  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 v << c for the validity of the
perturbation expansion is replaced by the more restrictive criterion v << c/Q.
Our numerical model is available in the supplemental material in the form of
Comsol model files and Matlab scripts.
