Interpretación crítica (Lecciones aprendidas)

Bjerknes

force

under

dual-frequency

excitation

In this section, a numerical approach is employed to study the

characteristics of the secondary Bjerknes force. The basic features of the

secondary Bjerknes force under dual-frequency excitation are investigated.

For comparison, the predictions of the secondary Bjerknes force under

single-frequency excitation are also shown.

Figure 4.3 shows the variation of the secondary Bjerknes force coefficient

B

f in the R₀₁R₀₂ plane under low sound pressure amplitude (P P_{e 0} 0.03). The repulsive forces (i.e., f_B 0) are represented by red areas while the attractive forces (i.e., f_B 0) are represented by grey scales. The darker the colour is, the higher the absolute value of f_B is.

Figures 4.3(a) and 4.3(b) are the predictions of f_B under single-frequency excitation ( f_s 100 kHz and f_s 200 kHz respectively). As shown in these figures, there are four regions in

01 02

R R planes, divided by the “boundaries” corresponding to the resonance radius of the driving frequency (for a simpler description, the

equilibrium radius of the bigger bubble is represented as R_{0 max} and the equilibrium radius of the smaller bubble is represented as R_0min):

a) Repulsive regions ( f_B 0): R_0min R_rs R_{0 max}. b) Attractive regions ( f_B 0): R_rs R_{0 min} R_{0 max}.

Figure 4.3(c) shows the f_B between two bubbles under the dual-frequency excitation ( f₁ 100 kHz and f₂ 200 kHz). Divided by the resonance bubble radii corresponding to the component driving

frequencies, there are nine regions in the R₀₁R₀₂ plane, which could be classified into three categories:

a) Repulsive regions ( f_B 0): R_{0 min} R_r2 R_r1R_0max.

b) Attractive regions ( f_B 0 ): R_r2R_r1R_{0 min} R_0max ; 2 0 min 0max 1

r r

R R R R ; R_{0 min} R_0max R_r2 R_r1.

c) Uncertain regions (where f_B could be positive or negative): 0 min r2 0max r1

R R R R ; R_r2 R_{0 min} R_r1R_0max.

This classification could also be explained by Eq. (4.36). In regions a) and

b), the values of f_B under the two component single-frequency excitation have the same sign. According to Eq. (4.36), the values of f_B

under the dual-frequency excitation can be considered as a linear

combination of the values of f_B under the two component single-frequency excitation if the value of P_e is limited. Hence, in

regions a) and b), the sign of f_B remains unchanged under dual-frequency excitation. In region c), f_B could be enforced or suppressed by adding the second acoustic excitation, leading to the sign

change, which depends on the relative values of f_B the secondary Bjerknes force corresponding to the two component frequencies.

Figure 4.3 The variations of the in the R₀₁R₀₂ 200 s f  kHz (b)] and dual kHz (c)] excitation.

corresponding resonance radii of the driving frequencies respectively. The

repulsive forces (i.e.,

attractive forces (i.e.,

bars are located at the bottom right corner. The white points and the

arrows indicate the two

μm.

variations of the secondary Bjerknes force coefficient

01 02

R R plane under single-frequency [ f_s 100 kHz (b)] and dual-frequency [ f₁ 100 kHz and

kHz (c)] excitation. P P_{e 0} 0.03 . R_r1 and R_r2 indicate the corresponding resonance radii of the driving frequencies respectively. The

repulsive forces (i.e., f_B 0) are represented by red areas while the attractive forces (i.e., f_B 0) are represented by grey scales. The scale at the bottom right corner. The white points and the

arrows indicate the two-bubble system with R₀₁ 16 μm and

secondary Bjerknes force coefficient f_B

100 kHz (a),

kHz and f₂ 200 indicate the

corresponding resonance radii of the driving frequencies respectively. The

) are represented by red areas while the

scales. The scale

at the bottom right corner. The white points and the

For further illustration, the values of f_B versus the equilibrium radius of bubble 2 ( R₀₂ ) is shown in Figure 4.4 for three typical cases:

01 r2 r1

R R R ( R₀₁ 10 μm), R_r2 R₀₁ R_r1 ( R₀₁25 μm) and

2 1 01

r r

R R R ( R₀₁40 μm). For the cases under single-frequency excitation, the absolute values of f_B rise significantly near the resonance. Furthermore, the sign of f_B would change near the resonance. For dual-frequency excitation, there are peaks near both resonance bubble

radii corresponding to the two frequencies, which means that the sign of

B

f may change two or three times in the full range of R₀₂ (10-50 μm).

