NUMERICAL STUDY OF THERMOVISCOUS EFFECTS IN . . . PHYSICAL REVIEW E 90, 043016 (2014)
FIG. 6. (Color online) Time-averaged heat current densities in
the channel cross section. (a) Sketch of the heat currents (arrows)
with indication of the responsible terms in the time-averaged energy
conservation Eq. (15). (b) Heat current density (arrows) and its
magnitude color plot ranging from zero (light green) to 7 × 102 W
m−2 (dark red) in the bulk of the channel. The strong currents inside
the boundary layers are not shown. (c) Magnitude of the y component
of the heat current density color plot ranging from zero (light green)
to 5 × 104 Wm−2 (dark red) inside the boundary layer at the bottom
wall. (d) Magnitude of the z component of the heat current density
color plot ranging from −6 × 102 Wm−2 (dark blue) to zero (light
green) inside the boundary layer at the bottom wall.
The dissipation of mechanical energy happens primarily in
the viscous boundary layers of thickness δs due to the work
done by the viscous stress force density (∇ · τ 1) with power
density (∇ · τ 1) · v1
. As the gradient of v1 perpendicular to
the wall inside the boundary layer is large, the dominant term
is η0
∂2vy1
∂z2 vy1
∼ η0
1
4 (va
s , where two factors of 12
y1)2/δ2
enter
from spatial and time averaging. The total power dissipation
is given by the product of the power density and the volume
of the boundary layers, Ps
∼ 2δswη0
1
4 (va
y1)2/δ2
s .
In steady state, Ps equals Pin, from which we find the
magnitude va
y1 of the resonant field in terms of vbc to be
va
y1
∼ 2
π
h
δs
λ
w vbc
∼ 500vbc for our system, which is in good
agreement with the numerical result for va
y1/vbc plotted in the
inset of Fig. 3.
To rationalize the magnitude of the second-order temperature
shift, we consider the diffusive energy transport through
the top and bottom walls. The diffusive energy current density
FIG. 7. (Color online) Color plot of the time-averaged secondorder
temperature T2 from zero (black) to maximum (white) in
three cases. (a) T2 calculated from the complete governing equations,
identical to Fig. 5(a). (b) T2 calculated without bulk viscosity ηb = 0.
(c) T2 calculated with zero viscous stress τ = 0 in the bulk defined
by |y| < (w/2 − 4 μm) and |z| < (h/2 − 4 μm).
is −kth
0
∇T2, and as heat diffuses to the perfectly conducting
walls on a length scale of δt, the outgoing power is Pout ∼
2wkth
0 ( 1
2T a
2 /δt). Here, the spatial average of T2 just outside
12
the thermal boundary layers along the top and bottom walls
has been approximated to T a
2 . In steady state, Pout equals Pin
and the magnitude of the second-order temperature becomes
T a
∼ 2
2
π2
h2
δtδs
( λ
w )2 1
cp
(vbc)2 ∼ 0.13 mK, which is comparable to
the numerical result in Fig. 5.
From the simplified picture of strong heat generation inside
the boundary layers, it may seem odd that the second-order
temperature field in Fig. 5 has two global maxima in the bulk
of the channel. This effect is due to the absorption in the bulk
of the channel originating from the nonzero divergence of the
stress force term v1
· τ 1
in Eq. (15c) as shown in Fig. 7. In
Fig. 7(a), we showthe complete second-order temperature field
T2. Fig. 7(b) shows an artificial temperature field calculated
without bulk viscosity, ηb = 0. No maxima appear in the bulk,
and the temperature field looks more as expected from the
simplified view of heat generation in the boundary layers.
However, there is still a small amount of heat generation in
the bulk of the channel from the shear viscosity. In Fig. 7(c)
this heat generation is suppressed by setting τ 1
= 0 in the bulk
more than 4 μm from the walls, while maintaining the full τ 1
in the boundary layers. The resulting plot of T2 shows howheat
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