European Journal of Mechanics B/Fluids 29 (2010) 430–434

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European Journal of Mechanics B/Fluids

journal homepage: www.elsevier.com/locate/ejmflu

The influence of thermodynamic gas parameters on laser-induced bubbledynamics in waterBing Han, Bo Yang, Rui Zhao, Hong-Chao Zhang, Zhong-Hua Shen, Jian Lu, Xiao-Wu Ni ∗Department of Applied Physics, Nanjing University of Science & Technology, Nanjing 210094, People’s Republic of China

a r t i c l e i n f o

Article history:Received 30 April 2009Received in revised form15 March 2010Accepted 10 June 2010Available online 19 June 2010

Keywords:Cavitation bubbleLaserSubstance inside bubbleFiber-optic sensor

a b s t r a c t

The oscillating properties of laser-induced cavitation bubbles in water are investigated by means of afiber-optic sensor based on optical beam deflection. The experimental results show two important points.One is that the smaller the bubble radius the more quickly the bubble surface moves. Thus, the variationsof the temperature and the pressure inside the bubble will be close to those of an adiabatic process. Theother is that the high-energy vapor inside the newborn bubble diffuses and coagulates rapidly throughviolent expansion and thermal conduction. Thus, the gas content of the bubble reduces significantly inthe first oscillation. Numerical simulation is made for the bubble model with consideration of liquidviscosity, surface tension, and gas content. Through modification of the polytropic index and the gascontent parameter, two parameters of this model, the numerical results fit the experimental results well.

© 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

In 1917, Rayleigh developed the spherically symmetric bubblemodel in an incompressible liquid [1]. By taking into account theenvironmental viscosity, Plesset built the Rayleigh–Plesset model(R–P model) in an incompressible liquid [2] in 1949. The scale ofthe bubble oscillation is the micron and the microsecond, whichmake the detecting technique of an oscillating bubble the difficultyin experimental investigations. Up to the present, detecting andmeasuring techniques have been developed for three primaryaspects of the bubble parameters, including the time-varyingbubble shape (e.g., interferometry, high-speed photography [3],fiber-optic sensor based on optical beam deflection [4]), thetime-varying pressure field around the oscillating bubble (e.g.,piezoelectricity sensor, force sensor based on a fiber-opticsensor [5], hydrophone), and the temperature field [6] inside andoutside the oscillating bubble. Profiting from the development ofcomputer technology, numerical simulation has become one ofthe most important investigating methods [7,8]. Furthermore, theassociation of the experimental investigation and the numericalsimulation has promoted the improvement of the cavitationbubble model. Due to the efforts of scholars, including Cole [9],Noltingk and Neppiras [10], Gilmore [11], and Poritsky [12], amodel with consideration of liquid viscosity, surface tension andgas content is available today. The plot of the time-varying bubble

∗ Corresponding author. Tel.: +86 25 84315075; fax: +86 25 84318430.E-mail addresses: hanbingnjust@gmail.com (B. Han), nxw@mail.njust.edu.cn

(X.-W. Ni).

0997-7546/$ – see front matter© 2010 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechflu.2010.06.004

radius calculated from thismodel is an attenuating undulate curve.However, compared with the experimental results, this kind ofoscillating model cannot describe laser-induced bubbles in water.First, the attenuation is too weak. Second, the time evolution ofthe min-radius (the minimum radius of the bubble) conflicts withthe experimental results. Through numerical simulation, Popinetand Zaleski [13] concluded a relationship between the liquid-jetvelocity and the environmental viscosity. Based on the fact thatthe calculated max-radius (the maximum radius of the bubble)is bigger than the experimentally measured value, Popinet alsoconcluded that liquid compressibility and bubble–environmentheat exchange should be taken into consideration in the bubblemodel. Prosperetti [14] discussed the strong dependence of theeffective polytropic exponent and the thermal damping constanton the driving frequency in the forced radial oscillations ofgas bubbles. An interesting behavior of the variable polytropicexponent proposed in [14] is that the bubble area experiences anadiabatic process when the driving frequency reaches a specificvalue. A small decrease or increase in the driving frequencywill induce a rapid change in the thermodynamic characteristicsof the bubble toward the isothermal process. Prosperetti givesa highly effective way to investigate the acoustic-associatedcavitations when the driving frequency is the key factor. Kellerand Miksis [15] developed a new equation for the forced radialoscillations of bubbles by considering the compressibility of thesurrounding liquid, which induced the sound emission from theoscillating bubble. This new bubble model works well when theforcing amplitude varies in a large range. Löfstedt et al. [16] alsodiscussed the important effect of compressibility based on theimprovedR–Pmodel for bubbles collapsing at largeMachnumbers.

