34 CHAPTER 3. FULL THERMODYNAMIC THEORY
thus be rewritten as r2T1 = ��k2
0T1, and (3.16c) becomes
0 r2T1 ) (3.23a)
��i!0cp T1 + i!pT0 p1 = kth
h
1 �� i
kth
0
0cp!
(��k2
i
T1 =
0 )
pT0
0cp
p1 ) (3.23b)
h
1 + i��t
ipT1
��1
= sp1; ��t =
1
2
k2
02
t ; (3.23c)
where we have substituted t and ��1 dened in Eq. (3.17) and Eq. (3.7b), respectively. For
MHz acoustics in water the thermal damping factor ��t 10��6. From the thermodynamic
relation Eq. (3.4b), using the identity T = s Eq. (3.7a), we can now express 1 in terms
of p1,
1
0
1 = T p1 �� pT1 ) (3.24a)
h
i
1
1 = sp1 ��
1 �� i��t
(��1)sp1 ) (3.24b)
0
1
0
1 =
h
1 + i(��1)��t
i
sp1: (3.24c)
The adiabatic approximation Eq. (2.2) is thus a good approximation for the bulk acoustic
wave when (��1)��t 1, which is case for the uids and operating conditions treated in
this theses. However, near walls of high thermal conductivity relative to the uid, where
thermal boundary layers are present, the Laplace term r2T1 becomes much larger, such
that j2
tr2T1j jT1j. Consequently, the adiabatic approximation is only valid inside the
thermal boundary layer if ��1 1, which is still a fairly good approximation for water
at 25 C for which = 1:01.
3.4.3 Magnitude of the acoustic streaming ow
The order of magnitude of the steady streaming ow can be estimated from dimensional
analysis of the governing equations. We rst consider the time-averaged second-order
momentum equation (3.20b) without perturbation of the viscosity, 1 = 0,
r
2
��
p2
1 �� 0
v1v1
= 0 ) (3.25a)
0jr2
v2
j jr
p2
j + 0jr
v1v1
j; (3.25b)
where we have only included the larger component of r
2
. r
p2
is a reaction to
the gradient component of the source term 0r
v1v1
and can be disregarded in our
determination of the order of magnitude of
v2
. Assuming the acoustic propagation to
be parallel to a wall, the leading order of jr2
v2
j near the wall is jr2
v2
j
1
2
s jv2j. On
the other hand jr
v1v1
j kjv1j2, because the leading order component of v1 is parallel
to the wall and thus jr v1j kjv1j. With these orders of magnitude and the denition