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On the pressure of cavitation bubbles
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Experimental Thermal and Fluid Science 32 (2008) 1188–1191 www.elsevier.com/locate/etfs
On the pressure of cavitation bubbles E.A. Brujan a,*, T. Ikeda b, Y. Matsumoto b b
a Department of Hydraulics, University Politehnica Bucharest, 060042 Bucharest, Romania Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Received 19 November 2007; received in revised form 17 January 2008; accepted 17 January 2008
Abstract Shock wave emission upon the collapse of a cavitation bubble attached to a rigid wall is investigated using high-speed photography with 200 million frames/s and 5 ns exposure time. At a distance of 68 lm from the bubble wall, the shock pressure is 1.3 ± 0.3 GPa. The shock pressure decays proportionally to r1.5 with increasing distance from the bubble. An estimation of the peak pressure at the bubble wall reveals a pressure of about 8 GPa. A major part of the shock wave energy is dissipated within the first 100 lm from the bubble wall. Ó 2008 Elsevier Inc. All rights reserved. Keywords: Cavitation; Shock wave
1. Introduction Ultrasonic irradiation of liquids produces cavitation: the formation, growth, and implosive collapse of bubbles [1]. Cavitation involves either rapid growth and collapse of bubbles (inertial cavitation) or sustained oscillatory motion of bubbles (stable cavitation). Stable oscillations of bubbles induce fluid velocities, emission of acoustic transients, and exert shear forces on the surrounding medium, whereas the collapse of inertial bubbles near a surface experience nonuniformities in their surroundings that results in the formation of high-velocity microjets and the emission of shock waves that propagate through the liquid. This violent collapse is predicted to generate extremely large pressure within the bubble, but, to date, there have been only a limited number of experimental measurements of the shock wave pressure in the close proximity of the bubble wall [2–5]. Given the importance of cavitation in sonochemistry, sonography, gene transfection and drug delivery [6,7], we decided to investigate the final collapse stage of inertial bubbles with high temporal and spatial resolution. We have analyzed the propagation of the shock wave emitted *
Corresponding author. E-mail address: [email protected] (E.A. Brujan).
0894-1777/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2008.01.006
upon the collapse of an inertial cavitation bubble attached to a rigid wall as this case has the largest damage potential. 2. Experimental A schematic diagram depicting the experimental arrangement used for investigating the final stage of the collapse of cavitation bubbles near a rigid boundary is shown in Fig. 1. A sinusoidal ultrasound pulse (1.08 MHz, 30 cycles) is created by the function generator. The output of ±1 V signal is amplified by 60 dB radio frequency amplifier and transmitted to the concave PZT transducer. In the degassed water, ultrasound is gathered at the focus of the transducer where the aluminum block as the rigid wall is placed. Then, in the localized focal volume (<2 mm3) near the rigid wall surface, the high amplitude pressure fluctuation (the absolute values of the peak positive and negative pressure are greater than 50 MPa and 10 MPa) is obtained. After the ultrasound is stopped, the bubble expands to its maximum volume and under the static pressure it collapses violently accompanied by strong shock wave emission [8,9]. The shock wave emitted upon bubble collapse is captured by a high speed camera IMACON 200 with 200 millions/s framing rate and 5 ns exposure time. The image on the fluorescent screen was
E.A. Brujan et al. / Experimental Thermal and Fluid Science 32 (2008) 1188–1191
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3. Results and discussion
Fig. 1. Schematic diagram of the experimental set-up.
recorded with an intensified scan ICCD camera system (Photometrics AT200A) with a 4800 3920 pixel array. The signal from the ICCD camera was then digitized with 12-bit resolution (256 grey levels) and passed to a computer. The illumination for the shadowgraph photography is obtained by a short arc strobe with 200 J/flash. The 1 mm gap between the electrodes is used as a point light source and the light is collimated to the parallel illumination by a convex lens (f = 60 mm). A Nikon lens (Ai-Micro Nikkor, 105 mm, F2.8S) with 10 close up rings enabled a field view of 2.53 mm 2.06 mm.
Fig. 2. A high-speed photographic sequence in side view showing the propagation of the shock waves emitted upon the collapse of a cavitation bubble attached to rigid boundary. The boundary is located in the righthand side of each frame. The white arrowheads indicate the location where the distance traveled by the bubble-collapse-induced shock wave was measured. Sequence taken with 200 million frames/s and an exposure time of 5 ns.
