Oscillations, Waves, and Interactions - GWDG

Acoustic cavitation 185
A convenient notation for more general (single frequency) pressure fields is obtained
by introducing a spatially varying amplitude and phase: p(x, t) = pa(x) cos[ωt +
φ(x)]. Then the primary Bjerknes force reads
F B1 = −∇pa(x) 〈 V (t) cos (ωt + φ(x)) 〉
+ pa(x)∇φ(x) 〈 V (t) sin (ωt + φ(x)) 〉 . (3)
Now we can associate the first term on the right-hand side as a standing wave contribution,
and the second term as a travelling wave part. For the ideal cases of a
plane standing (resp. travelling) wave, pa(x) = cos(k · x) and φ = const (resp.
φ(x) = −k · x and pa = const) for some wave vector k. It is immediately clear that
then the respective other terms disappear. In some situations, the sound field has
the character of a decaying (or damped) travelling wave, and both terms are contributing.
Then they may be counteracting and leading to spatial locations of stable
force equilibrium. In the field in front of sonotrode tips, such stagnation points have
been observed and modelled (Ref. [39], Sect. 6.3).
4.2 Secondary Bjerknes forces
Oscillating bubbles radiate sound, and neighboured bubbles interact via the emitted
pressure field. The pressure itself adds to the acoustic driving of the adjacent bubble,
and the pressure gradient leads to a net force analogous to the primary Bjerknes
force in Eq. (2). It turns out that the driving coupling has often negligible effect
compared to the primary (incident) pressure, but the gradient force is essential at
short bubble distances. Its time average is called the secondary Bjerknes force, and
to some approximation, it can be written [40]
F 1,2 ρ
B2 = −
4π
(x2 − x1)
|x2 − x1| 3
�
˙V1(t) ˙ �
V2(t) . (4)
Here F 1,2
is the force of bubble 1 on bubble 2, and the time average is over the time
B2
derivatives of the product of both bubble volumes V1 and V2. It can be seen that
the force decays with one over distance squared, and this is exactly like for both
the electromagnetic and the gravitational force. The sign of the force, however, as
well as its absolute strength, is determined by the time-averaged term. This in turn
depends on several paramters like bubble size and acoustic pressure at the bubbles’
locations. Different situations are therefore possible, and in the linearized case, this
discussion can be reduced to the relative oscillation phases [4,5]: bubbles oscillating
in phase attract each other, while in antiphase, they repel. 11 In many cases, however,
the bubbles are of similar size and oscillating in similar phase, and thus the effect
of the secondary Bjerknes force is frequently a mutual bubble attraction. Indeed
this attracting force dominates over the primary Bjerknes force from some close
11 It might be noted that the equality of the force of bubble 1 on bubble 2 and vice versa is
an effect of the approximation of an instantaneous interaction, i. e., of infinite sound speed.
Taking into account a time delay in Eq. (4) destroys the symmetry, which leads to situations
where one bubble attracts the other while being itself repelled!