v1y, and vertical velocity v1z are shown. The amplitudes and
structures of p1, T1, and v1y relate to the acoustic resonance,
while v1z arises due to the viscous interaction of the horizontal
half-wave resonance in the bulk with the bottom and top walls.
Consequently, the magnitude of v1z is insignificant compared to
the magnitude of v1y. The amplitudes of p1 and T1 have the same
spatial structure, shifted horizontally by l/4 with respect to the
spatial structure of v1y.
In Fig. 6 the amplitudes of the first-order fields are plotted
along the dashed white lines at y = w/4 shown in Fig. 5. In
Fig. 6(a) we have plotted the relative pressure change, p1/pz¼0
1 2 1,
with respect to the pressure amplitude at the center (y,z) = (w/4,0).
This relative change is in the order of 1024, implying that p1 is
nearly independent of z. In particular, p1 shows no marked
variation on the length scale of the boundary layer thickness d as
opposed to the velocity v1y and temperature T1 of Fig. 6(b)–(c). To
fulfill the boundary conditions in eqn (4), the latter two decrease
from their bulk values to zero at the wall over a few times dth and
d, respectively, which defines the thicknesses of the thermal and
viscous boundary layers, respectively (dashed lines in Fig. 6).
Further, also v1z increases from zero at the wall, but then it
exhibits a slow linear decrease outside the boundary layer,
Fig. 6(d). T1, v1y, and v1z all overshoot slightly before settling at
their respective bulk values, an effect similar to that observed in
the classical problem of a planar wall executing in-plane
oscillations.20 While T1 and v1y show no variations in the height
of the channel outside the boundary layers, p1 and v1z do so, with
p1 being symmetric (nearly parabolic) and v1z being antisymmetric
(nearly linear). These variations of p1 and v1z result
from the viscous interaction between the horizontal acoustic
resonance and the bottom and top wall, bounding the acoustic
resonance fields.
Fig. 6 Line plots, along the dashed white lines at y = w/4 shown in
Fig. 5, of the amplitudes of the oscillating first-order fields: (a) relative
pressure change p1/pz¼0
1 2 1, (b) temperature T1, (c) horizontal velocity
v1y, and (d) vertical velocity v1z. The main plots (blue curves) show the
field amplitudes close to the bottom wall, while the insets (red curves)
show the field amplitudes along the entire height of the channel. The
characteristic length scales of the thermal and viscous boundary layers,
dth and d, are indicated by the green and magenta dashed lines,
respectively. T1, v1y, and v1z all show marked variations on the length
scale of the boundary layer, while p1 and v1z only show variations across
the full height of the domain.
Fig. 7 Time-averaged second-order fields in the water-filled channel
excited in the horizontal half-wave 1.97-MHz resonance by the side-wall
actuation shown in Fig. 4(a) and driven by the first-order fields plotted in
Fig. 5. (a) Color plot of the time-averaged second-order pressure Sp2T
with a magnitude approximately 2.5 6 1025 times smaller than the
amplitude of the oscillating first-order pressure p1 in Fig. 5(a). (b) Vector
plot (white arrows) of the time-averaged second-order streaming velocity
Sv2T and color plot of its magnitude Sv2T. Four bulk (Rayleigh)
streaming rolls are clearly seen having the maximum speed near the top
and bottom walls. (c) Zoom-in on the 0.4-mm-thick boundary layer near
the bottom wall exhibiting the two boundary (Schlichting) streaming
rolls that drive the bulk (Rayleigh) streaming rolls.
This journal is The Royal Society of Chemistry 2012 Lab Chip, 2012, 12, 4617–4627 | 4623
Downloaded by DTU Library on 27 February 2013
Published on 23 July 2012 on http://pubs.rsc.org | doi:10.1039/C2LC40612H
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