuri Pishchalnikov More articles by this author , William Behnke-ParksWilliam Behnke-Parks More articles by this author , and Marshall StollerMarshall Stoller More articles by this author View All Author Informationhttps://doi.org/10.1097/JU.0000000000003269.10AboutPDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareFacebookLinked InTwitterEmail Abstract INTRODUCTION AND OBJECTIVE: Ureteroscopic lithotripsy breaks urinary stones using laser pulses delivered to the stone via optical fibers. Degradation of fibers reduces the efficacy of lithotripsy and varies with fiber diameter, stone composition, laser pulse energy and duration. The goal of this work was to reveal the physical mechanisms of fiber-tip damage (burnback) in water and air. METHODS: The dynamics of fiber-tip damage was studied using a high-speed camera Shimadzu HPV-X2 at a frame rate up to two million frames per second. The camera was mounted to a microscope focused on the 242-µm glass-core-diameter fiber positioned in direct contact with a target—typically, a glass slide coated with a 25±5 µm layer of hydroxyapatite (Himed, NY). Selected experiments were conducted with BegoStones and whole surgically retrieved urinary stones. Damaged fibers were cleaved to start each series of 20 pulses with flat fiber tips. Laser pulses were produced using a Lumenis VersaPulse holmium:YAG laser. Temporal profiles of infrared laser pulses and flashes of visible light were resolved using photodetectors. Cross-sectional profiles of laser-beam energy were assessed using Zap-It paper. Microscope images were used to measure the burnback length of the fiber. RESULTS: Burnback of optical fibers was associated with flashes of light (sparks) observed both in water and in air. Sparks in air could crack and fragment the fiber during single laser pulse. Twenty pulses in air produced the burnback length of 0.37 mm at 1.0 J (95% CI: 0.11–0.64 mm) and 0.14 mm at 0.6 J (95% CI: 0.10–0.18 mm), smaller than that at 1.0 J (p=0.010). In water, the fibers were damaged by sparks during the laser pulse and after the pulse during the collapse of laser-induced cavitation bubbles. Contribution of cavitation varied from insignificant to most dramatic, breaking a 1.1-mm length of the fiber in a single laser pulse. Cavitation without sparks showed no burnback with only minor damage of cladding (500 pulses, 1.0 J). Sparks were associated with an increased absorption of laser energy and were observed at laser energies typically >0.5 J and only with certain target materials near the fiber tip. Cross-sections of laser beams were larger with fibers damaged in air than that in water (p=0.007) and both were larger than that with undamaged fibers (p<0.005). CONCLUSIONS: These results suggest that the burnback degradation of optical fibers is associated mainly with laser-induced heating, ionization, and optical breakdown of target material. A better understanding of the mechanisms of fiber-tip damage will aid in development of future improvements to minimize degradation of optical fibers. Source of Funding: Research reported in this publication was supported by the National Institute of Diabetes and Digestive and Kidney Diseases of the National Institutes of Health under Award Number R43DK129104. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health © 2023 by American Urological Association Education and Research, Inc.FiguresReferencesRelatedDetails Volume 209Issue Supplement 4April 2023Page: e473 Advertisement Copyright & Permissions© 2023 by American Urological Association Education and Research, Inc.MetricsAuthor Information Yuri Pishchalnikov More articles by this author William Behnke-Parks More articles by this author Marshall Stoller More articles by this author Expand All Advertisement PDF downloadLoading ...

Correlation of laser-induced single bubbles with cavitation damage via in-situ imaging

Experiments with series of laser-induced single cavitation bubbles were performed on Cu-, Fe-, and Al- base alloys, namely 316LVM, a nickel aluminum bronze (NAB), and pure aluminum. The bubble dynamics were recorded by two high-speed cameras, while the resulting surface damage was investigated by a microscope in situ, i.e. between two successive bubbles with the sample remaining in the water. Bubbles with stand-off distances γ (the ratio of distance of the surface from the bubble and the bubble maximum radius) of 1.35 ± 0.5 and bubble radii of about 1.2 mm were directly correlated with the damage they produced. We found a correlation between locally enlarged (“strong”) gas-filled collapse areas (SCAs) during the second bubble collapse and the resulting pits on the sample surface. While all pits could be correlated to a preceding SCA, not all SCAs resulted in pits. This is true on both NAB and 316L. On aluminum, despite the softness of the material, some bubbles did not produce pits, and these where the ones with the most symmetric second-collapse torus. The pits that constitute, in sum, the damage of the material where analyzed in their number and geometry. The pitting rate and the maximum pit depth were roughly inversely depended on the material hardness. Also, beginning wear mechanisms were identified, like emerged grain boundaries on aluminum and light slip lines on steel. Finally, the collapse of a very small, secondary, dissolved-gas bubble close to the surface is documented to cause a pit. This may be one mechanism by which the pits are formed.