Fig. 2 shows a high-speed photographic sequence of the final stage of bubble collapse. Since the photographic record of the bubble dynamics with a converter camera at this framing rate can only cover a small part of the collapse process (40 ns), neither the maximum bubble radius nor the distance between bubble and boundary are known exactly. However, it can be inferred from high-speed photographic sequences taken at smaller framing rates that the maximum bubble radius is about 1.7 mm and the bubble is attached to the rigid boundary; it probably originates from a gas nucleus trapped in a small crevice of the boundary. The minimum radius of the bubble was measured as Rmin = 30 lm (first frame). Two shock waves are emitted upon the first collapse of the bubble. The shock wave indicated by the black arrowhead is probably generated at the impact of a microjet developed in an earlier stage of the collapse onto the rigid wall (microjet-induced shock wave) [10]. The shock wave indicated by the white arrowheads is generated at the minimum bubble volume (bubble-collapse-induced shock wave). The microjet-induced shock wave is, however, so weak that it is barely visible on the photographic frames. Therefore, we concentrate on the motion of the bubble-collapse-induced shock wave as this wave gives an indication of the maximum pressure inside the collapsing bubble. The high-speed photographic sequence gives values for the propagation velocity of the shock front which can be used to calculate the corresponding pressure values at the shock front. The shock wave width cannot be inferred from the photographic images. The shock front can be seen, because the refractive index gradient induced by the shock front refracts the illumination light out of the imaging lens. The width of the shock wave image is determined by the pressure profile, the pressure amplitude, and the aperture of the imaging lens. Thus the width of the shock wave image should not be mistaken for the shock width. We measured the distance r traveled by the bubble-collapseinduced shock wave as a function of the time delay t, and a curve was fitted through the measurement points using the LAB Fit curve fitting software [11]. From the slope of the r(t) curve, the shock wave velocity was derived (Fig. 3). Initially, the shock wave propagates with a velocity of 2520 ± 210 m/s and ends up with one of 1500 m/s. The shock pressure ps can be determined through a measurement of the shock front velocity us if the equation of state of the medium is known. We used the equation of state determined by Rice and Walsh [12] up ¼ c1 10ðus c1 Þ=c2 1 ð1Þ which fits experimental data for pressure values of up to 25 GPa. Here up is the particle velocity behind the shock front, c1 is the sound velocity in water, c1 = 5190 m/s, and c2 = 25306 m/s. The pressure ps is then related to us by [13]
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E.A. Brujan et al. / Experimental Thermal and Fluid Science 32 (2008) 1188–1191
Fig. 3. Propagation of the shock front plotted as a function of time. The position of the shock front was extracted from the high-speed photographic record using image processing techniques (open symbols). A curve was then fitted through the measurement points (solid line). The first derivative of the curve corresponds to the shock velocity (dashed line). The standard deviation between data points and fit is smaller than 1 lm for the shock wave position.
ps ¼ c1 q1 us 10ðus c1 Þ=c2 1 þ p1
ð2Þ
where q1 denotes the density of water before compression by the shock wave and p1 is the hydrostatic pressure. We determined the shock wave pressure up to a distance of 110 lm from the bubble wall (Fig. 4). For larger distances the deviations of the shock wave velocity from sonic velocity become very small, which results in a large measurement uncertainty for ps. The maximum shock wave pressure at a distance of 68 lm from the bubble wall is 1.3 ± 0.3 GPa. Up to a distance of 70 lm from the bubble wall, the pressure decreases proportionally to r1.5. The shock pressure decreases faster with increasing propagation distance than in the acoustic limit, where a pressure decay proportional to r1 would be expected for a spherical source [3]. The fast pressure decay is caused by the energy dissipation at the shock front and the nonlinearity of propagation, which
Fig. 4. Shock wave pressure as a function of the propagation distance. The shock wave velocities were converted to shock pressure using Eq. (1). The dashed line indicates the position of the bubble wall at its minimum volume. The number indicates the average slope of the pressure versus distance curve at a distance between 68 lm and 70 lm from the bubble wall.
