Явления кавитации в микроскопическом объеме жидкости (Cavitation Phenomena in Microscale Confinement) тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Леонов Константин Васильевич

• Леонов Константин Васильевич
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Леонов Константин Васильевич

Contents

Introduction

Chapter 1. The model of rigid confinement

1.1 Quasi-static approximation

1.1.1 The Blake model

1.1.2 The modified Blake model: Bubble in a spherical liquid cell

1.2 Kinetic and potential energy. The generalized Rayleigh-Plesset equation

1.3 The natural frequency

1.4 Dynamic regimes of a bubble in a confined liquid

1.4.1 The non-cavitation regime

1.4.2 The single cavitation expansion

1.4.3 The multiple cavitation

1.5 The high-frequency dynamic regimes. Transition to the bulk liquid case

1.6 Conclusion

Chapter 2. The model of an elastic confinement

2.1 Mathematical model

2.1.1 Quasi-static model for a bubble in a liquid cell

2.1.2 Dynamic model accounting for elasticity of the surrounding solid

2.2 Quasi-static approximation

2.3 Linear analysis

2.3.1 Natural frequency

2.3.2 Quality factor

2.3.3 Amplitude and phase response

2.4 Nonlinear bubble dynamics

2.4.1 Dynamics of a small bubble

2.4.2 Dynamics of a large bubble

2.5 Comparison with bubble dynamics in a bulk liquid. Analysis of an adiabatic approximation

2.6 Conclusion

Chapter 3. The mass transfer problem

3.1 Bubble dynamics: Case of fixed mass of gas in a bubble

3.2 Formulation of mass transport problem

3.3 Splitting of the problem

3.3.1 The oscillatory problem

3.3.2 The smooth problem

3.4 Regimes of mass transfer

3.4.1 Regime 1: Growing depletion layer

3.4.2 Regime 2: Gas diffusion within a liquid cell

3.5 Bubble dynamics: Case of rectified diffusion

3.5.1 Analysis of the bubble dynamics on the slow timescale

3.5.2 The influence of gas concentration on a resulting stable bubble size

3.6 Bubble dynamics: Case of ordinary diffusion

3.7 Diffusion stability of a cavitation bubble

3.7.1 Dynamics regimes of a stable bubble

3.8 Potential applications

3.9 Conclusion

Conclusions

Acknowledgments

Bibliography

Введение диссертации (часть автореферата) на тему «Явления кавитации в микроскопическом объеме жидкости (Cavitation Phenomena in Microscale Confinement)»

Introduction

Relevance of the work

Cavitation can be defined as any stimulated bubble activity. The stimulation may be due to flow, decompression, acoustic waves, sudden deposition of electromagnetic or ionizing radiation, or heat. The activity can refer to bubble inception or dynamics. While cavitation bubbles were historically associated with negative outcomes, e.g. noise, vibration, and erosion in hydraulic systems [Blake and Gibson, 1987; Brennen, 2013; Reuter et al., 2022], more and more efforts are dedicated to take profit from their power in a variety of applications. For example, the ability to control bubble dynamics by ultrasound waves underpins a few industrial processes in which bubbles grow or undergo collapse within simple (water) and complex (e.g. polymeric, biological) fluids [Dollet et al., 2019]. Radially oscillating bubbles forced by a sound field are commonly encountered in acoustic cavitation and sonoluminescence experiments, where bubble oscillations in a liquid may result in emitting of light at the phase of bubble collapse (see [Lauterborn and Kurz, 2010] and refs therein). The ultrasound has been widely applied to biotechnology, medicine, food technology, and other fields, especially the chemical industry, which mainly uses the chemical and mechanical effects of ultrasound cavitation [Rooze et al., 2013; Chowdhury et al., 2020; Tamidi et al., 2021]. Acoustically excited bubbles can disturb local flow field resulting in enhancement of chemical reactions [Crum et al., 1999] and mixing in microfluidic systems [Wang et al., 2009]. The use of the high energy ultrasound can induce cavitation effect in extra heavy oil, which can tear large molecules in extra heavy oil into light hydrocarbon molecules reducing the viscosity of the solution and improving transportability [Avvaru et al., 2018]. Bubbles subject to acoustic waves are also associated with applications of liquid degassing by triggering cavitation of gas-containing voids [Gondrexon et al., 1997] and used in many ultrasonic cleaning processes, for instance, in the cleaning of silicon wafers and computer components [Kim et al., 2009; Lauterborn and Kurz, 2010]. Vascular plants contain fluid-filled vessels (xylem) to transport water and minerals from the roots of the plants to their leaves, while water shortage can bring air bubbles into xylem and the bubble growth in xylem may cause mortality during drought [Cochard, 2006]. Microbubbles are currently in clinical use as effective ultrasound contrast agents, which consist of a low solubility gas core with a stabilizing shell, typically made of a lipid, a polymer, or a protein [Qin et al., 2009; Batchelor et al., 2022]. Their size allows them free flow through the vasculature, while their gas core provides ultrasound imaging contrast enhancement owing to the mismatch in acoustic impedance between gas and surrounding blood and soft tissue. Further, the incident ultrasonic field can induce volumetric oscillations of microbubbles, leading to increased scattering and improved image contrast. This effect can be amplified by driving bubbles at their resonance frequency, which

