The 'acoustic scallop': a bubble-powered actuator

INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF MICROMECHANICS AND MICROENGINEERING

doi:10.1088/0960-1317/16/8/029

J. Micromech. Microeng. 16 (2006) 1653–1659

The ‘acoustic scallop’: a bubble-powered actuator R J Dijkink, J P van der Dennen, C D Ohl and A Prosperetti1 Faculty of Applied Sciences and Burgerscentrum, University of Twente, 7500 AE Enschede, The Netherlands

Received 19 January 2006, in final form 19 June 2006 Published 11 July 2006 Online at stacks.iop.org/JMM/16/1653 Abstract The device described here consists of a millimeter-size tube immersed in a liquid, closed at one end, and partially filled with gas. A sound field in the liquid causes the gas volume to pulsate alternately expelling and drawing liquid through the open end of the tube. According to general fluid mechanical principles, the liquid exits the tube as a jet, while it enters it from the entire solid angle available. Averaged over a cycle, this flow pattern results in a net source of momentum which, by reaction, exerts a force on the tube. Possible applications include the self-propulsion of the tube, a pump and a rotary actuator. Thereby a single device can be made to respond to different frequencies, for example, switching the direction of the force by changing the sound frequency. The acoustic power required is well below biologically hazardous levels, which would permit, among others, powering the device remotely through living tissue. M This article features online multimedia enhancements (Some figures in this article are in colour only in the electronic version)

1. Introduction It is a well-known fluid mechanical fact that, provided the Reynolds number is not too small, fluid pushed out of a tube forms a jet, while fluid sucked into a tube enters it more or less omni-directionally (compatibly with the geometry of the tube mouth)2 . This asymmetry is due to the inability of a viscous boundary layer to sustain the adverse pressure gradient that develops at the tube mouth when fluid leaves it. On the other hand, when the fluid enters the tube, the pressure gradient is favorable and the boundary layer remains attached. This phenomenon is responsible, among others, for the fact that a candle can be put off by blowing, but not by sucking3 . 1

Permanent address: Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA. 2 A short ‘pulse’ of fluid will generate a vortex ring upon exiting the tube which, just as a jet, also carries momentum. 3 There is a connection between the different behavior of fluid entering or exiting a tube and the so-called Feynman sprinkler (Feynman 1985, Jenkins 2004), i.e. a rotary garden sprinkler in which fluid is sucked in rather than pushed out. A recent review of the copious literature on the subject and a nice explanation of the issues involved is provided by Jenkins (2004). The principle behind the regular sprinkler was already discovered by Hero of Alexandria in the second century BCE. In a short note, Belson (1956) considers the

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The asymmetry in the flow behavior at the tube exit has the consequence that, upon integrating over a full cycle, an oscillatory flow alternatively exiting and entering a tube is a net source of momentum but not of mass. An important practical application, which has recently received considerable attention, is in flow control (Glezer and Amitay 2002), where it is referred to as ‘synthetic jet’. Another application to propulsion has been patented (Payne 1975, 1977), but does not seem to have found application beyond an old-fashioned toy, the ‘putt-putt boat’. In this device, the oscillatory flow is caused by the growth and collapse of vapor bubbles in a U-shaped tube and is able to propel a toy boat. Apparently this entertaining device was invented a very long time ago but, while some articles on it have been published in semi-technical journals (Miller 1958, Mackay 1958, Finnie and Curl 1963, Schlichting and Rodewald 1990), it does not seem to have been the object of a serious scientific investigation. reverse (or ‘empty’) Hero’s engine and correctly concludes that it will not turn. He concludes his note with the words ‘If one oscillates the pressure [in the chamber] . . . a rather mysterious continuous rotation is produced.’ This is precisely the ‘acoustic windmill’ described below in the concluding section of this paper.

© 2006 IOP Publishing Ltd Printed in the UK

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Figure 1. A photo of the device studied in this paper attached with a drop of glue (dark mass on the right) to the tip of a flexible cantilever (vertical line on the right). The teflon tube has a length of 3 mm, an outer diameter of 750 µm and an inner diameter of 250 µm. The dark cylinder in the tube is the air bubble.

