12
C. Multi-pulse numerical analysis
From the single pulse results shown in Fig. 7 there is no
indication of any optimum for the pulse duration or repetition
period, and in general it provides little hope that
pulsed actuation should lead to lower values of . Figure
8 shows
Eac
,
vstr
, and for three pulsed schemes
with pulse duration 500 t0, 200 t0, and 30 t0 and pause duration
500 t0, 200 t0, and 210 t0, respectively. For all three
pulsed schemes, increases faster than for the constant
actuation Fig. 7(a), thus not indicating any increased
suppression of the streaming.
VI. DISCUSSION
Solving numerically the time-dependent problem of the
acoustic cavity and the build-up of acoustic streaming,
presents new challenges, which are not present in the
purely periodic problem. Firstly, the numerical convergence
analysis now involves both the spatial and temporal
resolutions. This we addressed in a sequential process
by first analyzing the spatial mesh with the periodic
frequency-domain solution, and thereafter doing a thorough
convergence analysis with respect to the temporal
resolution. Secondly, the convergence of the transient
solution towards the periodic state was poor for actuation
frequencies away from the resonance frequency of
the system. This makes off-resonance simulation computationally
costly, as it requires a better temporal resolution,
and it complicates comparison of simulations at
resonance with simulations off resonance. Thirdly, small
numerical errors accumulate during the hundred thousand
time steps taken during a simulation from a quiescent
state to a purely periodic state. These errors need
to be suppressed by the numerical time-domain solver,
which in the generalized-alpha solver is done through the
alpha parameter. Simulation with higher temporal resolution
required lower values of the alpha parameter to
have more suppression of accumulated numerical errors.
The model system used in this study is a simplification
of an actual device. The vibration of only the side
walls, and not the top and bottom walls, stands in contrast
to the physical system, in which the whole device
is vibrating in a non-trivial way, difficult to predict, and
only the overall amplitude and the frequency of the actuation
is controlled experimentally. Furthermore, our
model only treats the two-dimensional cross section of a
long straight channel, whereas experimental studies have
shown that there are dynamics along the length of the
channel 23. Nevertheless, successful comparison, both
qualitatively and quantitatively, have been reported between
the prediction of this simplified numerical model
and experimental measurements of Rayleigh streaming
in the cross sectional plane of a microchannel 10, which
makes it reasonable to assume that the time-dependent
simulations also provide reliable predictions.
It is also important to stress that our model only describes
the fluid and not the motion of the suspended
particles. Integrating the forces acting on the particles
becomes vastly more demanding when the streaming flow
is unsteady, because the drag forces from the oscillating
velocity components v1 and v2!
2 do average out, as they
do in the case of a purely time-periodic state. To include
this contribution in the particle tracking scheme,
the forces on the particles need to be integrated with a
time step of a fraction of the oscillation period, which
makes the solution of particle trajectories over several
seconds a very demanding task using brute-force integration
of the equations of motion.
Our analysis of the pulsed actuation schemes showed
that the slow decay of the streaming flow makes pulsation
inefficient in reducing the streaming-induced drag
force compared to the radiation force. Such a reduction
may, however, be obtained by a rapid switching between
different resonances each resulting in similar radiation
forces but different spatial streaming patterns which on
averages cancel each other out, thus fighting streaming
with streaming. An idea along these lines was presented
by Ohlin et al. Ref. 32, who used frequency sweeping
to diminish the streaming flows in liquid-filled wells in a
multi-well plate for cell analysis. However, the prediction
of particle trajectories under such multi-resonance conditions
requires an extensive study as described above.
Experimentally, the use of pulsed actuation to decrease
streaming flow has been reported by Hoyos et al. Ref.
18. However, this study is not directly comparable to
our analysis, as we treat the build-up of Rayleigh streaming
perpendicular to the pressure nodal plane, whereas
Hoyos et al. study the streaming flow in this plane. Such
in-nodal-plane streaming flows have been studied numerically
by Lei et al. 12, 13, though only with steady actuation.
The contradicting results of our theoretical study
and the experimental study of Hoyos et al. may thus rely
on the differences of the phenomena studied.
VII. CONCLUSION
In this work, we have presented a model for the
transient acoustic fields and the unsteady time-averaged
second-order velocity field in the transverse crosssectional
plane of a long straight microchannel. The
model is based on the usual perturbation approach for
low acoustic field amplitudes, and we have solved both
first- and second-order equations in the time domain for
the unsteady transient case as well as in the frequency
domain for the purely periodic case. This enabled us to
characterize the build-up of the oscillating acoustic fields
and the unsteady streaming flow.
Our analysis showed that the build-up of acoustic energy
in the channel follows the analytical prediction obtained
for a single damped harmonic oscillator with sinusoidal
forcing, and that a quasi-steady velocity component
is established already within the first few oscillations
and increases in magnitude as the acoustic energy builds