P. B. MULLER et al. PHYSICAL REVIEW E 88, 023006 (2013)
FIG. 6. (Color online) (a) Color plot of the number of times the
velocity has been measured in each square bin. (b) Color plot of
standard error of the mean (SEM) particle velocity in each square
bin.
see when the experimental velocity respectively overshoots
and undershoots the analytical prediction. Figure 7(a) shows a
color plot of up in the channel cross section, while Fig. 7(b)
FIG. 7. (Color online) (a) Color plot of the difference between
the experimental and analytical acoustophoretic particle speeds up
Eq. (38). (b) Line plots of up along the dashed lines in (a), marked
A, B, C, D, E, and F, with error bars indicating the 1σ error of up.
The lines are positioned at y = 0 μm, y = ±91.7 μm, z = 0 μm, and
z = ±52.3 μm. The off-center lines go through the rotation centers
of the flow rolls, and consequently up ≈ up
y ey in B, D, and F, while
up ≈ up
z ez in A, C, and E.
FIG. 8. (Color online) Experimental data from Ref. 45 compared
with the theoretical predictions of Eqs. (36) and (37). μPIV
measurements, in the center plane z = 0, of the y component of
the acoustophoretic velocity up
y (y,0)x (open and closed dots) for
0.6-μm-diameter polystyrene particles in water, small enough that
streaming dominates and up ≈ v2. The observed motion (thick
arrows) in (a) and (b) resembles the analytical results shown in
Figs. 2(b) and 2(c), respectively. For each value of y, the measured
velocity up
y is averaged along the x coordinate, with resulting SEM
smaller than the size of the dots. The sinusoidal parallel-plate
prediction (thin line) Eq. (36) is fitted to the data points far from
the side walls (open dots), while the rectangular-channel prediction
(thick line) Eq. (37) is fitted to all data points (open and closed dots).
In both fits the acoustic energy Eac is treated as a free parameter.
(a) The half-wave resonance λ/2 = w (n = 1) with f = 1.940 MHz
and Upp = 1 V. (b) The full-wave resonance λ = w (n = 2) with
f = 3.900 MHz and Upp = 1 V.
shows line plots of up along the dashed lines in Fig. 7(a),
allowing for a more detailed study of the spatial dependence
of the difference. These lines are chosen to go through the
rotation centers of the flow rolls. The error bars in Fig. 7(b)
show the 1σ error of up, taking into account both the SEM
for the experimental measurements Fig. 6(b) and the error
of the analytical prediction (4%) inherited from the derived
value for E
0.5μm
ac . The experimental and analytical velocities
do not agree within the error of up, moreover, a trend of
the experimental velocities being larger than the analytical
predictions is seen.
A further comparison between the analytical model presented
in this paper and experimental measurements on
0.6-μm-diameter polystyrene particles from Ref. 45 is shown
in Fig. 8. These particles are dominated by the drag from
the acoustic streaming, and in this comparison we are only
interested in studying how the side walls influence the shape
of u
p
y ( ˜ y,0) Eq. (37). Consequently, the amplitude of the
streaming velocity, and thus the acoustic energy density, was
treated as a fitting parameter. The experimental results support
our analytical prediction (37) (thick line) for the rectangular
channel with side walls, which shows a suppression of u
p
y near
the walls compared to the sinusoidal form of u
p
y in Eq. (36)
(thin line) predicted for the parallel-plate channel without side
walls. This is particularly clear for the full-wave resonance
λ = w (n = 2) Fig. 8(b). The difference in the amplitude of
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