7
Based on Eqs. (22) and (23), the characteristic timescale
E for the build-up of the acoustic energy is found to be
E =
1
2��!0
=
Q
!0
. (24)
The build-up of the energy in the single harmonic oscillator,
calculated at ! = !0 with �� = 1.20 × 10−3,
is shown in the inset of Fig. 3 together with the buildup
of acoustic energy Eac(t) of the microfluidic channel
solved numerically at resonance, ! = 2fres. The analytical
and numerical results are in good agreement, and
we conclude that the build-up of acoustic energy in the
channel cavity can be modeled as a single harmonic oscillator.
The energy builds up to 95% of its steady value
in about 500 t0 ≈ 8 E.
B. Decomposition of the velocity field
The task of calculating the build-up of the acoustic
streaming flow is a multi-scale problem, because the amplitude
of the oscillating acoustic velocity field is several
orders of magnitude larger than the magnitude of the
streaming flow. This is indeed the very reason that we
can apply the perturbation expansion
v = v1 + v2, (25)
and decompose the non-linear governing equations into
a set of linear first-order equations and a set of secondorder
equations. However, there is also another level of
difference in velocity scaling. In the purely periodic state,
the velocity can be Fourier decomposed as
v(r, t) = v!
1 (r) sin(!t) + v2!
2 (r) sin(2!t) + v0
2(r), (26)
where v!
1 (r) is the steady amplitude of the first-order
harmonic component, v2!
2 (r) is the steady amplitude
of the second-order frequency-doubled component, and
v0
2(r) is the magnitude of the second-order steady velocity
component referred to as the acoustic streaming flow.
The orders of magnitude of the three velocity components
in the periodic state are given by
v!
1 ∼ Qvbc, v2!
2 ∼
Q3v2b
c
cs
, v0
2 ∼
Q2v2b
c
cs
. (27)
The order of v1 is derived in the one-dimensional acoustic
cavity example presented in Ref. 30, the order of v0
2
is given by the well-known Rayleigh theory, while the
order of v2!
2 is a new result derived in Appendix A. The
magnitude of v2!
2 is a factor of Q larger than what is
expected from dimensional analysis of the second-order
equation (8c). Consequently, the criterion |v2| ≪ |v1| for
the perturbation expansion becomes
Q2vbc ≪ cs, (28)
which is more restrictive than the usual criterion based on
the first-order perturbation expansion, Qvbc ≪ cs. Thus,
the perturbation expansion becomes invalid for smaller
values of vbc than previously expected.
In the transient regime we cannot Fourier decompose
the velocity field. Instead, we propose a decomposition
using envelope functions inspired by Eq. (26),
1 (r, t) sin(!t) + v2!
v(r, t) = v!
2 (r, t) sin(2!t) + v0
2(r, t).
(29)
Here, the amplitudes are slowly varying in time compared
to the fast oscillation period t0. We can no longer separate
v2!
2 and v0
2 before solving the second-order timedependent
equations (7) and (8). To obtain the timedependent
magnitude of the quasi-steady streaming velocity
mode v0
2, we need to choose a good velocity probe,
and we thus form the unsteady time-average of v2(r, t),
v2(r, t)
=
Z t+t0/2
t−t0/2
v2(r, t′) dt′. (30)
The time-averaging is done with a fifth-order Romberg
integration scheme 31 using data points with a uniform
spacing of t0/16 in the time interval of width t0.
C. Steady and unsteady streaming flow
In this section we compare the unsteady timeaveraged
second-order velocity field
v2(r, t)
, from
the time-domain simulations, with the steady timeaveraged
second-order velocity field
vfd
, from
2 (r,∞)
the frequency-domain simulation. Figure 4(a) and (b)
each shows a snapshot in time of the transient v1 and
v2, respectively. For v2(r, t), the oscillatory component
v2!
2 (r, t) sin(2!t) dominates, as it is two orders of magnitude
larger than the quasi-steady component v0
2(r, t).
However, at late times, here t =
3000 t0, the amplitude
v2!
2 (r, t) has converged, and in
v2(r, t)
the oscillatory
component average to zero and only the quasi-steady
component remains.
The unsteady time average
v2(r, t)
evaluated at
t = 3000 t0 is shown in Fig. 4(c), exhibiting a single flow
roll, in agreement with the classical Rayleigh streaming
flow. In Fig. 4(d) is shown the steady
vfd
2 (∞)
from the frequency-domain simulation. Figure 4(c) and
4(d) use the same color scaling for the velocity magnitude,
to evaluate the convergence of the unsteady streaming
flow
v2(3000 t0)
towards the steady streaming flow
vfd
, and the two solutions agree well both qualitatively
and quantitatively. The convergence parameter
2 (∞)
C, Eq. (19), of
v2(3000 t0)
with respect to
vfd
2 (∞)
is C = 0.01, and if we multiply
v2
by a free factor,
taking into account that
v2
has not fully converged at
t = 3000 t0, the convergence parameter can be reduced
to C = 0.008. The remaining small difference between
the unsteady
v2(3000 t0)
and the steady
vfd
is
2 (∞)
attributed to the finite temporal resolution of the time
marching scheme. We can thus conclude that the timedomain
streaming simulation converges well towards the