Oscillations, Waves, and Interactions - GWDG

186 R. Mettin
R 02 [µm]
R 02 [µm]
180
160
140
120
100
80
60
40
20
4
3.5
3
2.5
2
1.5
1
0.5
20 kHz 10 kPa
0
0 20 40 60 80 100 120 140 160 180
R01 [µm]
1 MHz 10 kPa
0.5 1 1.5 2 2.5 3 3.5 4
R01 [µm]
R 02 [µm]
R 02 [µm]
180
160
140
120
100
80
60
40
20
0
0 20 40 60 80 100 120 140 160 180
R01 [µm]
4
3.5
3
2.5
2
1.5
1
0.5
20 kHz 50 kPa
1 MHz 100 kPa
0.5 1 1.5 2 2.5 3 3.5 4
R01 [µm]
R 02 [µm]
R 02 [µm]
180
160
140
120
100
80
60
40
20
0
0 20 40 60 80 100 120 140 160 180
R01 [µm]
4
3.5
3
2.5
2
1.5
1
0.5
20 kHz 100 kPa
1 MHz 200 kPa
0.5 1 1.5 2 2.5 3 3.5 4
R01 [µm]
Figure 11. Sign of the secondary Bjerknes force between two bubbles of equilibrium
radius R01 and R02 for fixed pressure amplitude pa. White areas indicate an attractive
force between the bubbles, and dark areas indicate mutual repulsion. Top row: f = 20 kHz,
Rres = 138 µm, pa = 10, 50, and 200 kPa; bottom row: f = 1 MHz, Rres = 3.15 µm, pa = 10,
100, and 200 kPa (water under normal conditions, κ = 1, σ = 0.07275 N/m).
distance on, and bubble coalescence is typically the consequence. In Fig. 7 the
subsequent approach and merging of the microbubble after ejection is in fact due
to the secondary Bjerknes force. The figure also shows one reason which hinders a
limitless accumulation of gas by bubble coalescence: the loss of spherical stability.
If nonlinearity is taken into account, modifications of the simple picture of the
secondary Bjerknes force occur. Essentially, the absolute strength of the force can be
much larger than from linear calculation, and also the size can be inverted [40]. Examples
for sign distributions of the secondary Bjerknes force in the nonlinear oscillation
case are shown in Fig. 11. Again 20 kHz and 1 MHz are chosen for illustration, and
the driving pressures are stepwise elevated to highlight the nonlinear effects. While
for 10 kPa still the classic picture with respect to the linear resonance radius occurs
(a “checkerboard”), already moderately high pressure amplitudes lead to “stripes”
that reflect the nonlinear resonances. The strong bubble response at the dynamic
Blake threshold becomes more and more important and determines the sign of the
force for small bubbles: in a way it takes over the role of the linear resonance to
separate attractive and repulsive combinations of bubble sizes.
Further effects, like non-spherical distortions at close distances and coupling of
oscillation and translation, have to be taken into account for a more complete picture