30 CHAPTER 3. FULL THERMODYNAMIC THEORY
equations with respect to perturbation order, which represent a separation with respect
to the magnitudes of the terms. The inequalities in Eq. (3.13) are the criteria that have
to be fullled for the perturbation expansion of the governing equations to be valid. The
inequality cs v1 follow from 0 1, which is explained in Section 2.3.
3.3.1 First-order equations
The rst-order equations are obtained by inserting the perturbation expansion Eq. (3.13)
into the governing equations (3.9b), (3.10a), and (3.11d) and keeping only rst-order terms,
which corresponds to a linearization of the governing equations.
p @tT1 �� T @tp1 = r v1; (3.14a)
0@tv1 = r
1 �� p11
; (3.14b)
0 r2T1; (3.14c)
0cp @tT1 �� pT0 @tp1 = kth
where 1 is given by
1 = 0
rv1 + (rv1)T
+
b
0 ��
2
3
0
(r v1) 1: (3.14d)
When considering a steady periodic state, where all rst-order elds oscillate with constant
amplitude and angular frequency !, the rst-order equations (3.14) can be transformed to
frequency domain by describing physical elds through the complex notation
g1(r; t) = Re
h
g1(r)e��i!t
i
; (3.15)
where g1 represents any rst-order variable, and the value of g1(r; t) is real, whereas the
value of g1(r) is complex. This leads to the substitution @t ! ��i!, and the rst-order
equations in frequency domain become
��i!p T1 + i!T p1 = r v1; (3.16a)
��i!0 v1 = r
1 �� p11
; (3.16b)
��i!0cp T1 + i!pT0 p1 = kth
0 r2T1: (3.16c)
From the rst-order energy equation (3.16c), we dene the thermal boundary layer thickness
t, similarly to how we dened s in Eq. (2.64) from the rst-order momentum equation
(3.16b),
t =
s
2kth
0
0cp!
: (3.17)
3.3.2 Time-dependent second-order equations
The second-order equations are obtained by inserting the perturbation expansion Eq. (3.13)
into the governing equations (3.9b), (3.10a), and (3.11d) and keeping only second-order