results in a modification of the pressure profile during propagation [3]. Extrapolating the pressure obtained at a distance of 68 lm from the bubble wall down to the position of the minimum bubble radius with the help of the r1.5 law, the pressure at the bubble wall and thus the pressure acting on the rigid wall may be as high as 7.7 ± 1.6 GPa. It is interesting to note that a similar result was reported by Pecha and Gompf [2]. However, a direct comparison with their result is not appropriate here because they investigated the shock wave emission from stable cavitation bubbles whose behaviour is strongly dependent on the ultrasound amplitude and frequency. We also evaluated the energy dissipation, ED, at the shock front as a function of propagation distance and obtained the shock wave energy by integration over the dissipated energy [14]. The Rankine–Hugoniot equation relates the increase of internal energy per unit mass at a shock front to the change of pressure and density at the shock front [14]: 1 1 1 DðrÞ ¼ ð3Þ ðps ðrÞ þ p1 ðrÞÞ 2 q1 qs ðrÞ where qs is the density behind the shock front. The density is, through conservation of mass [14], us q1 ¼ ðus up Þqs , and momentum, ps p1 ¼ us up q1 , also linked with us by q1 ð4Þ qs ¼ 1 u2pqs s 1
The total change of internal energy during propagation of a spherical shock front from r0 to r1 is obtained by integration of Eq. (3) Z r1 ED ¼ 4pr2 qs ðrÞDðrÞdr ð5Þ r0
Fig. 5 shows the accumulated energy loss, ED, as a function of propagation distance from the bubble wall. The loss rate is highest close to the bubble wall and has considerably decreased at the end of the measurement range. A major part of the shock wave energy is, therefore, already dissipated within the first 100 lm from
Fig. 5. Accumulated energy loss at the shock front as a function of the propagation distance.
E.A. Brujan et al. / Experimental Thermal and Fluid Science 32 (2008) 1188–1191
the bubble wall. The large amount of the shock wave energy loss within the first 100 lm of its propagation indicates that the shock wave-induced biological effects on biological tissues are confined to very small dimensions on a cellular or sub-cellular level. Acknowledgements Supported by Grants from Study for Open and Integrated Research Program of the Science and Technology Agency of Japan. E.A.B. was supported by a Japan Society for the Promotion of Science Invitation Fellowship (Grant S-04206). References [1] C.E. Brennen, Cavitation and Bubble Dynamics, Oxford University Press, Oxford, 1995. [2] R. Pecha, B. Gompf, Microimplosions: cavitation collapse and shock wave emission on a nanosecond time scale, Phys. Rev. Lett. 84 (2000) 1328–1330. [3] A.G. Doukas, A.D. Zweig, J.K. Frisoli, R. Birngruber, T.F. Deutsch, Noninvasive determination of shock-wave pressure generated by optical breakdown, Appl. Phys. B 53 (1991) 237–245.
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[4] R. Petkovsek, J. Mozina, G. Mocnik, Optodynamic characterization of shock waves after laser-induced breakdown in water, Opt. Express 13 (2005) 4107–4112. [5] R. Petkovsek, P. Gregorcic, J. Mozina, A beam-deflection probe as a method for optodynamic measurements of cavitation bubble oscillations, Meas. Sci. Technol. 18 (2007) 2972–2978. [6] K.S. Suslick, Sonochemistry, fourth ed., Kirk-Othmer Encyclopedia of Chemical Technology, Wiley & Sons, NY, 1998, 517–541. [7] D. Dalecki, Mechanical bioeffects of ultrasound, Annu. Rev. Biomed. Eng. 6 (2004) 229–248. [8] E.A. Brujan, T. Ikeda, Y. Matsumoto, Dynamics of ultrasoundinduced cavitation bubbles in non-Newtonian liquids and near a rigid boundary, Phys. Fluids 16 (2004) 2402–2410. [9] E.A. Brujan, T. Ikeda, Y. Matsumoto, Jet formation and shock wave emission during collapse of ultrasound-induced cavitation bubbles and their role in the therapeutic applications of high-intensity focused ultrasound, Phys. Med. Biol. 50 (2005) 4797–4809. [10] A. Shima, K. Nakajima, Collapse of a non-spherical bubble attached to a rigid wall, J. Fluid Mech. 80 (1977) 369–391. [11] P.W. Silva, C. Silva, LAB Fit Curve Fitting Software (Nonlinear Regression and Treatment of Data Program) V 7.2.32. [12] M.H. Rice, J.M. Walsh, Equation of state of water to 250 kilobars, J. Chem. Phys. 26 (1957) 824–830. [13] A. Vogel, S. Busch, U. Parlitz, Shock wave emission and cavitation bubble generation by picosecond and nanosecond optical breakdown in water, J. Acoust. Soc. Am. 100 (1996) 148–165. [14] G.E. Duvall, G.R. Fowles, High Pressure Physics and Chemistry, Academic Press, NY, 1963, 209–291.