typically occurs within the clinically approved frequency range for diagnostic ultrasound. Microbubbles have also been widely studied for their potential to enhance drug delivery for treatment of diseases [Marmottant and Hilgenfeldt, 2003; Coussios and Roy, 2008; Stride and Coussios, 2019; Batchelor et al., 2022]. In combination with an ultrasound, they are capable of locally increasing cell membrane permeability to enhance drug uptake - the process is known as sonoporation. Therapeutic payloads can also be directly incorporated into bubbles themselves and their release triggered using high intensity ultrasound, which produces expansion and subsequent collapse of bubbles disrupting a stabilizing shell. The enhanced localized delivery has allowed therapeutic benefit to be observed with lower doses of drugs and hence reduced off-site toxicity.

In the most of the above-mentioned examples the acoustically driven cavitation activity takes place in highly confined spaces due to geometrical limitations of the used equipment (e.g. microfluidic systems) or natural dimensions of the environment (e.g. size of the vasculature). It should be mentioned that in the absence of the external sound field bubbles of any size are unstable because the pressure inside a bubble is larger than in a liquid due to surface tension, and therefore a bubble tends to dissolve slowly, which is accompanied by a continuous mass flux from the interior of a bubble into a liquid. On the other hand, bubble oscillations forced by the periodic acoustic field initiate gas flows in and out of a bubble. During the expansion period gas diffuses from a liquid into a bubble, and during the contraction cycle the diffusion takes place in the opposite direction. There is a net flow of gas into a bubble because the area of the bubble wall is greater during the expansion period and therefore more gas will enter during the expansion than will leave during the contraction cycle. This phenomenon is called the rectified diffusion and usually leads to a growth of a bubble [Hsieh and Plesset, 1961]. The above arguments would predict that an oscillating bubble always grows. However, surface tension still promotes dissolution, and a competition takes place between both phenomena.

One of the most representative examples of the influence of the rectified diffusion in a confined liquid on industrial applications is the inkjet printing technology (see [Lohse, 2022] and references therein). In industrial settings the drop-on-demand piezoacoustic inkjet printing is a widespread technological application of microfluidics, which is used in the graphic printing industry. In this technique a piezoacoustic printhead can jet single droplets on demand out of the micronozzle by driving the ink out of the nozzle thanks to deformation of a piezoelectric element and the resulting pressure field. Under certain conditions an air bubble can be entrained in the nozzle, in particular at large jetting frequencies beyond 20 kHz. The bubbles were shown to nucleate on inkophobic dirt particles suspended in the ink. Other potential entrainment mechanisms include the bubble pinch-off from an inward gas jet formed at the oscillating air-ink meniscus driven by flow focusing and the cavitation inception in the rarefaction phase of the pressure wave [Reinten et al., 2022]. Most inks are

not pure liquids, but have a very complex composition, containing multiple liquids with different material properties, pigments, other colloidal particles, latex, cross-linkers, surfactants, and polymers, which also increases the probability of bubble nucleation on suspended particles in the ink. After inception, a bubble grows by the rectified diffusion, as the ink contains some amount of dissolved gas, resulting in modification of the drop production process and breakdown of the jetting. Once a bubble is entrained, it can practically be flushed out together with the drop within a few actuation cycles or eliminated through diffusive dissolution, i.e., by switching off the piezoactuation for a period of the order of seconds to minutes [Fraters et al., 2019]. These methods consume both ink and time and are therefore highly undesirable. As a result, the bubble entrainment and following rectified diffusion in a confined liquid are the significant technological issues in modern industrial digital printing.