Here we describe a novel implementation of this synthetic jet concept of potential use in microfluidics and medicine. Our device, which may be called an ‘acoustic scallop’, consists of a small plastic tube (inner diameter 200–300 µm, length 2– 4 mm) closed at one end and immersed in water. The tube is partially filled with a gas bubble which executes resonant volume pulsations under the action of an ambient sound field in the liquid (frequency around 1 kHz, pressure amplitudes below 3 kPa) generated by a piezoelectric transducer. As the bubble compresses and expands, liquid enters and exits the open end of the tube exhibiting the flow asymmetry mentioned before. As a result, the device is able to propel itself if free or, if attached to the wall of a conduit, to act as a micropump and drive a flow. As another application, several of these devices mounted on spokes attached to a wheel can generate a rotary motion. The effect only requires that the Reynolds number of the flow exiting the tube be large enough for inertia to be important (Purcell 1977). Hence the principle appears to be applicable over a wide range of dimensions and frequencies. This implementation of the synthetic jet principle constitutes yet another application of bubbles in microfluidics (Beebe et al 2002, Nguyen et al 2002, Laser and Santiago 2004, Jun and Kim 1998, Chen and Santiago 2002, Hua et al 2002, Tsai and Lin 2002, Maxwell et al 2003, Wang et al 2004). While the present investigation was simply motivated by the intriguing physics of the device, for practical applications an interesting feature is its ability to be powered remotely via acoustic waves weak enough, for example, to propagate through living tissue without negative side effects.

2. Experiment We built our devices using 2–4 mm long sections of 750 µm outer diameter polyethylene or teflon commercial tubing, one end of which was closed with a drop of UV hardening glue (dark mass on the right in figures 1 and 2). The inner diameter of the tube was 250 µm. Initial tests showed that the strong affinity of water for polyethylene impeded the motion of the 1654

Figure 2. Detail of the previous image with the middle part removed. The dark mass on the right (indicated with a white arrow) is the glue used to close one end of the tube and to attach it to the cantilever. The free surface of the bubble (point B) bows outward under the action of the expanding phase of the sound field. The inner bright part of the bubble is an optical artifact due to the light being refracted at the curved teflon–gas interfaces.

Figure 3. A three-dimensional schematic of the setup used in the experiment with the liquid filled glass tank. The top of the tank is covered with a plexiglas lid making sure not to trap any gas bubble. The ‘acoustic scallop’ is the small tube attached to the cantilever near the bottom of the tank. The sound field is produced by the piezoelectric transducer attached to one of the side walls. An amplified signal from a function generator is used to drive the piezo and a stroboscopic LED which illuminates the device at specified instants of the oscillation period. Imaging is done with a CCD camera looking through the bottom of the glass tank. A delay generator between the LED and the function generator permits us to choose the phase of the oscillation cycle at which the image is recorded. The images are stored on a PC along with pressure measurements taken with a hydrophone mounted through the plexiglas lid. Sound-absorbing rubber sheets line the three walls not containing the piezo. The rubber opposite to the piezo is inclined at a 20◦ angle with a self-hardening foam filling up the space between the inclined rubber and the glass wall of the tank. The small construct in the lower corner allows for easy and fast mounting and removal of the cantilever and tube.

gas–liquid–solid contact line producing an irregular behavior and a weaker effect. Thus, only data obtained with the teflon tubes are described below. The device was powered by a sound field generated in a glass cell. For most of the experiments reported here the cell was cubic with an inner side of 50 mm and a wall thickness of 2.5 mm (see figure 3). A 40 mm diameter, 2 mm thick piezoelectric transducer, operated well below its resonance frequency, was glued to the outside of one of the vertical walls

The ‘acoustic scallop’: a bubble-powered actuator

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Figure 4. In the upper part of the figure a sinusoid is fitted to the measured effective displacement of the bubble surface (defined as the change of the bubble volume divided by the tube cross section). These measurements are taken from photos, examples of which are reproduced in the 4 images in the lower part of the figure. Here the frequency was 1.6 kHz and the acoustic pressure amplitude approximately 0.4 kPa. The dark mass is the front of the bubble bounded above and below by the inner wall of the tube, the open end of which is to the left outside the field of view. The light-colored regions are an optical artifact due to the refraction of light by the curved surfaces of the system (see also figures 2, 7 and 8). 5 1.2kHz 1.4kHz 4

