22 CHAPTER 2. ADIABATIC THEORY
The subscript s refer to shear, since s is related to the viscous shear at the wall. There
is also a thermal boundary layer of thickness t, due to thermal conduction at the wall.
s is essentially a momentum diusion length, describing how far the momentum diuses
within an oscillation period.
The y-component of Eq. (2.63) along with the boundary conditions of Eq. (2.61) are
satised to zeroth order in ��0 by the following solution,
vy1(y; z) = iva cos(k0y)(1 �� e��z); = (1 �� i)
1
s
; (2.65)
1(y; z) = ��
va
cs
0 sin(k0y); (2.66)
In the bulk of the uid (z ! 1) the complex phases of vy1 and 1 are i and ��1, respectively,
representing out of phase oscillations. However, inside the boundary layer the phase of vy1
changes due the complex part of ,
vy1(y; z) = iva cos(k0y)
h
1 �� ei z
s e�� z
s
i
; (2.67)
and the oscillation of vy1 and 1 are thus not completely out of phase inside the boundary
layer, and the magnitude of the time-averaged product
1v1
j1jjv1j agrees with the
prediction from dimensional analysis.
This phase shift and shear of v1 inside the boundary layer is what leads to the generation
of a steady acoustic streaming ow. To calculate the generated streaming ow,
the rst-order problem needs to be treated more thoroughly, including the orthogonal velocity
component vz1, after which the time-averaged second-order equations Eqs. (2.18a)
and (2.19a) need to be solved. This is outside the scope of this presentation, and the reader
is referred to the Master Thesis by Jonas T. Karlsen Ref. 42 and the study by Rednikov
and Sadhal Ref. 8. The non-linear interactions of the oscillatory rst-order elds inside
the boundary layers generate a steady rotation of the uid
v2
inside the boundary layers,
which in turn drives a steady rotational ow in the bulk. This mechanism is sketched in
Fig. 2.3 for a standing wave in-between two parallel plates, with the direction of propagation
parallel to the walls, in contrast to the example in Section 2.3, where the direction of
propagation was orthogonal to the walls. The classical results by Lord Rayleigh Ref. 3
for the steady streaming velocity
v2
near the wall, in our notation becomes
vy2
=
3
8
v2
a
cs
sin
��
2k0y
; near wall: (2.68)
2.5 Summary of adiabatic theory
In this chapter we have derived the rst- and second-order acoustic equations within the
framework of the adiabatic approximation. These equations have been solved for the
problem of an acoustic wave between two orthogonally oscillating parallel plates, including
derivations of the oscillating rst-order velocity component, the time-averaged second-order