6 CHAPTER 1. INTRODUCTION
pressure nodal plane. This is relevant for experimental applications, such as acoustic trapping
21, and they explained the puzzling streaming pattern observed in a at microuidic
chamber by Hagsäter et al. 36.
To develop an approach that avoids the computational expensive resolution of the
boundary layers but still predict the correct acoustic loss, Hahn et al. 37 developed a
numerical model which utilizes analytical analysis of the boundary layer stresses and other
loss mechanisms, and employ these loss factors in an articial bulk damping factor. This
results in a computational ecient model that correctly predicts the amplitude of the
resonant acoustic eld. Such a method was also employed by Hahn et al. 38 to make
an optimization scheme that maximizes the amount of acoustic energy in a microuidic
chamber with respect to the geometrical dimensions of the chip and position of the piezo
transducer.
1.5 Outline of the thesis
This thesis deals with the theory of uid dynamics, acoustics, and thermodynamics. There
are many good textbooks on these topics of which I can recommend; Theoretical Microudics
by Henrik Bruus 17, Course of Theoretical Physics volume 5 Statistical Physics and
volume 6 Fluid Mechanics by Landau and Lifshitz39, 40. For this thesis I wanted to
give a presentation of the theory of acoustic streaming, and I have divided this into two
chapters.
Chapter 2 is meant as an introduction to students not familiar with acoustics and
thermodynamics. The topics of acoustic resonances and acoustic streaming are treated
within the adiabatic acoustic theory, which simplies the problem while still containing
the fundamental parts of the theory. The problem of the acoustic wave between two
orthogonally oscillating parallel plates are thoroughly treated, including derivations of the
oscillating rst-order velocity, the time-averaged second-order velocity, and the oscillating
second-order velocity, for which the latter has not been treated before in the literature.
Chapter 3 treats the full thermodynamic theory in a rigorous way. It includes the
thermodynamic relations, perturbations in material parameters due to the acoustic perturbations
of the thermodynamic state, and governing equations for the second-order temperature
perturbation. The theory is discussed through dimensional analysis of the governing
equations, to estimate the orders of magnitude of the acoustic elds and the acoustic
streaming ow, and to evaluate the error done by the adiabatic approximation. Furthermore,
the possible ambiguity of the combination of linear thermodynamic relations and
second-order perturbation theory is discussed.
Chapter 4 outlines the analytical framework for the calculation of the forces exerted on
suspended particles by the scattering of the acoustic wave and the drag from the acoustic
steaming ow.
Chapter 5 gives a brief introduction to the numerical methods applied in the studies of
this thesis. The nite element method and the implementation of the governing equations
are described, including a practical example. The spatial and temporal resolutions are
discussed, with emphasis on how proper numerical convergence is ensured.