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of the cell to generate the acoustic field. In order to avoid possible contamination of the results by standing acoustic waves, the two vertical walls parallel to the transducer axis were coated with a 2 mm thick rubber sheet. The vertical wall opposite the transducer was covered by a similar rubber sheet inclined downward by 20◦ to the vertical; the wedgeshaped space between this rubber sheet and the vertical cell wall behind it was filled with a self-hardening foam. The sound field in the cell was characterized by a miniature hydrophone and was found to be fairly uniform in the frequency range of present interest. A feedback loop maintained the sound amplitude constant at the desired level as the frequency was varied. To avoid waves on the liquid surface, the cell was filled to the brim and covered by a glass lid taking care to prevent the entrapment of gas bubbles. The liquid used was demineralized water, in which the bubble volume was stable within 1% for 15–30 min necessary to take a complete series of data. In order to confirm that, in the lower range of acoustic power used, the bubble behaved as a linear damped oscillator and to find its resonance frequency, we measured from each frame of a high-speed movie the displacement of the point B of figure 2 as well as that of the gas–liquid–solid contact line. With these data, by assuming a truncated spherical shape for the gas–liquid interface, it was possible to measure the gas volume versus time during an acoustic cycle as in Geng et al (1999). Sample results of this procedure for a frequency of 1.6 kHz and an acoustic pressure amplitude of 0.4 kPa are shown in figure 4 where it is seen that a sinusoid provides an excellent fit to the data. The force exerted by the scallop was measured by attaching it to a flexible plastic cantilever (visible on the right in figure 1) calibrated by fastening to its tip small putty beads the weight of which was measured with an accuracy of ±100 µg. The deflection of the cantilever was measured by means of a laser beam; we estimate that the accuracy of the force measurements reported below was about 15%. Figure 5 shows the force exerted by the device attached to the tip of the cantilever versus the acoustic amplitude at frequencies of 1.2, 1.4, 1.7 and 2 kHz. The acoustic energy flux ranges from 0 to 0.27 mW cm−2 , well below biologically hazardous levels (above 94 mW cm−2 , see FDA 510(k) 1997 Guidance Report). The strong response at 1.4 kHz is indicative of resonance in the neighborhood of this frequency. In order to make sure that these effects were due to the bubble compressibility and not to acoustic streaming or other artifacts, we repeated the experiment with tubes completely filled with water, or air filled tubes plugged at both ends. No appreciable cantilever deflection was found in either case. Use was also made of a rectangular cell with dimensions 60 × 100 mm2 and a height of 60 mm, with the transducer attached to one of the larger vertical walls. In particular, we tested the ability of the scallop to ‘swim’ in this second cell. For this purpose, the scallop was put inside a 1.6 mm diameter glass tube placed horizontally in the cell. Figure 6 shows an example of the scallop position and velocity versus time. The sound field had a frequency of 1.55 kHz and a pressure amplitude of approximately 1 kPa. The mean velocity is about 1.35 mm s−1 and the peak velocity about 10 mm s−1 . It is seen that, while the mean displacement is an increasing function of

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Figure 5. Force exerted by the ‘acoustic scallop’ versus acoustic pressure amplitude at four different frequencies. The response is greatest at 1.4 kHz, which is close to the resonance frequency of the bubble in the tube.

time, the velocity oscillates so that there is a backward motion during part of each cycle. 1655

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beginning of the bubble compression stage, shows that, while the liquid in line with the tube axis is still moving outward, liquid starts moving in from the sides, a process that continues to be evident in the third frame. The last frame corresponds to the situation just after the reversal of the motion of the bubble surface and shows the beginning of the outward axial motion. A ‘summary’ image obtained by averaging over 300 complete cycles (3000 individual frames) is shown in figure 8. The average flow appears to be partitioned into distinct zones of inflow and outflow because, on the axis, there is a prevalence of outflow while the opposite happens off-axis. This image provides a clear illustration of the asymmetry on which the effect studied here rests. It is evident from these data that the reason for the non-monotonic displacement of the scallop is that, due to the relatively large thickness of the tube, when the bubble contracts, the fluid entering the tube approximately comes from 1/2 of the total solid angle, rather than from the entire solid angle. As a consequence, some unbalanced net momentum remains. This backward motion, which decreases the efficiency of the propulsion, could be decreased by having a very small wall thickness in the vicinity of the tube mouth.