Thus, in medicine and industrial settings the acoustic radiation is commonly applied to the complex dispersed systems, in which oscillating cavitation bubbles can occur in the fluidic inclusions. These bubbles are trapped in the confinement, which alters the bubble dynamics in general and the mass transport phenomena around them in particular. That is why it is important to study the dynamics of confined bubbles along with the diffusion problem associated with such dynamics.

Level of scientific development of the research area

It is known that the presence of gas bubbles in a liquid substantially changes its acoustic properties. There are numerous studies devoted to the theoretical investigation of the propagation of harmonic oscillations in such mixtures. Various problems of acoustics with gas or vapor bubbles have been considered in the known monographs [Nigmatulin, 1991; Nakoryakov et al., 1990], including wave processes in gas-liquid systems, acoustics and shock waves in a homogeneous gas- and vapor-liquid mixture, effects of non-sphericity, crushing and grinding of bubbles on the propagation of waves in a liquid with gas bubbles, and dynamics of gas and vapor bubbles. Non-spherical bubble oscillations are often caused by the near-resonance dynamic conditions. For example, in the work [Vanovskiy, 2019] the author proposed a resonant mechanism of bubble fragmentation in a liquid, based on a significant increase in the amplitude of deformation during its nonlinear interaction with the radial mode and the fulfillment of resonance conditions, and obtained quantitative estimates of the resonant conditions for bubble fragmentation in cases of fast and slow activation of an acoustic wave.

Bubble dynamics has been intensively investigated theoretically, experimentally, and numerically for a wide variety of geometries. These include single bubbles in a bulk (infinite) liquid, near a rigid surface, a free surface, an elastic surface, solid and liquid bodies as well as other bubbles ([Nigmatulin, 1991; Petrov, 2010; Wang et al., 2019] and references therein), while the theoretical studies on cavitation phenomena in microscale confinement have received some attention only recently

with regard to situations where bubbles pre-exist or appear by nucleation in the trapped liquid inside a solid medium. It is motivated by applications in biology [Cochard, 2006; Stroock et al., 2014], geology [Marti et al., 2012; Roedder and Bodnar, 1980; Caupin, 2022], the dynamics of porous media [Scherer and Smith, 1995; Vincent et al., 2014b], and medicine [Deweber et al., 2011; Kawchuk et al., 2015]. The situation is defined as fully confined if the medium containing a bubble is surrounded by walls in all directions of space, so that no flux in or out of the confining material is possible. The condition of full confinement thus also holds for situations where fluxes are possible in and out of the confining cavity (e.g. porous walls), if the typical timescale of these fluxes is large compared to the timescales of interest for the bubble dynamics.

Currently, there are several three-layer geometry models of confined bubbles, such as bubble-solid shell-liquid [Church, 1995], bubble-liquid-air [Obreschkow et al., 2006], bubble-liquid-solid shell [Fourest et al., 2015], and bubble-liquid-elastic solid [Vincent, 2012; Vincent et al., 2012; Vincent et al., 2014a; Vincent and Marmottant, 2017; Wang, 2017]. In the paper [Vincent et al., 2012] the system for generation of negative pressure (tension) in a microscale spherical water-filled cavity in a hydrogel is presented. The authors used the evaporation technique to generate static negative pressures following the method proposed in [Wheeler and Stroock, 2008]. This technique is based on capillary-driven liquid flow in porous medium towards to the air, if it is not saturated by water vapor: the pore liquid responds to the lowering of the vapor pressure with a reduction of its hydrostatic pressure, as predicted by the Kelvin equation, where this liquid stress has its dynamic origin in capillarity due to the increase of curvature r 1 of the liquid-air interface and corresponding Laplace pressure. The material used is a hydrophilic hydrogel, which a priori limits the mechanism of heterogeneous cavitation on the wall. The inclusion is filled by soaking in degassed water, which reduces or dissolves any gaseous nuclei. Moreover, the water before reaching the inclusion must cross the porous medium, which has a nanometric mesh and naturally filters impurities. The authors carried out a series of experiments for imaging of the dynamics of a cavitation bubble in a liquid, which was entrapped in a transparent elastic solid, using light scattering, laser strobe photography, and high-speed camera recordings. The results of the experiments showed unexpectedly fast bubble oscillations in terms of the volume, which depend on the confinement size and elasticity. In bulk or partially confined conditions, bubble volumetric deformations can be accommodated by pushing a liquid away. This is no longer true in a fully confined situation, where any expansion of the bubble must be accompanied by compression of the liquid and/or stretching of the surrounding material. In particular, the effective bulk modulus of the elastic solid medium was introduced in the paper [Vincent et al., 2014a], which allows to explain the bubble dynamics as a mechanical analog of springs oscillations. In the paper [Doinikov and Marmottant, 2018], the authors showed that the dynamics of a spherical gas bubble,

freely oscillating at the center of a microscale spherical liquid-filled cavity surrounded by an infinite viscoelastic medium, depends on the properties of the bubble and the confining solid medium. It was found that if the bubble radius is relatively small, the gas content plays a key role in the dynamics. On the contrary, if the bubble radius is comparable to the radius of the cavity, the dynamics is determined by the properties of the solid environment. The last statement is in good agreement with the results of the modeling of experimental observations in the paper [Vincent et al., 2014a].