B. Han et al. / European Journal of Mechanics B/Fluids 29 (2010) 430–434 431

Nomenclature

Glossary of symbols

A Gas-content coefficientC Constant depending on the molar quantity and the

temperature of the gasn Polytropic indexP Gas pressureP0 Initial gas pressure inside the bubblePR Pressure inside the bubblePv Saturated vapor pressureP∞ Environmental pressureR Bubble radiusR0 Initial bubble radiusR ∂R/∂tR ∂2R/∂t2t TimeV Gas volumeρ0 Liquid densityµ Liquid viscosityσ Surface tensionγ Ratio of specific heats, also adiabatic index

Valuable conclusions about bubbles experiencing the minimumvolume were proposed. The acoustic cavitations associated withsonoluminescence (SL) and shock waves are similar to thoselaser-induced cavitations in the fierceness of the collapse, butthe initial expansions of the laser-induced cavitations are alsovery fierce, which is different from the case for the SL-associatedcavitations. Lauterborn and Ohl [17] investigated laser-inducedas well as acoustic-induced cavitations. Oscillations of acoustic-induced cavitations are studied through numerical modelingand high-speed photography technology. Detailed experimentalresults on the collapsing shock waves from laser-induced singlebubbles were obtained. As [17] pointed out, modeling of thecavitation cluster is an interesting challenge awaiting furtherinvestigation.Amongparameters that affect the laser-inducedbubble oscillat-

ingmode, there are four that are themost important: liquid viscos-ity, surface tension, gas content, and thermodynamic characteristicof the substance in the bubble. As is well known, sound emission isone of the damping factors of an oscillating bubble, which meansthat absence of the compressibility of the surrounding liquid canbe substituted by other damping mechanisms. In this paper, a newdampingmode is proposed and integrated into the R–Pmodel. Thethermodynamic characteristic of the substance inside the bubbleis added to the model with consideration of liquid viscosity, sur-face tension, and gas content. The polytropic index n and the gascontent of different oscillating periods are discussed. Numericalsimulation shows that this new model can describe laser-inducedcavitation bubbles through parameter optimizations.

2. Theoretical bubble model

Rayleigh developed the first spherically symmetric bubblemodel in an incompressible liquid [1], which can be written as

RR+32R2 =

PR − P∞ρ0

. (1)

In order to improve the Rayleigh model, many scholars,including Plesset [2], Cole [9], Noltingk and Neppiras [10],Gilmore [11], and Poritsky [12], have made their contribution by

taking into consideration various parameters. One of the mostpopular models, including parameters such as liquid viscosity,surface tension, and gas content, can be written as

RR+32R2 +

1ρ0

[P∞ − Pv −

(AP∞ − Pv +

2σR0

)(R0R

)3γ

+ 4µRR+2σR

]= 0. (2)

Numerical simulation shows that this model does not describea laser-induced bubble in water properly. First, R is much biggerthan the experimental result around the max-radius in everyperiod. Afterwe replaced the adiabatic index γ with the isothermalindex in Eq. (2), R around the max-radius decreases noticeably.But compared to the experiment, R is too small around the min-radius. Second, the calculated min-radius increases in turn inevery period. This conflicts with the experimental results. Afterwe decreased A to a smaller constant, the min-radius decreasesnoticeably, although the increasing trend of the min-radius in thesequential periods does not change. Therefore, the properties of thelaser-induced bubble in water are investigated in two aspects, asfollows.

