Chapter 3
Full thermodynamic theory
In this chapter we derive the governing equations for the conservation of mass, momentum
and energy in a compressible Newtonian uid. The governing equations are expanded
following standard second-order perturbation theory. The treatment is based on the textbooks
Theoretical Microudics by Henrik Bruus 17, Course of Theoretical Physics volume
5 Statistical Physics and volume 6 Fluid Mechanics by Landau and Lifshitz39, 40. Part of
the theory presented in this chapter is also presented in Refs. 28, 11, 29 Appendixes A, B,
and D. 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.
3.1 Thermodynamics
In this treatment we describe changes to the thermodynamic state of a uid particle1 by
changes in the following ve variables, the temperature T, the pressure p, the mass density
, the internal energy " per mass unit, and the entropy s per mass unit, outlined in Table
3.1. These ve variables are interdependent and small changes to the state of the uid
can be described by changes in just two variables 39. We choose T and p to be our
1A uid particle is a collection of uid molecules large enough to have well-dened average values with
small uctuations, thus fullling the continuum hypothesis. The minimum dimension for a liquid particle
is approximately 10 nm 17.
Table 3.1: Thermodynamic variables and the chosen hierarchy of independent and dependent variables.
Name symbol unit hierarchy
Temperatur T K Independent
Pressure p Pa Independent
Density kgm��3 Dependent
Internal energy " J kg��1 Dependent
Entropy s JK��1 kg��1 Dependent
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