To model the cavitation phenomena in the hydrogel, the authors in the paper [Vincent and Marmottant, 2017] developed a theory, which includes the kinetic energy of the liquid flow induced by bubble motion and the potential energy of a confined bubble as a function of its radius, including contributions from gas compressibility, surface tension, liquid compressibility, and elastic deformation of the surrounding solid. The equations governing the free oscillatory dynamics of fully confined bubbles, extending the Minnaert formula [Minnaert, 1933] and the Rayleigh-Plesset equation [Rayleigh, 1917; Plesset, 1949], were considered, which allowed to evaluate the frequency of the bubble oscillations and the equilibrium radius reached by the growing cavitation bubble.

Vincent and Marmottant [Vincent and Marmottant, 2017] and Wang [Wang, 2017] derived the Rayleigh-Plesset-like equations, which describe free finite amplitude oscillations of a bubble in a microscale liquid-filled cavity confined by an infinite elastic solid. In both models, the cavitation threshold is the reference state of a bubble, leading to transient bubble dynamics. In this case, the authors of the paper [Vincent and Marmottant, 2017] show that the liquid compressibility and solid elasticity effects are predominant, whereas the surface tension on the gas/liquid interface, viscosity of a liquid, and trapped gas effects can be neglected in the validation of their model with experimental data. In [Wang, 2017] the effects of the surface tension, viscosity of a liquid, and trapped gas, as well as the liquid compressibility and solid elasticity, on transient bubble dynamics in a confined liquid cell are considered, which results in damped bubble oscillations until the equilibrium bubble radius is reached. Both models consider dissipation using well-known formulae from bubble dynamics in a bulk liquid. In [Vincent and Marmottant, 2017] the importance of dissipation mechanisms (viscous, acoustic, thermal) when considering cavitation dynamics in a confined liquid using the approximation of a bulk liquid was discussed.

It should be mentioned that the conditions for cavitation inception and dissipation mechanisms in the bubble-in-cell system were not discussed. In the paper [Vincent and Marmottant, 2017] the dynamics of a liquid cell is defined by bubble oscillations, whereas in the paper [Wang, 2017] bubble oscillations in a cavity subjected to an acoustic wave are considered. The dynamic regimes under an external driving were not studied and the gas diffusion was neglected in the above-mentioned models of a microscale confined liquid, assuming a fixed amount of gas in a bubble.

The problem of mass transport of gas dissolved in a bulk liquid around a bubble undergoing volume oscillations has been studied extensively [Blake, 1949; Strasberg, 1959; Hsieh and Plesset, 1961; Eller and Flynn, 1965; Fyrillas and Szeri, 1994; Fyrillas and Szeri, 1995; Brenner et al., 1996; Akhatov et al., 1997; Hilgenfeldt et al., 1998; Brenner et al., 2002]. Hsieh and Plesset [Hsieh and Plesset, 1961] proposed the first complete formulation of the problem and derived an approximate solution for linear oscillations, in good agreement with the measurements of Strasberg [Strasberg, 1961]. A theoretical breakthrough was performed by Eller and Flynn [Eller and Flynn, 1965], who obtained a solution of the problem usable for any bubble dynamics R(t) calculated separately and developed the theory for non-linear and/or large amplitude oscillations, using a boundary layer analysis in Lagrangian coordinates first introduced by Plesset and Zwick [Plesset and Zwick, 1952]. Fyrillas and Szeri [Fyrillas and Szeri, 1994] performed a theoretical study of the rectified diffusion, modifying the limiting assumptions inherent in the Eller and Flynn formulations. They split the convection-diffusion problem into two parts: the oscillatory and smooth problems. Both problems were treated by singular perturbation methods: the oscillatory problem was solved through the boundary-layer analysis, and the smooth problem was solved by the method of multiple scales in time. Meanwhile, the first experiments on single-bubble sonoluminescence (SBSL) revealed the actual existence of a diffusively stable cavitation bubble, neither growing, nor dissolving [Gaitan et al., 1992]. This unusual feature in cavitation was investigated by Brenner [Brenner et al., 1996] and Akhatov [Akhatov et al., 1997], where the authors used the theory of Fyrillas and Szeri [Fyrillas and Szeri, 1994] and found that diffusively stable bubbles can be obtained in degassed conditions. The authors showed that small gas bubbles in a liquid are stable in the presence of a strong sound field, which allows to explain permanent bubbles oscillations for several days without dissolution and without changing their size in the sonoluminescence experiments.

