36 CHAPTER 3. FULL THERMODYNAMIC THEORY
In conclusion, the magnitude of the acoustic streaming velocity predicted by the dimensional
analysis Eq. (3.26e) is in agreement with the order of magnitude of the exact
analytical result Eq. (3.27). The corrections to the magnitude of the acoustic streaming,
due to perturbation in the viscosity, is treated numerically in Ref. 29 Appendix D and is
on the order of 20% at room temperature.
3.4.4 Thermodynamics and perturbation theory
In this section the possible ambiguity of the combination of linear thermodynamic relations
and second-order perturbation theory is discussed. This topic is outside the scope of the
studies presented in this thesis, and the considerations presented here are meant as an
outlook on the further development of the theory of second-order perturbation theory
applied in acoustics.
Comparing the unperturbed governing equations (3.9b), (3.10a), and (3.11d), and the
second-order perturbation equations (3.18), it may seem inconsistent that we introduce
perturbations in the coecients , b, and kth, but not in the thermodynamic coecients
T , p, and cp. The explanation of this is grounded on how we treat the thermodynamics.
The thermodynamic relations in Section 3.1 are based on a linear Taylor expansions of
the equation of state about a thermodynamic equilibrium state, such as
d =
@
@T
!
p
dT +
@
@p
!
T
dp: (3.28)
The parameters T , p, and cp are some of the coecients that arise in these linear expansions
and are by denition evaluated at the equilibrium state. They are thus properties of
the equilibrium state, independent of the perturbed state of the uid.
The coecients , b, and kth are also coecients in linear constitutive relation, e.g.
the denition of the dynamic viscosity from the linear relation between shear stress and
shear velocity in a Couette ow. However, since they are not dened from thermodynamic
constitutive relations, but rather related to transport phenomena, such as viscous
and thermal diusion, they depend on the perturbed state of the uid, and not just the
equilibrium state.
This explains why we introduce perturbations in the transport coecients , b, and
kth, but not in the thermodynamic coecients T , p, and cp. However, it may still seem
inconsistent that we only apply linear relations between the thermodynamic variables, while
we are treating second-order perturbations of the independent thermodynamic variables.
To avoid this inconsistency we need a new set of thermodynamic relations based on secondorder
Taylor expansions, such as
d =
@
@T
!
p
dT +
@
@p
!
T
dp +
1
2
@2
@T2
!
p
(dT)2 +
1
2
@2
@p2
!
T
(dp)2 : (3.29)
This would introduce new expansion coecients, such as
@2
@p2
T
, which are also properties
of the equilibrium state, independent of the perturbed state of the uid. The values of the