2.5. SUMMARY OF ADIABATIC THEORY 23
velocity component, and the oscillating second-order velocity component. The treatment
of the oscillatory second-order velocity component is new, and our analysis shows that
its amplitude is a factor of Q larger than what is expected from dimensional analysis
of the second-order equations. Furthermore, we have discussed the acoustic boundary
layer, the shear stresses and the phase shift of the acoustic velocity oscillations within
it, and how this leads to the generation of a steady streaming velocity. In the following
chapter the governing equations are derived in a more rigorous way without the adiabatic
approximation.
(a) (b)
(c) (d)
Figure 2.3: Sketch of the boundary layer and the generated acoustic streaming ow between two parallel
plates. (a) Standing acoustic wave between two parallel plates, with the direction of propagation parallel
to the walls. Light green arrows indicate the oscillating velocity v1. (b) Zoon-in on the boundary layer
near the bottom wall. The amplitude of the velocity oscillations decay to fulll the no-slip condition
v = 0 at the wall. (c) Dark green arrows indicate the steady rotational streaming ow in the bulk of the
uid, often referred to as Rayleigh streaming, generated by the non-linear interactions of the oscillatory
rst-order elds inside the boundary layers. (d) Zoom-in on the boundary layer with dark green arrows
indicating the rotational motion of the steady streaming ow inside the boundary layer, often referred to
as Schlichting streaming. The gure is adapted from Ref. 43.