Experimental Thermal and Fluid Science 32 (2008) 1188–1191 www.elsevier.com/locate/etfs
On the pressure of cavitation bubbles E.A. Brujan a,*, T. Ikeda b, Y. Matsumoto b b
a Department of Hydraulics, University Politehnica Bucharest, 060042 Bucharest, Romania Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Received 19 November 2007; received in revised form 17 January 2008; accepted 17 January 2008
Abstract Shock wave emission upon the collapse of a cavitation bubble attached to a rigid wall is investigated using high-speed photography with 200 million frames/s and 5 ns exposure time. At a distance of 68 lm from the bubble wall, the shock pressure is 1.3 ± 0.3 GPa. The shock pressure decays proportionally to r1.5 with increasing distance from the bubble. An estimation of the peak pressure at the bubble wall reveals a pressure of about 8 GPa. A major part of the shock wave energy is dissipated within the first 100 lm from the bubble wall. Ó 2008 Elsevier Inc. All rights reserved. Keywords: Cavitation; Shock wave
1. Introduction Ultrasonic irradiation of liquids produces cavitation: the formation, growth, and implosive collapse of bubbles [1]. Cavitation involves either rapid growth and collapse of bubbles (inertial cavitation) or sustained oscillatory motion of bubbles (stable cavitation). Stable oscillations of bubbles induce fluid velocities, emission of acoustic transients, and exert shear forces on the surrounding medium, whereas the collapse of inertial bubbles near a surface experience nonuniformities in their surroundings that results in the formation of high-velocity microjets and the emission of shock waves that propagate through the liquid. This violent collapse is predicted to generate extremely large pressure within the bubble, but, to date, there have been only a limited number of experimental measurements of the shock wave pressure in the close proximity of the bubble wall [2–5]. Given the importance of cavitation in sonochemistry, sonography, gene transfection and drug delivery [6,7], we decided to investigate the final collapse stage of inertial bubbles with high temporal and spatial resolution. We have analyzed the propagation of the shock wave emitted *
Corresponding author. E-mail address: [email protected] (E.A. Brujan).
0894-1777/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2008.01.006
upon the collapse of an inertial cavitation bubble attached to a rigid wall as this case has the largest damage potential. 2. Experimental A schematic diagram depicting the experimental arrangement used for investigating the final stage of the collapse of cavitation bubbles near a rigid boundary is shown in Fig. 1. A sinusoidal ultrasound pulse (1.08 MHz, 30 cycles) is created by the function generator. The output of ±1 V signal is amplified by 60 dB radio frequency amplifier and transmitted to the concave PZT transducer. In the degassed water, ultrasound is gathered at the focus of the transducer where the aluminum block as the rigid wall is placed. Then, in the localized focal volume (<2 mm3) near the rigid wall surface, the high amplitude pressure fluctuation (the absolute values of the peak positive and negative pressure are greater than 50 MPa and 10 MPa) is obtained. After the ultrasound is stopped, the bubble expands to its maximum volume and under the static pressure it collapses violently accompanied by strong shock wave emission [8,9]. The shock wave emitted upon bubble collapse is captured by a high speed camera IMACON 200 with 200 millions/s framing rate and 5 ns exposure time. The image on the fluorescent screen was
E.A. Brujan et al. / Experimental Thermal and Fluid Science 32 (2008) 1188–1191
1189
3. Results and discussion
Fig. 1. Schematic diagram of the experimental set-up.
recorded with an intensified scan ICCD camera system (Photometrics AT200A) with a 4800 3920 pixel array. The signal from the ICCD camera was then digitized with 12-bit resolution (256 grey levels) and passed to a computer. The illumination for the shadowgraph photography is obtained by a short arc strobe with 200 J/flash. The 1 mm gap between the electrodes is used as a point light source and the light is collimated to the parallel illumination by a convex lens (f = 60 mm). A Nikon lens (Ai-Micro Nikkor, 105 mm, F2.8S) with 10 close up rings enabled a field view of 2.53 mm 2.06 mm.
Fig. 2. A high-speed photographic sequence in side view showing the propagation of the shock waves emitted upon the collapse of a cavitation bubble attached to rigid boundary. The boundary is located in the righthand side of each frame. The white arrowheads indicate the location where the distance traveled by the bubble-collapse-induced shock wave was measured. Sequence taken with 200 million frames/s and an exposure time of 5 ns.