2.1. Relationship between the polytropic index and the oscillatingbubble

To explain the investigating method proposed in this paperclearly, it is necessary to introduce the well-known ideal gasprocess equation, which can be given as

PV n = C . (3)

For gas inside a spherical bubble, Eq. (3) can be rewritten as(R0R

)3n=PP0. (4)

Obviously, in Eq. (2), n = γ , which means that the processinside the bubble is treated as an adiabatic process.When the bubble radius is very small, in other words when

the bubble is at the end of a compressing process or at thebeginning of an expanding process, the movement of the bubblesurface is very fast. We also observed that a pulse signal isreceived by a hydrophone placed near an oscillating bubble, everytime the bubble shrinks to the minimum volume. In fact, themovement of the bubble surface is so rapid in thewhole oscillationthat the substance inside the bubble hardly has a chance toexchange heat with the outside environment. The variations oftemperature and pressure inside the bubble approach those of anadiabatic process [18]. This is possibly the reason that the wholeoscillating process has been treated as an adiabatic process byother researchers.However, the movement of the bubble surface slows down

gradually during the expansion stage. As a result of the deceler-ation, exchange of heat between the substances inside and outsidethe bubble is inevitable. If there is enough time, the heat exchangewill make the process inside the bubble an isothermal one. Obvi-ously, it is inappropriate to fix the polytropic index n as the param-eter γ in the bubble model Eq. (2). Furthermore, the speed changeof the oscillating bubble surface is gradual, and thus the boundarybetween the adiabatic process and the isothermal process is notclear. That is to say, n should change between the adiabatic indexna (equal to γ ) and the isothermal index ni (equal to 1.0) as a func-tion of R. To summarize, n should have the following properties:(i) when R → 0, n → ni; (ii) when R → +∞, n → na; (iii) as R

432 B. Han et al. / European Journal of Mechanics B/Fluids 29 (2010) 430–434

increases, n increases monotonically. Thus, we propose an experi-mental equation (5) for n and R:

n = na − (na − ni) e−R. (5)

Substituting Eq. (5) into Eq. (2), we get

RR+32R2 +

1ρ0

[P∞ − Pv −

(AP∞ − Pv +

2σR0

)

×

(R0R

)3(na−(na−ni)e−R)+ 4µ

RR+2σR

]= 0. (6)

2.2. Relationship between the gas-content parameter and the oscillat-ing bubble

Plasma is induced in water when the energy intensity of thelaser focus exceeds the breakdown threshold of water. Because ofthe high temperature and pressure, the plasma expands violently.Thus a spherically symmetric shock wave is emitted with theplasma center as the core. In the meantime, the plasma heats theencircling water layer fiercely. Consequently, a spherically sym-metric bubble, which contains high-density and high-temperaturevapor, is formed with the plasma as the geometric center. Thepressure inside the newborn bubble could reach the order of108 Pa [19]. The high-energy vapor inside the newborn bubblediffuses and coagulates rapidly through violent expansion andthermal conduction. The ionization–recombination of the massivemicroscopic particles and the vaporization–diffusion of the mas-sive molecule clusters can be equivalent to the variation of the gascontent of the bubble.Let Ai be the gas-content parameter at moment ti. Ai+1 denotes

the gas-content parameter at moment ti+1. We suppose that theproportion of the particles which experience the physical andchemical changes is directly proportional to the time interval 1twith −δ (δ ≥ 0) as the proportional coefficient. Based on theanalysis given in the previous section, this hypothesis can beexpressed in the gas-content form as

Ai+1 − AiAi

= −δ(ti+1 − ti). (7)

Then we get

dAA= −δdt. (8)

Thus, we propose an experiential formula (9) for the temporalevolution of A. Formula (9) is the equivalent function to thephysical and chemical changes of the substance inside the bubble.

− δA = A, (9)

where δ can be named as the attenuation parameter. Experimentshows that the violent degree of the substance change inside thebubble is different in different oscillating periods. Thus δ should bedifferent for different oscillating periods. The exact value of everyδ can be obtained through fitting the experimental data.Substituting Eq. (9) into Eq. (6), the new modified integral

formula now becomes

RR+32R2 +

1ρ0

[P∞ − Pv −

(−AδP∞ − Pv +

2σR0

)

×

(R0R

)3(na−(na−ni)e−R)+ 4µ

RR+2σR

]= 0. (10)

It will be shown later that Eq. (10) imitates the experiment wellthrough proper treatment of the parameters.