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Figure 6. Position (top) and velocity (bottom) versus time of a freely ‘swimming’ scallop. The backward portion of the displacement during each cycle is caused by a net unbalanced momentum of the liquid entering the tube due to the tube wall thickness (see the text). The sound frequency was 1.55 kHz and the acoustic pressure amplitude 1 kPa. M An AVI movie showing the scallop moving is available from stacks.iop.org/JMM/16/1653

To better understand this feature, we carried out particle tracking velocimetry (PTV) measurements of the liquid flow velocity near the open end of the tube. This technique permits us to determine the liquid velocity by measuring the displacement of suspended neutrally buoyant particles in two images separated by a short time interval. The velocity field at four phases of the sound field (0◦ , 72◦ , ◦ 144 and 216◦ ) is shown in figure 7. The driving frequency was 1.5 kHz and the pressure amplitude 1 kPa. The shutter time was 1/40 000 s and the frame rate 15 000 fps. The total number of particles tracked was over 24 000. Each one of these images is the result of a phase average over 300 instantaneous frames, all corrected for the translation of the scallop. The first image is close to a maximum of the outward velocity. There are no data inside the tube or in front of the tube exit as the rapid motion of the particles blurred their image, which caused errors in the data processing. We estimate the average speed of the bubble interface to be about 0.6 m s−1 . The second image, at the 1656

A gas bubble partially filling a tube closed at one end and immersed in water (figure 1) behaves as a damped harmonic oscillator when subjected to an oscillating pressure field (O˜guz and Prosperetti 1998, Chen and Prosperetti 1998, Geng et al 1999). The resonance frequency f0 of this oscillator is approximately given by (O˜guz and Prosperetti 1998) 1 κP0 f0 = , (1) 2π ρL0 LB in which κ is a frequency-dependent parameter determined by the thermodynamic cycle executed by the gas in the course of the oscillations, P0 is the undisturbed pressure in the bubble, ρ the liquid density, LB the length of the bubble and L0 the length of the liquid column comprised between the bubble interface and the exit of the tube (figure 2). In general 1 κ γ , where γ ( 1.4 for air) is the ratio of the specific heats of the gas; small dimensions and high frequencies favor a nearly isothermal behavior (Chen and Prosperetti 1998). Figure 9 shows the resonance peak measured for a scallop with LB = 3.4 mm and L0 = 0.4 mm. With κ 1.2, equation (1) gives 1.50 kHz to be compared with the measured peak position of 1.55 kHz. The oscillation amplitude is maximum close to resonance where, approximately, it takes the value pQ p = (2) A= 2ρL0 ω0 b 8π 2 ρL0 f02 in which p is the acoustic pressure amplitude, b the damping parameter, ω0 = 2πf0 and Q = ω0 /b the so-called quality of the resonance (O˜guz and Prosperetti 1998, Chen and Prosperetti 1998). The peak Reynolds number of the oscillatory flow can be estimated as Re = 2Rω0 A/ν, where A is the amplitude of the bubble oscillations and ν the liquid kinematic viscosity. Thus RpQ . (3) Re 2πρνL0 f0


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Figure 7. The flow field obtained from PTV data at four phases of the acoustic cycle: 0◦ , 72◦ , 144◦ and 216◦ (left to right, top to bottom). The driving frequency is 1.5 kHz and the acoustic pressure amplitude 1 kPa. The first frame is close to a maximum of the outward velocity. The second image is taken at the beginning of the bubble compression stage, the third near the maximum inward velocity and the last one at the beginning of the bubble expansion. 0.09

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Figure 8. The PTV velocity field averaged over 300 cycles (3000 individual frames) for the conditions of the previous figure. The average flow appears to be partitioned into distinct zones of inflow and outflow because, on the axis, there is a prevalence of outflow while the opposite happens off-axis.