Moreover, in the region away from the resonance radius, the values of f_B

under dual-frequency excitation are between the corresponding values of

B

f under two component single-frequency excitation. However, the positions of the peaks of dual-frequency approach are slightly different

from the positions of single–frequency approach. Taking the case where

01 10

R  μm [Figure 4.4(a)] as an example. When R₀₂ R_r2 R_r1, all the B

f under single- and dual-frequency excitation are positive, corresponding to the “attractive regions” in Figure 4.3. When

02 r1 r2

R R R , all the f_B under single and dual-frequency excitation are negative, corresponding to the “repulsive regions” in Figure 4.3. When

2 02 1

r r

R R R , f_B in the low-frequency approach (100 kHz) is positive while f_B in the high-frequency approach (200 kHz) is negative.

Therefore, f_B under the dual-frequency excitation varies from negative to positive, corresponding to the “uncertain regions” in Figure 4.3(c).

Figure 4.4 The variations of the secondary Bjerknes force coefficient f_B

versus equilibrium bubble radius of bubble 2 when the radius of bubble 1

is fixed as: (a) R₀₁ 10 μm, (b) R₀₁ 25 μm, (c) R₀₁40μm. The bubbles are driven by single-frequency [ f_s 100 kHz (dashed line),

200 s

f  kHz (dotted line)] and dual-frequency [ f₁ 100 kHz and 2 200

f  kHz (solid line)] excitation respectively. P P_{e 0} 0.03. The horizontal line indicates f_B 0.

When the pressure amplitude is higher, the influence of the nonlinearity

plane under a relatively high sound pressure amplitude (P P_{e 0} 0.3). Here, the driving frequencies are 100 kHz and 150 kHz respectively.

Comparing these results with the cases under the low pressure amplitude,

there are still boundaries near the resonances radii corresponding to the

driving frequencies. Furthermore, new peaks of f_B and new “repulsive regions” appear in the original “attractive regions”. These phenomena are

marked by white circles in Figure 4.5. They can also be classified as

below:

a) Harmonics (marked by solid lines) occur near the corresponding

resonance radii corresponding to the frequency nf₁ or mf₂, where

n=2,3 andm=2 in Figure 4.5.

b) Subharmonics (marked by dashed lines) occur near the resonance

radii corresponding to the frequency f n₁ or f₂ m. Limited by the range of the bubble radii, only the subharmonic of the high

frequency component (150 kHz) is shown. In Figure 4.5 (b) and (c),

m= 2.

c) Combination resonances (marked by the dash dotted line) occur near

the resonance radii corresponding to the frequency nf₁mf₂. In Figure 4.5(c),n=m=1.

Figure 4.5 The variations of the in the R₀₁R₀₂ 150 s f  kHz (b)] and dual (c)] excitation. P P_e represented by red

represented by grey scales. The scale bars

corner. The resonances marked with white circles

line), subharmonics (dashed line) and

line).

variations of the secondary Bjerknes force coefficient

01 02

R R plane under single-frequency [ f_s 100 kHz (b)] and dual-frequency [ f₁ 100 kHz and f₂ 

0 0.3 e

P P  . The repulsive forces (i.e., f_B

represented by red areas while the attractive forces (i.e., f

represented by grey scales. The scale bars are located at the bottom right

The resonances marked with white circles are harmonics (solid

line), subharmonics (dashed line) and combination resonance (dash secondary Bjerknes force coefficient f_B

100 kHz (a), 2 150 f  kHz 0 B f  ) are 0 B f  ) are at the bottom right

harmonics (solid

Like the main resonances, there are peaks of f_B near the harmonics, subharmonics and combination resonances. And the sign of f_B changes when the bubble radius cross over these resonance radii. In particular, for

harmonics of higher order [i.e., the second harmonic in Figure 4.5(a)],

only a peak of f_B appears while the sign of f_B does not change. Doinikov (Doinikov, 1999) derived an analytical solution of the secondary

Bjerknes force by including the first harmonic of bubble oscillation under

the single-frequency excitation [Doinikov, 1999, Eq. (37)]. If the second

harmonic included, it will be

(1) (2) (3)

1 1 1

B

F _F F F _ e₁₂. (4.43) Here, F₁(1) represents the force induced by the linear component of bubble oscillation, which is of order 2 . (2)

1

F represents the force induced by the first harmonic component of bubble oscillation, which is of

order 4. F₁(3) represents the force induced by the second harmonic component of bubble oscillation, which is of order 6. Therefore, as shown in Figure 4.5, when  P P_{e 0} 0.3, the effect of the first harmonic will become important and can change the value significantly as

well as the sign of f_B. However, under this pressure, the effect of the second order harmonic is not strong enough, so it can only change the

By comparing Figures 4.5(a) and 4.5(b) with 4.5(c), we can conclude that

the secondary Bjerknes forces in the R₀₁R₀₂ plane under dual-frequency excitation involve all the harmonics and subharmonics

corresponding to the two component frequencies. Meanwhile, there are

unique combination resonances in the R₀₁R₀₂ plane under dual-frequency excitation. Therefore, the variation of the sign of f_B in the R₀₁R₀₂ plane under the dual-frequency excitation shows much more complicated patterns.