Theoretical studies on the rectified diffusion in microscale confinement have received little attention so far. The typical mass transfer problem is usually considered in the cylindrical confinement geometry. For example, the results of the experiments showed that the gas diffusion across ultrasound-driven microbubbles in the confinement of nano- and micro-channels can (partially) block them and impede their proper use (see [Moreno Soto et al., 2020] and references therein). In this case, bubbles are usually formed either of dissolved gas in the liquid, or liquid vapor originating from boiling or chemical processes occurring along the channel [Ajaev and Homsy 2006; Gedupudi et al., 2011; Zu et al., 2011]. However, to the best of our knowledge, no solution has yet been presented in the open literature for the rectified diffusion in the condition of full confinement. In the dissertation we derive such a solution to show the possible diffusion regimes in the bubble-in-cell system.

Dissertation goal

The goal of this dissertation is the development of theory of the dynamics and diffusion phenomena related with a single cavitation bubble in a microscale liquid volume under an external driving.

To achieve this goal, the following problems were addressed:

1. The model of behavior of a spherical bubble with a fixed mass of gas inside in a microscale confined compressible liquid (spherical liquid cell) in the presence of an external driving force was developed for two surrounding media: perfectly rigid medium and an elastic solid medium, and corresponding governing equations for bubble dynamics were derived.

2. The model was demonstrated for the statics and dynamics of a confined bubble under different forms of driving and different control parameters. The obtained results were compared with the available theoretical/experimental results in the literature.

3. The problem of the coupled bubble oscillations and mass transfer, driven by a periodic external pressure in a microscale compressible spherical liquid cell surrounded by an infinite elastic solid, was formulated, assuming that a liquid cell has some amount of dissolved gas, which is involved in dissolution or growth of a gas bubble due to mass transfer in a liquid.

4. The diffusion regimes of a bubble in a liquid cell were studied for two different cases: 1) a bubble is a small bubble nucleus, which grows/dissolves due to the rectified diffusion, and 2) a bubble is a relatively large bubble, which has a constant size during volume bubble oscillations on average.

Scientific novelty

1. The volumetric effect of microscale confinement on bubble dynamics was analyzed. The criteria of cavitation inception and cavitation vanishing of a spherical bubble in a compressible externally driven spherical liquid cell were obtained in the assumption of a fixed mass of gas inside a bubble.

2. The three possible regimes of bubble dynamics in a confined liquid volume at different forms of forced oscillations were for the first time described.

3. The novel mass transport model for a single cavitation bubble in a microscale compressible externally driven spherical liquid cell surrounded by an infinite elastic solid was developed.

4. It was found that gas diffusion in a confined liquid volume contributes to a change in the cavitation threshold and the manifestation of the dynamic regimes of a bubble.

5. The three possible diffusion regimes in the bubble-in-cell system under an external driving were first described and conditions of the diffusion stability of a bubble in a liquid cell were obtained.

Theoretical and practical significance

This work provides a theoretical framework to investigate, monitor, and control outcomes associated with cavitation phenomena in a microscale confined liquid volume. The obtained results can be used in future studies of bubbles in confined systems together with acoustic applications, as well as for validation of the more complex numerical models and codes. The study of the bubble dynamics in a confined liquid, along with the mass transport problem, may offer potential for further study and use in applications of medicine, where microbubbles are often used as an ultrasound contrast agents and drug delivery agents in vasculature, liquid degassing, ultrasonic cleaning processes, enhancement of chemical reactions, mixing in microfluidic systems, and reducing the viscosity of extra heavy oil. The theoretical findings of the present dissertation can also be proposed as the conceptual guidance for improvement of the modern applications of ultrasound technology.