From results such as those shown in figure 9 it is possible to estimate that Q 2. If we use Q = 2 in (3), for p = 1 kPa, we find Re 63. By writing Re = 2Ru/ν, where u is the peak velocity of the liquid jet, for Re = 63 we find u 0.13 m s−1 , in good agreement with the PTV results of figure 7. An estimate of the magnitude of the force exerted by the device can be obtained as follows (Finnie and Curl 1963, Schlichting and Rodewald 1990). The mean speed of the liquid expelled during the expansion phase is of the order of the peak-to-peak amplitude 2A of the oscillations of the bubble interface divided by the duration 1/(2f ) of the expansion phase, in which f is the sound frequency. If S = πR 2 is the cross section of the tube, the impulse of the ejected liquid can therefore be estimated to be (2ASρ)(4f A) = 8ρf SA2 . The estimation of the impulse of the entering liquid is harder.

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Figure 9. Measured resonance response of a scallop with LB = 3.4 mm and L0 = 0.4 mm. The theoretical resonance frequency is estimated from equation (1) to be about 1.50 kHz; the measured peak position is 1.55 kHz.

Suppose that the liquid enters the region of the tube mouth by flowing with a uniform radial velocity u through a hemisphere of radius Rh during a time 1/(2f ). The impulsein the direction of the axis of the tube is then (1/2) 2πRh2 ρu2 (1/2f ) where the prefactor 1/2 accounts for the projection of the velocity on the tube axis. The mass of fluid entering is 2πRh2 ρu (1/2f ), which must equal 2ASρ. Using this continuity constraint, the impulse entering the tube is then estimated as ρSAu. Combining the two estimates of the impulse we find for the force 1 u . (4) F = 8ρf 2 SA2 1 − 8 Af 1657

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Figure 10. Comparison of the measured force (black line) with the theoretical expression (5) as a function of the pressure amplitude at 1.4 kHz.

Using the continuity constraint for the entering liquid one finds u/(Af ) = 2(R/Rh )2 . If we take the estimate Rh R, the previous expression gives F 6ρf 2 SA2 .

(5)

In order to compare this result with experiment we used the measured values of the oscillation amplitude A. An example of such a comparison is shown in figure 10 for the f = 1.4 kHz data of figure 5. Although the orders of magnitude and the slope of the lines are comparable, two evident differences emerge from this comparison. Firstly, the data seem to exhibit a threshold effect and, second, a quadratic dependence on p is not clear. The threshold is probably a Reynolds-number effect. As already mentioned, the flow asymmetry over the cycle is a consequence of fluid inertia and, in particular, of the detachment of the streamlines from the contour of the nozzle during the expansion phase of the bubble. No such phenomenon would take place at low Reynolds number. The dependence of F on p raised to a power lower than 2 could be understood if the oscillation amplitude were to increase less than linearly with the pressure amplitude. This phenomenon would occur, e.g., if the damping were not simply proportional to the velocity, as might be expected to happen due to the contact line motion inside the tube. By impeding the free oscillation of the bubble surface, this effect would also increase the threshold. It may be noted that some of the results shown before, such as the swimming of figure 6 or the PTV of figure 7, were taken with a pressure amplitude of 1 kPa, while, according to the results of figure 10, there does not seem to be much of an effect at this pressure amplitude. The explanation of this apparent contradiction lies in the use of different scallops and different cells in the two sets of experiments.

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Figure 11. An ‘acoustic windmill’ formed by attaching the devices described in this paper to the spokes radiating from a pivoted hub. The dark mass in the lower part of the image is the slanted wall of the cell which partially obscures the view (see figure 3) M An AVI movie of this figure is available from stacks.iop.org/JMM/16/1653.

ambient sound field. An aspect of this arrangement of potential practical interest is that the actuator is powered remotely by the sound source with no need for a direct physical contact with it. The presence of a fluid medium capable of supporting sound waves is all that is required. In particular, sound waves of sufficient amplitude to produce the observed effect would readily propagate in biological tissues and would be well below hazardous levels. The principle described in this work can be exploited in a variety of ways. For example, by attaching a tube to the end of a series of spokes radiating out of a hub as shown in figure 11, we made an ‘acoustic windmill’ rotating at a speed of a few radians per second. By joining two oppositely oriented tubes with different resonance frequencies to the tip of a cantilever, we have demonstrated the possibility of bi-directional motion of the ‘scallop’ by switching the frequency of the drive. By attaching the tube at the center of a rectangular channel, the net impulse imparted to the liquid may result in a flow—an acoustic pump. While this study has focused on a demonstration of the principle, we have made no attempt to develop a more quantitative theory or to optimize the device. For the former, it would appear that a numerical simulation would be the most direct step to be taken. The latter objective would be greatly facilitated by a more systematic way to make the scallops. For this study they were made by hand with an inherent strong variability among devices.

