8
frequency-domain simulation, and this constitutes the
primary validation of the unsteady non-periodic simulations.
FIG. 4. (Color online) (a) Snapshot of the oscillatory firstorder
velocity field v1 (vectors) and its magnitude color plot
ranging from 0 m/s (black) to 0.7 m/s (white) at t = 3000 t0.
(b) Snapshot of the oscillatory second-order velocity field v2
(vectors) and its magnitude color plot ranging from 0 m/s
(black) to 0.02 m/s (white) at t = 3000 t0. (c) Snapshot of the
unsteady time-averaged second-order velocity field
v2
(vectors),
Eq. (30), and its magnitude color plot ranging from 0
mm/s (black) to 0.1 mm/s (white) at t =
3000 t0. (d) Steady
time-averaged second-order velocity field
vfd
2 (∞)
(vectors),
Eqs. (11) and (12), and its magnitude color scaling as in (c).
In both the time-domain and the frequency-domain simulations
the parameters of the oscillating velocity boundary condition
was ! = 2fres and vbc = !d, with wall displacement
d = 1 nm.
D. Build-up of the velocity field
To visualize the build-up of the acoustic fields over
short and long timescales, we have chosen the three point
probes shown in Fig. 1(b). The oscillating first-order velocity
field is probed in the center of the channel (0, 0),
far from the walls in order to measure the bulk amplitude
of the acoustic field. The horizontal component of
the second-order velocity vy2 is probed on the horizontal
symmetry axis at ( 1
4w, 0), where the oscillatory component
v2!
2 has it maximum amplitude. The vertical component
of the second-order velocity vz2 is probed on the
vertical symmetry axis at (0, 1
4h) where the oscillatory
component v2!
2 is small and of the same order as the
quasi-steady component v0
2, making the unsteady timeaveraged
second-order velocity at this point a good probe
for the quasi-steady streaming velocity.
In Fig. 5 is shown the build-up of the velocity probes
(a-c) and their time-averages (d-f) for the first 20 oscillations.
The thick lines are the oscillating velocities
while, the thin lines are the envelopes of the oscillations.
Already within the first 20 oscillation periods we see in
Fig. 5(f) the build-up of a quasi-steady velocity component.
The unsteady time-averaged horizontal velocity
vy2
, Fig. 5(e), is still primarily oscillatory, showing
that for this probe the oscillatory component v2!
2 is much
larger than the quasi-steady component v0
2.
The temporal evolution of the velocity probes on the
longer time scale up to t = 1500 t0 is shown in Fig. 6.
In Fig. 6(a) and (b) the amplitudes of the oscillatory
first- and second-order velocity components are seen to
stabilize around t = 700 t0 ∼ 10 E. The steady amplitudes
of the velocity probes Fig. 6 agree with the theoretical
predictions of Eq. (27), yielding orders of magnitude
v!
1 /cs ∼ 3 × 10−4 (Fig. 6(a)), v2!
2 /cs ∼ 5 × 10−5
2/cs ∼ 1 × 10−7 (Fig. 6(e) and 6(f)).
(Fig. 6(b)), and v0
The time-average of vy1 tends to zero for long times
as it is purely oscillatory, whereas the time-average of
vy2 tends to the magnitude of the quasi-steady component
v0
2, because the large but now steady oscillatory
component v2!
2 average to zero. The dashed lines in
Fig. 6(e) and (f) represent the magnitude
of
the steady
time-averaged second-order velocity
vfd
from the
2 (∞)
frequency-domain simulation.
V. ACOUSTIC STREAMING GENERATED BY
PULSED ACTUATION
In the following we study the effects of switching the
oscillatory boundary actuation on and off on a timescale
much longer than the oscillation period t0 in either
single- or multi-pulse mode. The aim is to investigate
whether such an approach can suppress the influence of
the streaming flow on suspended particles relative to that
of the radiation force.