Dynamics of laser-induced shock wave and cavitation during laser ablation of aliphatic nitroalkanes

Nanosecond time-resolved schlieren imaging is performed to investigate laser-induced breakdown in nitromethane, nitroethane and 1-nitropropane, and the associated dynamics of laser-induced shock waves and cavitation bubbles. The exothermic reactions upon decomposition and the chemical reactions in the vapor plume behind the shock front increase the shock wave velocity. The correlations of oxygen balance with shock wave velocity, bubble radius, lifetime and bubble energy are established. For instance, shock wave velocity is proportional to the negative inverse of oxygen balance. In addition, the bubble energy of nitroethane tends to saturate at high pulse energy.

Temporal evolution of pressure profiles for laser-induced cavitation bubble on the metal surface

When a laser is focused on an underwater object, it experiences a large amount of pressure owing to the plasma confinement effect of water. A hemispherical bubble is generated on the surface of the object, and large pressure is generated when the bubble collapses. In this study, we conducted experiments using different laser energies to analyze the pressure–time histories associated with bubble contraction. The maximum pressure was 10%–40% of the laser ablation pressure, whereas the pressure pulse width was 5–10 times longer than the laser pulse width. Furthermore, the bubble motion could be adiabatically explained, except for the plasma interaction region. The results indicate that the pressure at which the bubble collapses does not depend on the maximum size of the generated bubble but depends on the energy of water vapor within the bubble.

Real-time passive cavitation mapping with high spatial-temporal resolution

Shock wave lithotripsy (SWL) and laser lithotripsy (LL) have been widely used for clinical treatment of kidney stones. Cavitation plays an important role in stone fragmentation in both SWL and LL, yet it may also contribute to renal tissue injury. It is therefore crucial to determine the spatiotemporal distributions of cavitation activities to maximize stone fragmentation while minimizing tissue injury. Passive cavitation mapping (PCM) has most practical applications in deep biological tissues and is most promising for clinical translation. We have developed a set of technologies for 2D/3D PCM that can be seamlessly integrated with ultrasound imaging and photoacoustic imaging. Our 2D/3D PCM has achieved a spatial resolution of hundreds of micrometers and a temporal resolution of several microseconds. We also developed a transient angular spectrum approach for PCM reconstruction, which is ten times faster than the traditional delay-and-sum method. Using the 2D/3D PCM system, we imaged shockwave- and laser-induced single cavitation bubbles in both free field and constricted space, as well as on large animal models. Collectively, our results have demonstrated the high reliability and spatial-temporal accuracy of the 2D/3D PCM approach, which paves the way for future in vivo applications and human studies during SWL and LL.

Dissimilar cavitation dynamics and damage patterns produced by parallel fiber alignment to the stone surface in holmium:yttrium aluminum garnet laser lithotripsy.

Recent studies indicate that cavitation may play a vital role in laser lithotripsy. However, the underlying bubble dynamics and associated damage mechanisms are largely unknown. In this study, we use ultra-high-speed shadowgraph imaging, hydrophone measurements, three-dimensional passive cavitation mapping (3D-PCM), and phantom test to investigate the transient dynamics of vapor bubbles induced by a holmium:yttrium aluminum garnet laser and their correlation with solid damage. We vary the standoff distance (SD) between the fiber tip and solid boundary under parallel fiber alignment and observe several distinctive features in bubble dynamics. First, long pulsed laser irradiation and solid boundary interaction create an elongated "pear-shaped" bubble that collapses asymmetrically and forms multiple jets in sequence. Second, unlike nanosecond laser-induced cavitation bubbles, jet impact on solid boundary generates negligible pressure transients and causes no direct damage. A non-circular toroidal bubble forms, particularly following the primary and secondary bubble collapses at SD = 1.0 and 3.0 mm, respectively. We observe three intensified bubble collapses with strong shock wave emissions: the intensified bubble collapse by shock wave, the ensuing reflected shock wave from the solid boundary, and self-intensified collapse of an inverted "triangle-shaped" or "horseshoe-shaped" bubble. Third, high-speed shadowgraph imaging and 3D-PCM confirm that the shock origins from the distinctive bubble collapse form either two discrete spots or a "smiling-face" shape. The spatial collapse pattern is consistent with the similar BegoStone surface damage, suggesting that the shockwave emissions during the intensified asymmetric collapse of the pear-shaped bubble are decisive for the solid damage.

