6.6. NUMERICAL MODELING OF TIME-DEPENDENT STREAMING 57
acoustophoresis, the streaming ow needs to be either diminished or altered in such a
way that it does not counteract the focusing. An example of the latter is presented in
Ref. 31 Appendix C, which demonstrates a single ow roll streaming pattern in a nearlysquare
channel cross section. The numerical model for the square channel streaming ow
was based on the studies of the fundamental one-dimensional half-wavelength resonance
in the rectangular channel Ref. 28 Appendix A. By using actuation of all four channel
boundaries in the model, we could engineer a streaming ow consisting of a single ow roll.
The drag force induced by the single roll streaming ow does not counteract the focusing
by the weak radiation force on the small particles. The single roll streaming ow was also
observed experimentally, resulting in a spiraling focusing of particles smaller than the usual
critical particle size of about 2 µm in standard half-wavelength acoustophoresis devices.
The single roll streaming ow was obtained by utilizing the two closely-spaced resonance
peaks of the fundamental half-wavelength resonances along the width and the height of
the nearly-square channel. Figure 6.4 shows the numerical predictions of the particle
trajectories for three actuation frequencies around the two resonance peaks. It was not
possible to make a detailed comparisons of theory and experiments because the numerical
results depended largely on the choice of boundary conditions, and the observed smallparticle
trajectories varied along the length of the channel. However, the nearly-square
channel did on average along the channel length enable two-dimensional focusing of E.
coli bacteria and 0.6 m polystyrene particles with recoveries above 95%, whereas for the
standard one-dimensional focusing in a rectangular channel, the recoveries were below 50%.
6.6 Numerical modeling of time-dependent streaming
All previous studies of acoustic streaming treats the steady uid ow, which can be observed
in various acoustic systems. In acoustouidic applications this is reasonable because
the streaming ow reaches steady state on a timescale of a few milliseconds, which is typically
much shorter than other relevant timescales, such as focusing time of suspended
particles. The initial purpose of studying the build-up of the acoustic streaming and its
response to a pulsed actuation, Ref. 30 Appendix E, was to diminish the magnitude
of the streaming ow. This was motivated by an experimental study of a similar system
by Hoyos and Castro 49, which indicated that a pulsation of the ultrasound actuation
could reduce the streaming ow relative to the radiation force, and thus allow for radiation
force dominated manipulation of sub-micrometer particles. Furthermore, scaling analysis
indicated that the timescale for the build-up of the acoustic resonance was approximately
ten times faster than the timescale for the build-up of the streaming ow. The more fundamental
purpose of the study in Ref. 30 Appendix E was to understand the physical
mechanisms involved in the build-up of the acoustic resonance and the streaming ow.
The study showed that the build-up of acoustic energy in the microuidic channel
Fig. 6.5(b) could be accurately described by the analytical solution for a sinusoidally-driven
under-damped harmonic oscillator Fig. 6.5(a). A distinct feature of the underdamped
harmonic oscillator is that it may transiently overshoot its steady energy, as shown in