2024. Vol. 19. Issue 1, Pp. 7–17

URL: http://mfs.uimech.org/mfs2024.1.002,en

DOI: 10.21662/mfs2024.1.002

URL: http://mfs.uimech.org/mfs2024.1.002,en

DOI: 10.21662/mfs2024.1.002

Theoretical analysis of linear oscillations of bubbles in a polydisperse cluster

E.Sh. Nasibullaeva

Mavlyutov Institute of Mechanics UFRC RAS, Ufa, Russia

The dynamics of a single air bubble in an unbounded liquid are significantly different from the dynamics of an individual bubble in clusters due to the hydrodynamic interaction between the bubbles. Studying the mechanism of this interaction is one of the important aspects in the study of the fundamental nature of acoustic and hydrodynamic cavitation. In this work, to analyze small oscillations of bubbles in a spherical cluster near a stable equilibrium position, the mathematical theory of a linear conservative system with several degrees of freedom is applied to explain the mechanism of interaction between bubbles of different sizes. Using this theory, in the general case, it has been proven that the number of resonance frequencies in a polydisperse cluster coincides with the number of fractions. It is shown that in the regions of the main resonance (at low frequencies) bubbles of different fractions oscillate in phase, and in the regions of secondary resonances (at high frequencies) the phases successively change to the opposite, starting with the fraction containing bubbles of the largest radius, and further changing in order of decreasing. Using the example of a two-fraction cluster, it was found that there is an inertial connection between the bubbles, but there is no force connection; when the number of bubbles of one of the fractions is small, the connection between them and the bubbles of the other fraction is weak, while the interaction between them can be strong. An analysis of the energy transfer between bubbles of different fractions showed that the change in the nature of bubbles oscillations in the fraction with a small radius, while the nature of vibration of bubbles in the other fraction does not change, is the result of dynamic damping.

bubble cluster,

amplitude-frequency characteristics,

linear conservative system with several degrees of freedom,

natural frequency of oscillations,

resonance,

partial system,

dynamic damping

The single bubble dynamics in a boundless liquid differs significantly from the dynamics of a individual bubble in clusters (called bubble cloud or bubble cluster in the literature) due to the hydrodynamic interaction between bubbles. Studying the mechanism of this interaction is one of the important aspects in studying the fundamental nature of acoustic and hydrodynamic cavitations. All theoretical works based on the assumption of the thermal nature of cavitations can be divided into two groups according to the approaches used in their studies: 1) analysis of the macroscopic motion of a bubble cluster by averaging the equations of conservation of momentum and mass of liquid and gas, when the mixture of liquid and gas bubbles is considered as continuous environment; 2) analysis of microscopic movements of individual bubbles in a cluster. In this work, the dynamics of spherical gas bubbles in a polydisperse cluster is analytically studied in the case of small oscillations of the bubbles based on a mathematical model of a bubble cluster developed by the author within the framework of the second approach. In accordance with this model, a polydisperse cluster was considered as a large drop containing liquid and a large number of microbubbles of different sizes, and the bubbles in the cluster were conditionally divided into a finite number of fractions, where each fraction was characterized by the initial radius of the bubbles.

The main **goal** of this work is to explain the mechanism of interaction between bubbles of different sizes in a polydisperse cluster under the influence of an external field.

**Methodology.** The mathematical theory of a linear conservative system with several degrees of freedom is applied to the analysis of bubble oscillations in a cluster near a stable equilibrium position.

**Findings.** Based on the amplitude-frequency characteristics obtained earlier by the author, corresponding graphs were constructed for two- and three-fraction clusters. It was shown, that

- in addition to the main resonance in the low-frequency region, there are secondary resonances at higher frequencies, and the number of these resonances is 1 less than the number of fractions. In the region of the main resonance, bubbles of different fractions oscillate in phase, and in the regions of secondary resonances, the phases successively change to the opposite, starting with the fraction containing bubbles of the largest radius, and then change in descending order;
- there is an inertial connection between the bubbles, but there is no force connection;
- with a small number of bubbles of one fraction, the connection between them and the bubbles of another fraction is weak, but the interaction between them can be strong.

**Originality.** Using the theory of a linear conservative system with several degrees of freedom, in the general case it has been proven that the number of resonant frequencies in a polydisperse cluster coincides with the number of fractions, and the nature of the oscillations of bubbles of different fractions (in phase or antiphase) in different resonant regions has been explained. An analysis of the energy transfer between bubbles of different fractions showed that a change in the nature of oscillations of bubbles in a fraction with a small radius (without changing the nature of vibration of bubbles in another fraction) is the result of dynamic damping.

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