Methodology and research methods

The main governing equations of the bubble dynamics are derived within the framework of the Lagrangian formalism, where the kinetic and potential energy of a bubble in a confined liquid are used. To study the natural frequency of the bubble oscillations and the amplitude-phase frequency response of the bubble-in-cell system, the linear stability analysis is used, where the linearized system of equations of bubble motion in the liquid cell is considered neglecting higher-order terms. The bifurcation diagrams are used to give an overview of the occurrence of resonances and bifurcations, and to illustrate some typical bifurcation scenarios of the dynamics of the bubble-in-cell system. The strongly nonlinear ordinary differential equation governing the bubble dynamics in a confined liquid is solved numerically employing the Wolfram Mathematica software1 and built-in solver based on LSODA approach, automatically switching between nonstiff and stiff methods [Petzold, 1983; Radhakrishnan and Hindmarsh, 1993]. This approach uses Adams methods (orders 1-12) as the family of nonstiff methods, and backward differentiation formulas (BDF) (orders 1-5) as the family of stiff methods, where both the step size and the method order are dynamically varied throughout the problem to meet local error requirements and to ensure stability. The equations governing the convection and diffusion of dissolved gas in the liquid cell are solved within the framework of the

1 https://www.wolfram.com

2 LSODA is a variant of LSODE (Livermore Solver for Ordinary Differential Equations) with Automatic method switching

approximation proposed in [Fyrillas and Szeri, 1994], using the multiple-time-scale approach, the method of the integral balance, and the nonlinear time averages.

Scientific results submitted for the defense

1. The generalization of the Blake model, which predicts the rapid growth of a spherical cavitation bubble nucleus and transient cavitation process in an infinite incompressible liquid, was extended to the case of a confined geometry of a compressible spherical liquid cell (microscale confinement). It was shown that cavitation is completely suppressed by confinement at relatively small liquid cell sizes.

2. Within the framework of the generalized model the Rayleigh-Plesset model was broaden to account for interaction with an elastic solid and an external pressure forcing. Depending on the form of the external driving, the three possible regimes of bubble dynamics in a confined liquid volume are realized:

• The non-cavitation regime, which corresponds to the bubble oscillations below the Blake threshold;

• The single cavitation expansion, which occurs due to the stepwise tension of a liquid cell exceeding the Blake threshold;

• The multiple cavitation, which is defined by the periodic driving at tension of a liquid cell exceeding the Blake threshold, where a bubble is exposed to cavitation expansion followed by cavitation vanishing to a finite radius.

3. The rectified diffusion theory of Fyrillas and Szeri [Fyrillas and Szeri, 1994] was extended for the case of a single cavitation bubble in a microscale compressible externally driven spherical liquid cell surrounded by an infinite elastic solid. The influence of dissolved gas on bubble behavior in a liquid cell results in the three possible diffusion regimes:

• The total bubble dissolution, when a gas concentration in a liquid is not enough to compensate mass fluxes in and out of a bubble;

• The partial bubble dissolution, when a small gas concentration in a liquid is enough to compensate mass fluxes in and out of a bubble, which shrinks to a resulting stable bubble size;

• The partial bubble growth, when a concentration of dissolved gas is enough to provide a resulting stable bubble with a larger size than the initial one.

4. The diffusion stability in the bubble-in-cell system is provided by the regimes of the partial bubble dissolution and the partial bubble growth, where the formation of a stable bubble is

accompanied by a transient dynamics, which includes the dynamic regimes in the

presence/absence of cavitation inception or their combination.

Validity of the research results

The validity and reliability of the results are ensured using the classical approaches and equations of continuum mechanics, and detailed description of the developed models. One of the dynamic regimes of a bubble, which is described in the dissertation, has a qualitative agreement with the published results of cavitation experiments conducted in a stiff polymer hydrogel by other authors, including also quantitative agreement with the number of oscillations of a bubble before complete damping. The developed models of the bubble dynamics and mass transport in a liquid cell can be reduced to the well-known theoretical models in an infinite liquid. The dissertation materials have been widely discussed at specialized seminars, international and national scientific conferences, and confirmed by publications in the peer-reviewed scientific journals.

List of scientific conferences and seminars

The results of the present study were discussed at the following scientific conferences and seminars: Gen-Y 2.0 (Skoltech Young Scientists Cross-Disciplinary Conference, Sochi, March 13-17, 2019); The XII All-Russian Congress on Basic Problems of Theoretical and Applied Mechanics (Ufa, August 19-24, 2019); Virtual Technical Meeting of the Society of Engineering Science 2020 (September 29-October 1, 2020); The International Autumn School-Conference "Advanced Problems in Mechanics - 2020" (St. Petersburg, November 9-13, 2020); 73rd Annual Meeting of the APS Division of Fluid Dynamics (November 22-24, 2020); The Skoltech Center for Design, Manufacturing and Materials (CDMM) seminar (2021); The 11th International Symposium on Cavitation 2021 (CAV2021) (May 10-13, 2021); Research seminar of the Department of Gas and Wave Dynamics (Faculty of Mechanics and Mathematics, Moscow State University) (2024); Research seminar of the Multiphase systems modeling laboratory (Project Center for Energy Transition and ESG, Skoltech) (2024); The Skoltech Center for Materials Technologies (CMT) seminar (2024); Research seminar of the Laboratory of Biomaterials (Vladimir Zelman Center for Neurobiology and Brain Rehabilitation, Skoltech) (2024).

