10
TABLE II. Characteristic timescales. The values are obtained
by using the kinematic viscosity = /0 = 8.93×10−7 m2/s
(Table I), the Q-factor Q = 416 (Fig. 3), and the channel
height h = 160 μm (Fig. 1).
Timescale Expression Value
Oscillation time t0 5.1 × 10−7 s ≈ 1 t0
Resonance
relaxation time
2 t0 3.4 × 10−5 s ≈ 66 t0
E = Q
Momentum
diffusion time
= 1
2
�� h
8
2 2.8 × 10−4 s ≈ 558 t0
A. Single-pulse scaling analysis
A striking feature of Fig. 6, is the separation of
timescales between the roughly exponential build-up of
the acoustic resonance in Fig. 6(a) and of the streaming
flow in Fig. 6(f). It appears that the resonance, and
hence the acoustic radiation force on a suspended particle,
is fully established almost ten times faster than
the streaming flow and the resulting drag force on a suspended
particle. To investigate this further, we look at
the scaling provided by the three timescales relevant for
the problem of transient acoustic streaming, all listed
in Table II: the oscillation time t0 of the acoustic wave,
the resonance relaxation time E of the acoustic cavity,
and the momentum diffusion time governing the quasisteady
streaming flow.
The momentum diffusion time is = 1
2
�� 1
8h
2
, where
=
0
is the kinematic viscosity, and 1
8h is approximately
half the distance between the top boundary and
the center of the streaming flow roll. Inserting the relevant
numbers, see Table II, we indeed find that E ≈ 66 t0
is much faster than ≈ 558t0. However, this separation
in timescales does not guarantee a suppression of
streaming relative to the radiation force. One problem is
that the streaming is driven by the shear stresses in the
boundary layer, and these stresses builds up much faster
given the small thickness of the boundary layer. This we
investigate further in the following subsection. Another
problem is that the large momentum diffusion time
implies a very slow decay of the streaming flow, once it is
established. The latter effect, we study using the following
analytical model. Consider a quantity f (streaming
velocity or acoustic energy), with a relaxation time and
driven by a pulsed source term P of pulse width tpw. The
rate of change of f is equivalent to Eq. (14b),
@tf = P −
1
f, (31a)
P =
(
1
f0, for 0 < t < tpw,
0, otherwise,
(31b)
where 1
f0 is a constant input power. This simplified
analytical model captures the roughly exponential buildup
and decay characteristics of our full numerical model,
and allows for analytical studies of the time-integral of
f(t). For a final time t > tpw we find
Z t
0
f(t′) dt′ = f0tpw − f0
h
e−1
(t−tpw) − e−1
t
i
. (32)
From this we see that when t ≫ +tpw the time-integral
of f(t) is approximately f0tpw and not dependent on the
relaxation time . Consequently, if both the acoustic energy
and the acoustic streaming can be described by exponential
behavior with the respective relaxation times
E and , the ratio of their time-integrated effects is
the same whether the system is driven by a constant actuation
towards their steady time-periodic state or by
a pulsed actuation with pulse width tpw. This simplified
analytical model indicates that there is little hope
of decreasing acoustic streaming relative to the acoustic
radiation force by applying pulsed actuation, in spite of
the order of magnitude difference between the relaxation
times for the acoustic energy and the streaming.
B. Single-pulse numerical analysis
We investigate the features of pulsed actuation more
detailed in the following by numerical analysis. In Fig. 7
is shown the temporal evolution of the total acoustic energy
Eac
and the magnitude of the acoustic streaming
flow
vstr
for the three cases: (i) the build-up towards
the periodic state, (ii) a single long actuation pulse, and
(iii) a single short actuation pulse. The magnitude of the
acoustic streaming is measured by the unsteady timeaveraged
velocity probe
vstr
=
vz2(0,
1
4
, (33)
h)
and the unsteady energy and streaming probes obtained
from the time-domain simulation are normalized by
their corresponding steady time-averaged values from the
frequency-domain simulation.
We introduce the streaming ratio to measure the influence
of streaming-induced drag on suspended particles
relative to the influence of the acoustic radiation force
for the unsteady time-domain solution, in comparison to
the periodic frequency-domain solution. To calculate the
relative displacement s of particles due to each of the
two forces, respectively, we compare their time integrals.
Since the radiation force scales with the acoustic energy
density, we define the streaming ratio (t) as
(t) =
Z t
0
vstr(t′)
vfd
str(∞)
dt′
Z t
0
Eac(t′)
Efd
ac(∞)
dt′
∼
sstr
sfd
str
srad
sfd
rad
, (34)
where sstr and srad are the total particle displacements
in the time from 0 to t due to the streaminginduced
drag force and the acoustic radiation force, respectively.
In the periodic state = 1, and to obtain