Fig. 2 shows a high-speed photographic sequence of the final stage of bubble collapse. Since the photographic record of the bubble dynamics with a converter camera at this framing rate can only cover a small part of the collapse process (40 ns), neither the maximum bubble radius nor the distance between bubble and boundary are known exactly. However, it can be inferred from high-speed photographic sequences taken at smaller framing rates that the maximum bubble radius is about 1.7 mm and the bubble is attached to the rigid boundary; it probably originates from a gas nucleus trapped in a small crevice of the boundary. The minimum radius of the bubble was measured as Rmin = 30 lm (first frame). Two shock waves are emitted upon the first collapse of the bubble. The shock wave indicated by the black arrowhead is probably generated at the impact of a microjet developed in an earlier stage of the collapse onto the rigid wall (microjet-induced shock wave) [10]. The shock wave indicated by the white arrowheads is generated at the minimum bubble volume (bubble-collapse-induced shock wave). The microjet-induced shock wave is, however, so weak that it is barely visible on the photographic frames. Therefore, we concentrate on the motion of the bubble-collapse-induced shock wave as this wave gives an indication of the maximum pressure inside the collapsing bubble. The high-speed photographic sequence gives values for the propagation velocity of the shock front which can be used to calculate the corresponding pressure values at the shock front. The shock wave width cannot be inferred from the photographic images. The shock front can be seen, because the refractive index gradient induced by the shock front refracts the illumination light out of the imaging lens. The width of the shock wave image is determined by the pressure profile, the pressure amplitude, and the aperture of the imaging lens. Thus the width of the shock wave image should not be mistaken for the shock width. We measured the distance r traveled by the bubble-collapseinduced shock wave as a function of the time delay t, and a curve was fitted through the measurement points using the LAB Fit curve fitting software [11]. From the slope of the r(t) curve, the shock wave velocity was derived (Fig. 3). Initially, the shock wave propagates with a velocity of 2520 ± 210 m/s and ends up with one of 1500 m/s. The shock pressure ps can be determined through a measurement of the shock front velocity us if the equation of state of the medium is known. We used the equation of state determined by Rice and Walsh [12] up ¼ c1 10ðus c1 Þ=c2 1 ð1Þ which fits experimental data for pressure values of up to 25 GPa. Here up is the particle velocity behind the shock front, c1 is the sound velocity in water, c1 = 5190 m/s, and c2 = 25306 m/s. The pressure ps is then related to us by [13]
1190
E.A. Brujan et al. / Experimental Thermal and Fluid Science 32 (2008) 1188–1191
Fig. 3. Propagation of the shock front plotted as a function of time. The position of the shock front was extracted from the high-speed photographic record using image processing techniques (open symbols). A curve was then fitted through the measurement points (solid line). The first derivative of the curve corresponds to the shock velocity (dashed line). The standard deviation between data points and fit is smaller than 1 lm for the shock wave position.
ps ¼ c1 q1 us 10ðus c1 Þ=c2 1 þ p1
ð2Þ
where q1 denotes the density of water before compression by the shock wave and p1 is the hydrostatic pressure. We determined the shock wave pressure up to a distance of 110 lm from the bubble wall (Fig. 4). For larger distances the deviations of the shock wave velocity from sonic velocity become very small, which results in a large measurement uncertainty for ps. The maximum shock wave pressure at a distance of 68 lm from the bubble wall is 1.3 ± 0.3 GPa. Up to a distance of 70 lm from the bubble wall, the pressure decreases proportionally to r1.5. The shock pressure decreases faster with increasing propagation distance than in the acoustic limit, where a pressure decay proportional to r1 would be expected for a spherical source [3]. The fast pressure decay is caused by the energy dissipation at the shock front and the nonlinearity of propagation, which
Fig. 4. Shock wave pressure as a function of the propagation distance. The shock wave velocities were converted to shock pressure using Eq. (1). The dashed line indicates the position of the bubble wall at its minimum volume. The number indicates the average slope of the pressure versus distance curve at a distance between 68 lm and 70 lm from the bubble wall.
