PETER BARKHOLT MULLER AND HENRIK BRUUS PHYSICAL REVIEW E 90, 043016 (2014)
builds up and a Poiseuille backflow is established, which by
mass conservation and no-slip boundary conditions becomes
vpoi(z) ≈ −v0
{6(1/4 − z2/h2)}. By a first-order Taylor expansion
of vpoi(z) at the wall z = −h/2, we can determine the
maximum vstr of vplug(z) + vpoi(z) near the wall to first order
in δs/h,
vstr
≈ v0
1 − 6
δs
h
1 + ln
h
6δs
. (21)
This provides an estimate for the magnitude of the acoustic
streaming shown by dashed lines in Fig. 4(d), with the viscous
boundary-layer thickness Eq. (12a) calculated for each of the
three temperatures. This simple one-dimensional analytical
model captures the trend of the numerical data well, though
overall it predicts slightly lower streaming amplitudes. The
deviation from the numerical data is ascribed primarily to
the monotonic approximation vplug(z) of the z dependence
of the velocity inside the viscous boundary layer. The full
z dependence of the streaming velocity inside the viscous
boundary layer is nonmonotonic and overshoots slightly before
leveling. This can be seen in Fig. 4(b), and thus the maximum
velocity occurs at this overshoot and is consequently slightly
larger than predicted by the approximate analytical model.
For channel heights below 10 μm, the assumptions of a
boundary-driven plug flow with a superimposed Poiseuille
backflow begin to collapse as the height of the channel
becomes comparable to the boundary-layer thickness, and a
more elaborate analytical calculation of the streaming velocity
field is necessary 12.
C. Time-averaged second-order temperature
In Fig. 5(a), we show the time-averaged second-order
temperature field T2 calculated for the default 380-μm-by-
160-μm geometry at the resonance frequency. In Figs. 5(b)
and 5(c), we show line plots of T2 along the horizontal and
vertical dashed lines in Fig. 5(a). T2 has a saddle point in the
center of the channel (y = 0,z = 0), two global maxima on the
horizontal centerline z = 0, and a wide plateau on the vertical
center line y = 0. The temperature field is forced to be zero at
all boundaries due to the boundary condition of infinite heat
conduction. The gradient of T2 along line C indicates a decline
in heat generation inside the boundary layer going from the
center toward the left and right walls. The global maxima in
the bulk result from heat generation in the bulk as discussed
in Sec. V.
V. DISCUSSION
In Fig. 6(a) we provide an overview of the energy transport
and dissipation in the system by showing a sketch of the
energy currents in the channel cross section. To explain
the convection of energy, we consider the first-order velocity
to be composed of a weak nonresonant part vbc
1 , which fulfills
the oscillating velocity boundary conditions, and a strong
resonant part vres
1 , which has zero amplitude at all walls 32.
In Fig. 6(b), we show the total energy current density, given by
all the terms inside the divergence in Eq. (15c), in the bulk of
the channel, thus not including the thin boundary layers at the
top and bottom walls. The plot shows how mechanical energy
is entering the system at the left and right walls, due to the
FIG. 5. (Color online) Time-averaged second-order temperature
T2 calculated for the default 380-μm-by-160-μmgeometry, actuation
frequency fres
= 1.966 992 3 MHz, and equilibrium temperature
T0
= 25 ◦C. (a) Color plot (black 0 mK to white 0.17 mK) of T2
in the channel cross section. (b) and (c) Line plots of T2 along the
horizontal and vertical dashed lines in (a), respectively.
oscillating boundary condition, and is convected toward the
top and bottom walls. This transport is dominated by the term
p1v1
in Eq. (15c), particularly the nonresonant part p1vbc
1
,
since vres
1 is out of phase with p1 in the bulk. The y and z
components of the energy current density inside the boundary
layer at the bottom wall are shown in Figs. 6(c) and 6(d). The
transport parallel to the wall, Fig. 6(c), results from p1vres
1
,
which is large, since vres
1 is phase-shifted inside the boundary
layer. The transport perpendicular to the wall, Fig. 6(d), results
predominantly from the thermal diffusion term −kth
0
∇T2.
To rationalize the amplitudes of the fields, we estimate the
order of magnitude of the energy transport and dissipation
in the system. The incoming energy current density from
the oscillating velocity boundary condition at the left and
right walls is given by the time-averaged product of the
local pressure and velocity p1vbc
y1
. Multiplying this by
the area 2h, we obtain the magnitude of the incoming
power Pin
∼ 2h12
pa1
vbc. Here, the factor 12
enters from time
averaging, is the channel length, and the superscript “a”
denotes the amplitude of the resonant field. From the inviscid
part of the first-order momentum conservation Eq. (9a), we
estimate the magnitude pa1
∼ ρ0csva
y1 and therefore obtain
Pin
∼ hρ0csva
y1vbc.
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