Fig. 1. Experimental setup based on optical beam deflection. 1: Nd:YAG laser; 2,3: beam splitter; 4: attenuation group; 5: concave–convex lens group; 6: focusinglens; 7: cube target; 8: glass cuvette; 9: He–Ne laser; 10, 11: focusing lens; 12: five-dimensional fiber positioner; 13: single-mode optical fiber; 14: photomultiplier;15: oscilloscope; 16: PIN photodiode; 17: energy meter.

3. Experiment and discussion

3.1. Experimental setup

The experimental layout based on the optical beamdeflection isillustrated in Fig. 1. The bubbles were generated in a glass cuvetteby a Q-switched Nd:YAG laser. The laser delivered single-mode(TEM00) light pulses (1.06 µm, 10 ns), which ensured the fixedand controllable bubble center. The split laser beam collected bythe PIN photodiode (rise time 100 ps) was used as the burst signalof the oscilloscope to collect data from the photomultiplier. Thedirection of the laser beamwas perpendicular to a polished surfaceof the rigid cube target placed in the glass cuvette. The target couldbe moved with a resolution of 10 µm along the x (change thedistance between the detect beam and the bubble center) and y(change the laser–target interaction point) directions, as shownin Fig. 1. In order to prevent linear breakdown in the water, thepulsed laser beam was first expanded by a concave–convex lensgroup, then focused to the surface of the target by a biconvex lens.Since the energy intensity in the focus area (with a radius of about50 µm) was several orders higher than the breakdown thresholdofwater, a series of phenomenawould happen on thewater–targetinterface, such as laser-induced plasma, shock wave, cavitationbubble and target erosion. Thus, a hemispherical concave regionwith a radius of about 1 mm (bigger than the bubble radius) couldbe produced by several scores of laser pulses (18 mJ/pulse). Thecentre of the hemisphere was the same as the laser focus, andthis point was also the initial centre of the bubble. The incidentlaser beam was reflected and thus refocused to the centre bythe hemisphere, which formed a semi-free focusing system. Themeasurement origin was fixed on the interaction target surface, asshown in Fig. 2.The detecting section was formed by elements 9–14 in Fig. 1.

The continuous detecting He–Ne laser beam (0.6328 µm) was fo-cused before being passed through the front of the bubble centre.And finally the He–Ne laser was focused on the end surface of thesignal fiber fixed on the five-dimensional fiber positioner. The de-tecting light signal put out by the fiber was first transformed intoan electric signal and amplified by the photomultiplier, then sentinto the oscilloscope. Before the experiment, the fiber should beadjusted properly by coupling to the detecting light. That is to say,when there is no plasma, shock wave, or bubble in the path of thedetecting light, the output energy of the fiber is half themaximum,

B. Han et al. / European Journal of Mechanics B/Fluids 29 (2010) 430–434 433

Table 1Typical data of the oscillating bubble (shown in Fig. 4) induced in water.

Typical data The first oscillation The second oscillation The third oscillation

Max-radius/mm 1.473 0.710 0.497Min-radius/mm 0.250 0.300 0.157Oscillating period/µs 273 139 98

Fig. 2. The initial placement of the laser-induced bubble.

Fig. 3. Bubble radius R plotted as a function of time in water (25.27 mJ/pulse):(+) experimental data detected by the experimental setup shown in Fig. 1; (—)simulation result for Eq. (2); (-◦-) simulation result for Eq. (6).

which can be judged by reading the oscilloscope wave amplitudedirectly. If satisfied, this condition would ensure that the fiber waspositioned at the linear section of the Gauss beam of the detectinglight.

3.2. Experimental results and analysis

The simulating results for Eqs. (2) and (6) by means of the finitedifferencemethod and the experimental results are shown in Fig. 3.Since the content of the laser-induced bubble in water is mainlyvapor, the adiabatic index na was fixed to 1.33. It is obvious thatthe simulations deviate from the experiment badly.

a

b

Fig. 4. (a) Bubble radius R and the polytropic index n plotted as functions of timein water (25.27 mJ/pulse): (+) experimental data detected by the experimentalsetup shown in Fig. 1; (—) fitting curve calculated from Eq. (10) by means of thefinite difference method; (-◦-) n plotted as a function of time. (b) Oscilloscopewave detected 400 µm off the target surface; the detecting position is shownin (a).

