32 CHAPTER 3. FULL THERMODYNAMIC THEORY
3.3.3 Time-averaged second-order equations
We now considering a steady oscillatory state and apply the time-averaging over one oscillation
period
:::
to the equations to treat only the steady components of the density,
velocity, and temperature. Since the time average of a time derivative of a steady periodic
eld is zero by denition,
@t(:::)
= 0, it is advantages not to take the time average of
the second-order time-dependent equation Eq. (3.18), but instead take the time average of
the governing equations (3.9a), (3.10a), and Eq. (3.11a), and then use the thermodynamic
relations Eqs. (3.4b) and (3.5b) to end up with the following second-order time-averaged
equations,
r
v2
+ T
(v1 r)p1
�� p
(v1 r)T1
= 0; (3.20a)
r
2
��
p2
1 �� 0
v1v1
= 0; (3.20b)
r
h
kth
0 r
T2
+
kth
1 rT1
+
v1 1
�� (1 �� pT0)
p1v1
�� 0cp
T1v1
i
= 0; (3.20c)
where
2
is given by
2
= 0
r
v2
+
r
v2
T
+
b
0 ��
2
3
0
r
v2
1
+
*
1
rv1+ (rv1)T
+
+
*
b
1��
2
3
1
+
: (3.20d)
(rv1) 1
Solving the time-averaged second-order governing equations (3.20) together with a set of
boundary conditions, we are able to predict the steady streaming ow
v2
in the general
thermodynamic case and the steady temperature perturbation
T2
, which are both studied
in Ref. 29 Appendix D.
3.4 Discussion of thermodynamic theory
In this section we discuss some aspects of the thermoviscous acoustic theory presented
in this chapter. Based on dimensional analysis of the perturbation equations, we discuss
the orders of magnitude of the rst-order elds, the adiabatic approximation, and the
magnitude of the acoustic streaming ow. Furthermore, the possible ambiguity of the
combination of linear thermodynamic relations and second-order perturbation theory is
discussed.
3.4.1 Magnitudes of rst-order elds
We want to examine the validity of the adiabatic approximation used in Chapter 2, but rst
we need to determine the order of magnitude of the rst-order elds in the steady periodic
state. Considering the rst-order momentum equation in frequency domain Eq. (3.16b) we