Study on pulsation characteristics of laser-induced cavitation bubble and its degradation of acid black solution

Laser cavitation degradation is a new technology for dye wastewater treatment. By designing and building a laser cavitation experimental platform, the acid black water solution was degraded by laser cavitation under different operating conditions, and the degraded solution was analyzed and characterized by ultraviolet spectrophotometer to obtain the influence of laser energy and initial concentration of the solution on the degradation ability. The generation process of hydroxyl radical was summarized through the study of laser-induced cavitation bubble pulsation, and the degradation mechanism of acid black was further revealed. The results show that with the increase of laser energy, the degradation rate of acid black solution with initial concentration of 20 mg/L gradually increases from 0.78% to 27.28%. When the laser energy of is 100 mJ, the solution with the initial concentration of acid black in the range of 5–20 mg/L is selected, and the degradation rate decreases from 98.55% to 7.63%. The higher the laser energy is, the greater the degradation rate is. However, if the initial concentration is higher, it is difficult to degrade the acid black solution by laser cavitation.

Cavitation bubble dynamics and microjet atomization near tissue-mimicking materials

In recent years, considerable interest has been devoted to the interactions between cavitation bubbles and tissue-mimicking materials due to their promising applications in medicine and biomedical sciences. The strong fluid–structure interaction between a cavitation bubble and these elastic surfaces triggers unique collapse dynamics, characterized by bubble splitting and subsequent microjetting phenomena that can damage adjacent boundaries. In this work, we investigate how the elasticity of the boundary and the distance between the bubble and the elastic surface affect the bubble dynamics and the velocity of its microjet. To this end, we generate single laser-induced cavitation bubbles in the vicinity of agarose hydrogels with different degrees of elasticity and follow the bubble dynamics using high-speed imaging techniques, with a special focus on the formation and evolution of the microjets. We provide a time-resolved evidence of the atomization of the liquid microjet within the bubble, which precedes the establishment of a fully liquid microjet. The atomized portion of the microjet can reach supersonic velocities of up to 2000 ms−1, while the ensuing fully developed liquid microjet travels at averaged speeds of up to 1000 ms−1. To gain further insight into the bubble dynamics leading to the formation of these very fast microjets, we also propose a numerical model based on the boundary integral method and observe a remarkable agreement between the numerical simulations and the experimental observations.

Influence of an external magnetic field on laser-induced plasma and cavitation bubbles in submerged targets

A pulsed Nd:YAG laser is tightly focussed on a metal target immersed in distilled de-ionized water. The resultant laser-induced plasma and subsequent cavitation bubble behavior are studied under the influence of an external magnetic field that is varied from 700 to 1000 Gauss. The study is conducted using a beam deflection probe arrangement. In addition, laser-induced breakdown spectroscopy is also employed to study the plasma spectrum. Furthermore, three different magnetic materials are employed for this investigation: ferromagnetic nickel, paramagnetic gadolinium, and diamagnetic copper. The studies revealed that cavitation bubble radii and collapse durations increased considerably as the magnitude of the external magnetic field was increased. This effect was prominent in the case of nickel and less so in the case of gadolinium and copper. For nickel, collapse times increase when the magnetic field was applied, whereas for gadolinium and copper, significant changes were not observed. The differences observed in collapse times showed that magnetic properties of the targets played a vital role in this phenomenon. The process of pulsed laser ablation in liquid also led to the respective generation of metallic nanoparticles from individual materials. Characterization of the generated nanoparticles revealed size reduction when synthesized under the influence of an external magnetic field. These characterizations were performed using transmission electron microscopy and UV-Vis spectroscopy.

Modeling acoustic emissions and shock formation of cavitation bubbles

Despite significant progress in understanding and foretelling pressure-driven bubble dynamics, models that faithfully predict the emitted acoustic waves and the associated shock formation of oscillating or collapsing bubbles have received comparably little attention. We propose a numerical framework using a Lagrangian wave tracking approach to model the acoustic emissions of pressure-driven bubbles based on the Kirkwood–Bethe hypothesis and under the assumption of spherical symmetry. This modeling approach is agnostic to the equation of the state of the liquid and enables the accurate prediction of pressure and velocity in the vicinity of pressure-driven bubbles, including the formation and attenuation of shock fronts. We validate and test this new numerical framework by comparison with solutions of the full Navier–Stokes equations and by considering a laser-induced cavitation bubble as well as pressure-driven microbubbles in excitation regimes relevant to sonoluminescence and medical ultrasound, including different equations of state for the liquid. A detailed analysis of the bubble-induced flow field as a function of the radial coordinate r demonstrates that the flow velocity u is dominated by acoustic contributions during a strong bubble collapse and, hence, decays predominantly with u∝r−1, contrary to the frequently postulated decay with u∝r−2 in an incompressible fluid.