arXiv:1509.02554v1 physics.flu-dyn 8 Sep 2015
A theoretical study of time-dependent, ultrasound-induced
acoustic streaming in microchannels
Peter Barkholt Muller∗ and Henrik Bruus†
Department of Physics, Technical University of Denmark,
DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark
(Dated: Submitted to Phys. Rev. E, 8 September 2015)
Based on first- and second-order perturbation theory, we present a numerical study of the temporal
build-up and decay of unsteady acoustic fields and acoustic streaming flows actuated by vibrating
walls in the transverse cross-sectional plane of a long straight microchannel under adiabatic conditions
and assuming temperature-independent material parameters. The unsteady streaming flow
is obtained by averaging the time-dependent velocity field over one oscillation period, and as time
increases, it is shown to converge towards the well-known steady time-averaged solution calculated
in the frequency domain. Scaling analysis reveals that the acoustic resonance builds up much faster
than the acoustic streaming, implying that the radiation force may dominate over the drag force
from streaming even for small particles. However, our numerical time-dependent analysis indicates
that pulsed actuation does not reduce streaming significantly due to its slow decay. Our analysis
also shows that for an acoustic resonance with a quality factor Q, the amplitude of the oscillating
second-order velocity component is Q times larger than the usual second-order steady time-averaged
velocity component. Consequently, the well-known criterion v1 ≪ cs for the validity of the perturbation
expansion is replaced by the more restrictive criterion v1 ≪ cs/Q. Our numerical model is
available in the supplemental material in the form of Comsol model files and Matlab scripts.
PACS numbers: 43.25.Nm, 43.20.Ks, 43.25.+y
I. INTRODUCTION
Acoustophoresis has successfully been used in many
applications to manipulate particles in the size range
from about 0.5 mm down to about 2 μm 1. However, for
smaller particles, the focusing by the acoustic radiation
force is hindered by the drag force from the suspending
liquid, which is set in motion by the generation of
an acoustic streaming flow 2, 3. This limits the use of
acoustophoresis to manipulate sub-micrometer particles,
relevant for application within medical, environmental,
and food sciences, and it underlines a need for better understanding
of acoustic streaming and ways to circumvent
this limitation.
The phenomenon of acoustic streaming was first described
theoretically by Lord Rayleigh 4 in 1884, and
has later been revisited, among others, by Schlicting 5,
Nyborg 6, Hamilton 7, 8, Rednikov and Sadhal 9, and
Muller et al. 10, to extend the fundamental treatment
of the governing equations and to solve the equations for
various open and closed geometries.
Numerical methods have been applied in many studies
to predict the streaming phenomena observed in various
experiments. Muller et al. 2 developed a numerical
scheme to solve the acoustic streaming in the
cross section of a long straight microchannel, which resolved
the viscous acoustic boundary layers and described
the interplay between the acoustic scattering force and
the streaming-induced drag force on suspended particles.
∗ peter.b.muller@gmail.com
† bruus@fysik.dtu.dk
This scheme was later extended to take into account the
thermoviscous effects arising from the dependence of the
fluid viscosity on the oscillating temperature field 11.
Lei et al. 12, 13 have developed a numerical scheme
based on the effective slip-velocity equations, originally
proposed by Nyborg in 1953 14, 15, which avoid the
resolution of the thin boundary layers but still enable
qualitative predictions of the three-dimensional streaming
flows observed in microchannels and flat microfluidic
chambers. To obtain quantitative results from such models
that does not resolve the acoustic boundary layers,
Hahn et al. 16 developed an effective model to determine
the loss associated with the viscous stresses inside
the thermoacoustic boundary layers, and apply this loss
as an artificial bulk absorption coefficient. This enables
the calculation of correct acoustic amplitudes, without
resolving the thin acoustic boundary layers. Acoustic
streaming in the cross section of a straight PDMS microchannel
exited by surface acoustic waves was studied
numerically by Nama et al. 17, describing the influence
of the acoustically soft PDMS wall on the particle focusability,
and examining the possibilities of having two
tunable counter-propagating surface acoustic waves.
All of the above mentioned studies consider steady
acoustic streaming flows. This is reasonable as the
streaming flow reaches steady state typically in a few
milliseconds, much faster than other relevant experimental
timescales. Furthermore, this allows for analytical
solutions for the streaming velocity field in some special
cases, and it makes it much easier to obtain numerical solutions.
However, an experimental study by Hoyos and
Castro 18 indicates that a pulsed actuation, instead of
steady, can reduce the drag force from the streaming flow