Fig. 4(a) shows the 25.27 mJ/pulse laser-induced bubble radiusR and the polytropic index n plotted as functions of time in water.The experimental data is detected by the setup shown in Fig. 1. Thefitting curve and the n–time curve are calculated from Eq. (10) bymeans of the finite difference method. Typical data in Fig. 4(a) isshown in Table 1. Fig. 4(b) shows the oscilloscope wave detected400 µm off the target surface. The detecting position is shown inFig. 4(a).Let Ai0 be the initial gas-content parameter of the ith oscillation.

δi denotes the attenuation parameter of the ith oscillation. It can beseen from Fig. 4(a) that the fitting curve imitates the experimentwell by choosing different δi for different periods and a proper A10,as shown in Table 2.Two conclusions can be made from Fig. 4 and Table 1 together.

First, the first and the second expansion shock waves (strong peakpulses at the beginning of the first and the second expansion inFig. 4(b)) show that most of the bubble energy is released violently

434 B. Han et al. / European Journal of Mechanics B/Fluids 29 (2010) 430–434

Table 2Optimal δi for different periods and optimal A10 for the oscillating bubble (shown inFig. 4) induced in water. A20 and A

30 are calculated from Eq. (9).

i = 1 i = 2 i = 3

Ai0 3000 12.32 3.08

δi 2× 104 1× 104 1× 104

in the early stage of the laser-induced bubble. Second, the max-radius of the second period is about 53% smaller than that of thefirst period, and the max-radius of the third period is about only28% smaller than that of the second period. This result is consistentwith [20]. It is known that the decrease in the gas content leadsto the decrease in the max-radius. Ai0 and δ

i in Table 2 show thatthe gas content decreases 99.6% after the first oscillation, whilethe decrease slows down in the later periods. Thus the majority ofthe internal substance disappears rapidly in the first period. Suchcontent loss abates in the later periods.It can also be seen in Fig. 4(a) that the polytropic index n

decreases from the adiabatic index to the isothermal index sharplyaround the max-radius. Except for the short moment around themax-radius, n stays at the adiabatic-index level, and this leveldoes not decrease in the following periods. That is to say, thebubble surface moves very slowly around the max-radius, whileit moves significantly fast during the rest of the oscillation. So itcan be concluded that a laser-induced bubble in water undergoesan adiabatic or near-adiabatic process for the majority of its life,about more than 80%.

4. Conclusions

A function between the polytropic index n and R and a temporalevolution form of the gas-content parameter A are proposedin this paper. By substituting the two modified parameters intothe bubble model with consideration of liquid viscosity, surfacetension, and gas content, a newmodel especially for laser-inducedbubbles in water is obtained. Numerical simulation shows thatthis new model can describe laser-induced cavitation bubblesthrough choosing different values of the attenuation parameter δifor different periods and a proper initial gas-content parameter A10.The majority of the bubble energy is released violently in the

first oscillation period, so it is at the very beginning that the most

violent substance change happens inside a newborn bubble. Exceptfor the short moment around the max-radius, a laser-inducedbubble in water undergoes an adiabatic or near-adiabatic processfor the majority of its life.The properties of laser-induced bubbles in water can be

investigated through the research method discussed in this paper.There is no guarantee that the most proper fitting parameters forthe experimental results in this paper are the most proper ones forother experiments, but we can use the same investigating methodto find different fitting parameters for different experiments. Sinceit is hard to get the exact gas content and the exact polytropicindex of a laser-induced bubble through experiments, the fittingparameter method proposed in this paper may be useful.

Acknowledgements

This project is supported by the National Natural ScienceFoundation of China under Grant No. 60208004, Natural ScienceFoundation of Jiangsu Province under Grant No. BK2001056, andthe Teaching and Research Award Program for Outstanding YoungProfessor in Higher Education Institute, MOE, P. R. China.

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