4. Conclusions Acknowledgments In this paper, we have described some preliminary experiments on a novel bubble-powered actuator which may be called an ‘acoustic scallop’. The force exerted by the actuator is due to the asymmetry in the flow entering and exiting a tube. The oscillating flow is produced by subjecting a gas bubble to an 1658

This work was supported by the Stichting voor Fundamenteel Onderzoek der Materie of The Netherlands (FOM) and the Nederlandse Organisatie voor Wetenschappelijke Onderzoek (NWO).


References Beebe D, Mensing G and Walker G 2002 Physics and application of microfluidics in biology Ann. Rev. Biomed. Eng. 4 261–86 Belson H S 1956 ‘Empty’ Hero’s engine Am. J. Phys. 24 413–4 Chen X and Prosperetti A 1998 Thermal processes in the oscillations of gas bubbles in tubes J. Acoust. Soc. Am. 104 1389–98 Chen C and Santiago J 2002 A planar electroosmotic micropump J. Microelectromech. Syst. 11 672–83 FDA 1997 Information for Manufacturers Seeking Marketing Clearance of Diagnostic Ultrasound Systems and Transducers, available online http://www.fda.gov/cdrh/ode/ulstran.pdf Feynman R P 1985 Surely You’re Joking Mr. Feynman (New York: Norton) Finnie I and Curl R L 1963 Physics in a toy boat Am. J. Phys. 31 289–93 Geng X, Yuan H, O˜guz H and Prosperetti A 1999 The oscillations of gas bubbles in tubes: experimental results J. Acoust. Soc. Am. 106 674–81 Gleick J 1992 Genius: The Life and Science of Richard Feynman (New York: Pantheon) Glezer A and Amitay M 2002 Synthetic jets Annu. Rev. Fluid Mech. 34 503–29 Hua S, Sachs F, Yang D and Chopra H 2002 Microfluidic actuation using electrochemically generated bubbles Anal. Chem. 74 6392–6 Jenkins A 2004 An elementary treatment of the reverse sprinkler Am. J. Phys. 72 1276–82

Jun T K and Kim C J 1998 Valveless pumping using traversing vapor bubbles in microchannels J. Appl. Phys. 83 5658–64 Laser D J and Santiago J G 2004 A review of micropumps J. Micromech. Microeng. 14 R35–R64 Mackay R S 1958 Boat driven by thermal oscillations Am. J. Phys. 26 583–4 Maxwell R, Gerhardt A, Toner M, Gray M and Schmidt M 2003 A microbubble-powered bioparticle actuator J. Microelectromech. Syst. 12 630–40 Miller J S 1958 Physics in a toy boat Am. J. Phys. 16 199 Nguyen N, Huang X and Chuan T 2002 MEMS-micropumps: a review J. Fluids Eng. 124 384–92 O˜guz H and Prosperetti A 1998 The natural frequency of oscillation of gas bubbles in tubes J. Acoust. Soc. Am. 103 3301–8 Payne P R 1975 Heat engine in the form of a water pulse-jet US Patent 3,898,800 Payne P R 1977 Pulse-jet water propulsor US Patent 4,057,961 Purcell E M 1977 Life at low Reynolds number Am. J. Phys. 45 3–11 Schlichting H and Rodewald B 1990 Physikalische ph¨anomene am dampf-jet-boot Praxis Naturwissenschaften—Phys. 39 19–25 Tsai J and Lin L 2002 Active microfluidic mixer and gas bubble filter driven by thermal bubble micropump Sensors Actuators A 97–98 665–71 Wang G R, Santiago J G, Mungal M G, Young B and Papademetriou S 2004 A laser induced cavitation pump J. Micromech. Microeng. 14 1037–46

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