Chapter 2
Adiabatic theory
In this chapter we treat the theory of acoustic resonances and the generated acoustic
streaming using the adiabatic thermodynamic approximation. 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 Ref. 30 Appendix E.
The purpose of this chapter is to give students, not familiar with the topics of acoustic
resonances and acoustic streaming, an introduction to the topic in its most simple form and
with some practical examples. After deriving the governing equations, the problem of the
acoustic wave between two orthogonally oscillating parallel plates is thoroughly treated,
including derivations of the rst-order oscillating velocity, the second-order time-averaged
velocity, and the second-order oscillating velocity, for which the latter has not previously
been treated in the literature. Finally, we discuss the acoustic boundary layer and its role
in the generation of boundary-driven acoustic streaming.
For an adiabatic process the entropy is conserved for each part of a system 39. Thus
for every little uid volume we consider the entropy to be constant, and consequently the
temperature and pressure oscillations of the acoustic wave can be expressed solely in terms
of the density oscillations, and thus we need only consider acoustic perturbations in the
density and the velocity v.
2.1 Governing equations
Without deriving it, we will take the NavierStokes equation, describing the conservation
of momentum, and the continuity equation, describing the conservation of mass, as our
starting point,1
@tv + (v r) v = ��rp + r2v +
1
3 + b
r(r v) ; (2.1a)
@t = ��r (v) ; (2.1b)
1In Chapter 3 the derivations of the conservation equations for mass, momentum, and energy are
presented.
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