3.4. DISCUSSION OF THERMODYNAMIC THEORY 35
s =
q
0
0! Eq. (2.64), Eq. (3.25b) becomes
0
1
2
s jv2j 0kjv1j2 ) (3.25c)
jv2j jv1j2
cs
; (3.25d)
which is consistent with Lord Rayleigh's result of jv2j = 3
8
jv1j2
cs
in Ref. 3.
We now include the perturbation of the viscosity Eq. (3.19a), and the equivalent of
Eq. (3.25c) now becomes
0
1
2
s jv2j j1j
1
2
s jv1j + 0kjv1j2 ) (3.26a)
0 jv1j + jv1j2
jv2j j1j
cs ) (3.26b)
jv2j
1
0
"
@
@T
T1
+
@
@
1
#
jv1j + jv1j2
cs ) (3.26c)
jv2j
1
0
"
@
@T
��1
p
jv1j
cs
+
@
@
0 jv1j
cs
#
jv1j + jv1j2
cs ) (3.26d)
jv2j
(
1 +
@
@T
��1
0p
+
@
@
0
0
)
jv1j2
cs
: (3.26e)
The estimated order of magnitude of jv2j Eq. (3.26e) is as far as we can get with only dimensional
analysis. The magnitude of the acoustic streaming near a planar wall, including
eects of viscosity perturbations, have been solved analytically by Rednikov and Sadhal
Ref. 8, which in our notation becomes
jv2j =
(
1 +
2
3
q
cp0
kth
0
1 + cp0
kth
0
"
(��1) ��
��1
0p
@1
@T
p
#)
3
8
jv1j2
cs
: (3.27)
The fraction cp0
kth
0
is known as the Prandtl number and is the ratio of momentum diusivity
and thermal diusivity, and it can also be expressed in terms of the viscous and thermal
boundary layers thicknesses as 2
s
2
t
. For water at room temperature the Prandl number is
approximately 4. The term ��1
0p
��@1
@T
p in Eq. (3.27) agrees with the second term in the
result from the dimensional analysis Eq. (3.26e). In the analytical calculation of Ref. 8,
they show that the pressure dependency of the viscosity has no inuence in the case of a
planar wall, and thus the only viscosity derivative that enters the nal expression Eq. (3.27)
is
��@1
@T
p. Furthermore, the exact numerical result includes another term, scaling by ��1
in Eq. (3.27), which enters due to non-adiabatic compressibility, and which is not predicted
by the dimensional analysis Eq. (3.26e).