P. B. MULLER et al. PHYSICAL REVIEW E 88, 023006 (2013)
and less than ±0.1 μm in the x and y directions. Two scan
positions along the z direction were used to cover the whole
cross-sectional area of the channel.
Monodisperse spherical polystyrene particles with nominal
diameters of 5.33 μm (SD 0.09) and 0.537 μm (PDI
0.005) were used for the experiments (ρps = 1050 kgm−3 and
κps = 249 TPa−1). For simplicity we will refer to them as
5-μm-diameter and 0.5-μm-diameter particles, respectively.
The particles were fabricated and labeled with a proprietary
fluorescent dye by Microparticles GmbH to be visualized with
an epifluorescent microscopy system. The illumination was
provided by a continuous diode-pumped laser with 2 W at
532 nm wavelength.
Once the particle 3D positions had been detected using
the APTV technique, their trajectories and velocities were
calculated. Due to the low seeding density in the experiments,
the particle interdistance was large enough to employ a simple
nearest-neighbor approach in which the particle in one frame
is identified with the closest particle in the next frame.
The method was compared with more sophisticated ones as
predictors and probabilistic algorithms with identical results.
Trajectories composed by less than five particle positions
were rejected. From the obtained trajectories the velocities
could be calculated given the capture rate of the camera.
Different approaches have been followed depending on the
type of trajectories expected. For particles following almost
straight paths as those dominated by radiation force, a simple
two-position approach was used and the velocities were
calculated based on the frame-to-frame particle displacement.
For particles with highly curved paths, like those present in
streaming-dominated flows, a more sophisticated multiframe
approach has been followed, as those reported already by
Hain and K¨ahler 46 for μPIV. In our case, each velocity
data point was calculated from a trajectory segment composed
by four consecutive points. Such a multiframe approach
applied for PTV has been shown to better solve the velocity
vector positions and values when the trajectories present large
curvatures and for high-shear flows 47.
IV. RESULTS
A. APTV measurements
Examples of the measured 3D trajectories of the 5-μmdiameter
particles are shown in Fig. 4(a). The data were
collected from 10 consecutive experiments with the piezo
operated at 1.94 MHz and peak-to-peak voltage of Upp =
0.91V. An overall number of 111 trajectories were determined.
The 5-μm-diameter particles are affected mainly by the
acoustic radiation force Frad
y that quickly pushes them to the
p
y 24,45.
center of the channel with a horizontal velocity u
At the vertical pressure nodal plane y = 0, Frad vanishes and
the hitherto negligible drag force from the acoustic streaming,
shown in Fig. 2(b), slowly drags the particles towards the top
and bottom of the channel.
Examples of the measured 3D trajectories of the 0.5-μmdiameter
particles are shown in Fig. 4(b). The data were
collected from four consecutive experiments with the piezo
operated at 1.94 MHz and peak-to-peak voltage of Upp =
1.62V. An overall number of 731 trajectories were determined.
FIG. 4. (Color online) Measured particle trajectories (thin black
lines) obtained using the 3D-APTV technique in the microchannel
(gray walls) actuated at the 1.94-MHz horizontal half-wave resonance.
For selected trajectories, the particle positions are represented
by dots. (a) 5-μm-diameter particles moving (thick arrows) to
the vertical center plane y = 0, and (b) 0.5-μm-diameter particles
exhibiting circular motion as in Fig. 2(b).
The acoustic radiation force Frad
y is in this case minute and the
particles are primarily transported by the acoustic streaming
v2 of the fluid resulting in particle trajectories following the
four vertical vortices in the bulk, shown in Fig. 2(b). The setup
and results are illustrated in the entry for the APS-DFD 2012
Video Gallery 48.
B. Comparison of theory and experiments
Theoretically, the acoustophoretic particle velocity up
is given by Eq. (32) combined with the expressions for
the streaming velocity of the liquid Eqs. (25) and (31)
and the expression for the radiation force on the particles
Eq. (33). The amplitudes of both the acoustic streaming
and the radiation force depend linearly on the acoustic energy
density Eac through Eqs. (12) and (33). To make a theoretical
prediction of the motion of the 0.5-μm-diameter particles,
we need to determine the acoustic energy density E
0.5μm
ac .
This calibration was done in situ based on the measurements
of the 5-μm-diameter particles, by the following three-step
procedure.
First, we determined the acoustic energy density E
5μm
ac for
the experiment with the 5-μm-diameter particles. This was
done by fitting the sin(π˜y
)-dependent expression (36) for
u
p
y ( ˜ y,0) to the measured instantaneous velocities, using the
amplitude as the only fitting parameter 26,45. The small
contribution from the acoustic streaming to the 5-μmdiameter
particle velocity was taken into account although
it constituted only 6% of the total particle velocity. The fit
showed good agreement between theory and experiment, and
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