PETER BARKHOLT MULLER AND HENRIK BRUUS PHYSICAL REVIEW E 90, 043016 (2014)
The momentum Eq. (6b) likewise becomes
ρ0∂tv1
= ∇ · τ 1
− p11, (9a)
where τ 1 is given by
τ 1
= η0∇v1
+ (∇v1)T +
ηb
0
− 23
η0
(∇ · v1) 1. (9b)
The energy Eq. (6c) requires a little more work. To begin
with, it can be written as
ρ0∂t ε1
+ ε0∂tρ1
= kth
0
∇2T1
− p0
∇ · v1
− ε0ρ0
∇ · v1. (10a)
The two terms containing ε0 cancel out due to the continuity
Eq. (8a), and the term ρ0∂tε1 is rewritten using Eq. (1a),
whereby
ρ0T0∂t s1
+ p0
ρ0
∂tρ1
= kth
0
∇2T1
− p0
∇ · v1. (10b)
The two terms containing p0 cancel out due to the continuity
Eq. (8a), and the term ρ0T0∂t s1 is rewritten using the time
derivative of Eq. (3a). This leads to
ρ0cp ∂tT1
− αpT0 ∂tp1
= kth
0
∇2T1. (10c)
Equations (8c), (9a), and (10c) are the resulting firstorder
thermoviscous equations for conservation of mass,
momentum, and energy, respectively. In the frequency domain,
they become
− iωαp T1
+ iωκT p1
= ∇ · v1, (11a)
−iωρ0 v1
= ∇ · τ 1
− p11, (11b)
−iωρ0cp T1
+ iωαpT0 p1
= kth
0
∇2T1. (11c)
From Eqs. (11b) and (11c), neglecting the pressure terms,
we can derive the length scales δs and δt for diffusion of
momentum and heat, respectively,
δs
=
2η0
ρ0ω
=
2ν
ω
= 0.38 μm, (12a)
δt =
2kth
0
ρ0cpω
=
2Dth
ω
= 0.15 μm, (12b)
where the subscript “s” indicates shear stress, subscript
“t” indicates thermal, and ν = η0/ρ0 and Dth
= kth/(ρ0cp) are
the momentum and thermal diffusivities with numerical values
derived from the parameter values at ambient temperature and
pressure listed in Table I.
E. Second-order time-averaged equations of
thermoviscous acoustics
Moving on to second-order perturbation theory, writing
the fields as g = g0
+ g1
+ g2, we note that the second-order
acoustic perturbation g2 may contain both oscillating terms and
a time-constant term. The time averaging over one oscillation
period of a field g(t ) is denoted g. We note that all full time
derivatives average to zero, ∂g(t ) = 0, and that in our work
t|g2
|/|g1
| 10−3.
In the following, all pure second-order fields are taken to be
time-averaged and thus written plainly as g2 without the angled
brackets. With this notation, the second-order time-averaged
continuity Eq. (6a) becomes
∇ · ρ0v2
+ ρ1v1
= 0, (13)
while the momentum Eq. (6b) takes the form
∇ · τ 2
− p21 − ρ0
v1v1
= 0, (14a)
where τ 2 is given by
τ2
= η0∇v2
+ (∇v2)T +
ηb
0
− 23
η0
(∇ · v2) 1
+η1∇v1
+ (∇v1)T +
ηb
1
− 23
η1
(∇ · v1) 1
. (14b)
It is in the two last terms that the first-order temperature
and density dependence of the viscosities come into play
through the perturbations η1 and ηb
1. Note that the secondorder
temperature perturbation does not enter the governing
equations for v2. Note also that both shear-induced (Rayleigh)
streaming and bulk-absorption-induced Eckart streaming are
included through the shear and bulk viscosity η and ηb,
respectively.
The energy Eq. (6c) in its second-order time-averaged form
is initially written as
∇ ·
v1
· τ 1
+ kth
0
∇T2
+
kth
1
∇T1
− p0 v2
−p1v1
− ε0ρ0v2
− ε0
ρ1v1
− ρ0
ε1v1
= 0. (15a)
The two terms with ε0 cancel due to the continuity
Eq. (13). Next, using Eq. (1a), we obtain the expression ρ0ε1
=
ρ0T0s1
+ (p0/ρ0)ρ1, which upon insertion into Eq. (15a) leads
to
∇ ·
v1
· τ 1
+ kth
0
∇T2
+
kth
1
∇T1
− p0 v2
−p1v1
− ρ0T0
s1v1
− p0
ρ0
ρ1v1
= 0. (15b)
The two p0 terms cancel by the continuity Eq. (13). Then,
from Eq. (3a) we find ρ0T0s1
= ρ0cp T1
− αpT0 p1, which by
substitution into Eq. (15b) yields
∇ ·
kth
0
∇T2
+
kth
1
∇T1
+ v1
· τ 1
−(1−αpT0)p1v1
− ρ0cp
T1v1
= 0. (15c)
Equations (13), (14a), and (15c) are the resulting timeaveraged
second-order thermoviscous acoustic equations for
conservation of mass, momentum, and energy, respectively.
The time-averaged acoustic energy density Eac in the fluid
is given by 23
Eac
= 12
κs
p
2
1
+ 12
ρ0
v
2
1
, (16)
where κs
= κT /γ is the isentropic compressibility and γ =
cp/cV is the ratio of specific heat capacities.
For a product of two time-harmonic fields in the complexvalued
representation Eq. (7), the time average can be
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