results in a modification of the pressure profile during propagation [3]. Extrapolating the pressure obtained at a distance of 68 lm from the bubble wall down to the position of the minimum bubble radius with the help of the r1.5 law, the pressure at the bubble wall and thus the pressure acting on the rigid wall may be as high as 7.7 ± 1.6 GPa. It is interesting to note that a similar result was reported by Pecha and Gompf [2]. However, a direct comparison with their result is not appropriate here because they investigated the shock wave emission from stable cavitation bubbles whose behaviour is strongly dependent on the ultrasound amplitude and frequency. We also evaluated the energy dissipation, ED, at the shock front as a function of propagation distance and obtained the shock wave energy by integration over the dissipated energy [14]. The Rankine–Hugoniot equation relates the increase of internal energy per unit mass at a shock front to the change of pressure and density at the shock front [14]: 1 1 1 DðrÞ ¼ ð3Þ ðps ðrÞ þ p1 ðrÞÞ 2 q1 qs ðrÞ where qs is the density behind the shock front. The density is, through conservation of mass [14], us q1 ¼ ðus up Þqs , and momentum, ps p1 ¼ us up q1 , also linked with us by q1 ð4Þ qs ¼ 1 u2pqs s 1
The total change of internal energy during propagation of a spherical shock front from r0 to r1 is obtained by integration of Eq. (3) Z r1 ED ¼ 4pr2 qs ðrÞDðrÞdr ð5Þ r0
Fig. 5 shows the accumulated energy loss, ED, as a function of propagation distance from the bubble wall. The loss rate is highest close to the bubble wall and has considerably decreased at the end of the measurement range. A major part of the shock wave energy is, therefore, already dissipated within the first 100 lm from
Fig. 5. Accumulated energy loss at the shock front as a function of the propagation distance.
E.A. Brujan et al. / Experimental Thermal and Fluid Science 32 (2008) 1188–1191
the bubble wall. The large amount of the shock wave energy loss within the first 100 lm of its propagation indicates that the shock wave-induced biological effects on biological tissues are confined to very small dimensions on a cellular or sub-cellular level. Acknowledgements Supported by Grants from Study for Open and Integrated Research Program of the Science and Technology Agency of Japan. E.A.B. was supported by a Japan Society for the Promotion of Science Invitation Fellowship (Grant S-04206). References [1] C.E. Brennen, Cavitation and Bubble Dynamics, Oxford University Press, Oxford, 1995. [2] R. Pecha, B. Gompf, Microimplosions: cavitation collapse and shock wave emission on a nanosecond time scale, Phys. Rev. Lett. 84 (2000) 1328–1330. [3] A.G. Doukas, A.D. Zweig, J.K. Frisoli, R. Birngruber, T.F. Deutsch, Noninvasive determination of shock-wave pressure generated by optical breakdown, Appl. Phys. B 53 (1991) 237–245.
1191
[4] R. Petkovsek, J. Mozina, G. Mocnik, Optodynamic characterization of shock waves after laser-induced breakdown in water, Opt. Express 13 (2005) 4107–4112. [5] R. Petkovsek, P. Gregorcic, J. Mozina, A beam-deflection probe as a method for optodynamic measurements of cavitation bubble oscillations, Meas. Sci. Technol. 18 (2007) 2972–2978. [6] K.S. Suslick, Sonochemistry, fourth ed., Kirk-Othmer Encyclopedia of Chemical Technology, Wiley & Sons, NY, 1998, 517–541. [7] D. Dalecki, Mechanical bioeffects of ultrasound, Annu. Rev. Biomed. Eng. 6 (2004) 229–248. [8] E.A. Brujan, T. Ikeda, Y. Matsumoto, Dynamics of ultrasoundinduced cavitation bubbles in non-Newtonian liquids and near a rigid boundary, Phys. Fluids 16 (2004) 2402–2410. [9] E.A. Brujan, T. Ikeda, Y. Matsumoto, Jet formation and shock wave emission during collapse of ultrasound-induced cavitation bubbles and their role in the therapeutic applications of high-intensity focused ultrasound, Phys. Med. Biol. 50 (2005) 4797–4809. [10] A. Shima, K. Nakajima, Collapse of a non-spherical bubble attached to a rigid wall, J. Fluid Mech. 80 (1977) 369–391. [11] P.W. Silva, C. Silva, LAB Fit Curve Fitting Software (Nonlinear Regression and Treatment of Data Program) V 7.2.32. [12] M.H. Rice, J.M. Walsh, Equation of state of water to 250 kilobars, J. Chem. Phys. 26 (1957) 824–830. [13] A. Vogel, S. Busch, U. Parlitz, Shock wave emission and cavitation bubble generation by picosecond and nanosecond optical breakdown in water, J. Acoust. Soc. Am. 100 (1996) 148–165. [14] G.E. Duvall, G.R. Fowles, High Pressure Physics and Chemistry, Academic Press, NY, 1963, 209–291.