Publications

The main results on the topic of the dissertation were published in the following peer-reviewed scientific journals:

1. Leonov K., Akhatov I. Towards a theory of dynamics of a single cavitation bubble in a rigid micro-confinement // International Journal of Multiphase Flow. 2020. V. 130. P. 103369. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103369

2. Leonov K., Akhatov I. Dynamics of an externally driven cavitation bubble in an elastic microconfinement // Physical Review E. 2021. V. 104(1). P. 015105. https://doi.org/10.1103/PhysRevE.104.015105

3. Leonov K., Akhatov I. The influence of dissolved gas on dynamics of a cavitation bubble in an elastic micro-confinement // International Journal of Heat and Mass Transfer. 2022. V. 196. P.123295.

https://doi .org/10.1016/j.ij heatmasstransfer.2022.123295

4. Leonov K. V., Akhatov I. Sh. The diffusion stability of an externally driven cavitation bubble in micro-confinement // Fluid Dynamics. 2024. V. 59(1). P. 60-73. https://doi.org/10.1134/S0015462823602413

Personal contribution

All the results presented in the dissertation were obtained by the candidate personally. Scientific supervisor took part in the problems setting, discussion of the results and preparation of scientific publications.

Dissertation structure

The dissertation consists of an introduction, 3 main chapters, and the last section - Conclusions. It is written on 126 pages, including 38 figures and 1 table. The list of references contains 87 titles including 4 of publications by the author.

Заключение диссертации по теме «Другие cпециальности», Леонов Константин Васильевич

Conclusions

In this work the generalization of the Blake model, which predicts the rapid growth of a spherical cavitation bubble nucleus and transient cavitation process in an infinite incompressible liquid, is extended to the case of an externally driven microscale confined liquid volume (spherical liquid cell). The liquid is assumed compressible and surrounded by a perfectly rigid/elastic solid medium. It is shown that the bubble nucleus with a fixed mass of gas inside expands to the finite radius when liquid cell tension exceeds the modified critical Blake threshold in a confined liquid. This finite radius is defined by the initial radius of the bubble nucleus and the liquid cell size:

• The smaller size of the bubble nucleus, the higher tension has to be applied for cavitation inception, which also leads to a larger resulting bubble size;

• The larger size of the initial liquid cell, the larger radius of the bubble will be achieved after cavitation.

In contrast to the case of an unbounded liquid, for a bubble nucleus in a confined liquid there is the critical size of the liquid cell RfontIcal at which cavitation does not occur. For R0 > RfQnt,cal the bubble experiences abrupt growth up to a finite bubble size under liquid cell tension. At liquid cell size R0 smaller than the critical one, the cavitation is completely suppressed by confinement.

Within the framework of the generalized model the Rayleigh-Plesset model is broaden to account for interaction with an elastic solid and an external pressure forcing. Using the linear stability analysis, we show that the natural frequency of the bubble in a confined liquid is defined by the bubble properties for the equilibrium states below the Blake threshold, whereas after cavitation inception, the natural oscillations depend on the parameters of the surrounding medium.

The three possible regimes of the bubble dynamics in the confined liquid volume under different forms of the external driving are shown for the first time:

• The non-cavitation regime, which corresponds to the bubble oscillations below the Blake threshold;

• The single cavitation expansion, which occurs due to the stepwise tension of the liquid cell exceeding the Blake threshold;

• The multiple cavitation, which is defined by the periodic driving at tension of the liquid cell exceeding the Blake threshold. In this case, the bubble is exposed to cavitation expansion followed by cavitation vanishing to a finite radius.

