The `acoustic scallop`: a bubble

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
0960-1317/06/081653+07$30.00
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
1653
R J Dijkink et al
0.25m m
0.75m m
L0
LB
B
3m m
1mm
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
(a )
(b)
( c)
(d )
0.25 mm
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
1.7kHz
2.0kHz
3
Force [ µN]
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
2
1
0
0
0.5
1.0
1.5
Pressure amplitude [kPa]
2.0
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
R J Dijkink et al
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.
40
35
Displacement [µm]
30
25
20
15
10
5
0
−5
0
5
10
15
Time [ms]
20
25
15
10
3. Theoretical considerations and results
Velocity [mm/s]
5
0
−5
−10
0
1
2
3
4
Time [ms]
5
6
7
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
(Og̃uz and Prosperetti 1998, Chen and Prosperetti 1998, Geng
et al 1999). The resonance frequency f0 of this oscillator is
approximately given by (Og̃uz 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 (Og̃uz 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
The ‘acoustic scallop’: a bubble-powered actuator
120mm/s
120mm/s
100µm
100µm
120mm/s
120mm/s
100µm
100µm
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
0.08
Amplitude [mm]
0.07
120mm/s
0.06
0.05
100µm
0.04
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.
0.03
800
1000
1200
1400
1600
Frequency [Hz]
1800
2000
2200
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
R J Dijkink et al
7
6
Force [µN]
5
4
3
2
1
0
0.2
0.4
0.6
0.8
1
1.2
Pressure amplitude [kPa]
1.4
1.6
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.
10mm
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).
The ‘acoustic scallop’: a bubble-powered actuator
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