In the case of an infinite elastic solid the bifurcation diagrams are considered with the radius of the bubble plotted versus the driving frequency, used as a control parameter. We use these diagrams to

give an overview of the occurrence of resonances and bifurcations, and to illustrate some typical bifurcation scenarios of the dynamics of the bubble-in-cell system at high driving frequencies. In our study we consider the bubble dynamics in a liquid cell for small and large bubbles, which correspond to the states below and above the cavitation threshold, respectively. In the case of a small bubble, the bubble dynamics is characterized by period-1 oscillations with typical peaks of the frequency response, which correspond to the orders of resonance. The dynamics of a large bubble has complex periodic behavior with a transition to chaos. The non-isothermal effects of the bubble oscillations in a confined liquid are discussed and comparison of the bubble dynamics with the case of a bulk liquid is provided in the adiabatic approximation for ultrasound driving frequencies. The analysis shows that within the physical parameters used in the dissertation, only the isothermal approximation model is applicable.

The rectified diffusion theory of Fyrillas and Szeri [Fyrillas and Szeri, 1994] is extended for the case of a single cavitation bubble in a microscale compressible externally driven spherical liquid cell surrounded by an infinite elastic solid, assuming that a liquid cell has some amount of dissolved gas, which is involved in dissolution or growth of a gas bubble due to mass transfer in a liquid. The mass transport problem is studied for two different cases: 1) a bubble is a small bubble nucleus, which grows/dissolves due to the rectified diffusion, and 2) a bubble is a relatively large bubble, which has a constant size during volume bubble oscillations on average.

For the first time, it is demonstrated that the mass transfer in the bubble-in-cell system leads to the three possible diffusion regimes of the bubble in the confined liquid volume under the external driving:

• The total bubble dissolution, when a gas concentration in a liquid is not enough to compensate the mass fluxes in and out of the bubble;

• The partial bubble dissolution, when a small gas concentration in a liquid is enough to compensate the mass fluxes in and out of a bubble, which shrinks to a finite stable bubble size;

• The partial bubble growth, when a concentration of dissolved gas is enough to provide a resulting stable bubble with a larger size than the initial one.

The last two regimes provide the diffusion stability in the bubble-in-cell system. The analysis shows that the regimes of the partial dissolution and partial growth can be formed under the condition of the three dynamic regimes of the bubble: 1) in the presence of cavitation inception, where both cavitation inception and cavitation vanishing are realized; 2) in the absence of cavitation, where cavitation is completely suppressed by confinement and the bubble oscillations occur in a nonlinear (non-explosive) way, and 3) combination of the previous two dynamic regimes, which corresponds to transient bubble dynamics with activation/deactivation of cavitation inception. In this case, concentration of dissolved

gas in the liquid plays a role of the governing parameter, which allows to control a dynamics regime during the formation of a stable bubble size.

This work provides a theoretical framework to investigate, monitor, and control outcomes associated with cavitation phenomena in a microscale confined liquid volume. The obtained results can be used in future studies of bubbles in confined systems together with acoustic applications, as well as for validation of the more complex models. Although a number of theoretical and numerical studies on bubble dynamics in a confined liquid have been reported, experimental data are yet lacking, and thus limiting model validation as well as quantitative evaluation of the effects of confinement on bubble dynamics. In particular, acoustically induced nonlinear oscillations of a bubble in a microscale confined liquid have not been systematically observed, and the obtained results can provide a theoretical guidance for future experiment planning.

Among the most promising directions for further development of the research one can mention the following:

• The issue of the bubble nucleation in a confined liquid

• The shape-stability analysis of a bubble in a confined liquid

• The heat transfer problem in a confined liquid

• The bubble dynamics in the case of different configurations of the system (e.g. the emulsion model: bubble-water-oil)

Also, an interesting issue here is the formulation of a general theory, which describes the physical analogy of the boiling and cavitation processes in a microscale confined liquid.

Список литературы диссертационного исследования кандидат наук Леонов Константин Васильевич, 2025 год

Bibliography

Ajaev, V. S. Modeling shapes and dynamics of confined bubbles / V. S. Ajaev, G. M. Homsy // Annu. Rev. Fluid Mech. - 2006. - Vol. 38. - P. 277-307.

Akhatov, I. The role of surface tension in stable single-bubble sonoluminescence / I. Akhatov, N. Gumerov, C.-D. Ohl, U. Parlitz, W. Lauterborn // Phys. Rev. Lett. - 1997. - Vol. 78, no. 2. - P. 227230.

Avvaru, B. Current knowledge and potential applications of cavitation technologies for the petroleum industry / B. Avvaru, N. Venkateswaran, P. Uppara, S. B. Iyengar, S. S. Katti // Ultrason. Sonochem. -2018. - Vol. 42. - P. 493-507.

Batchelor, D. V. B. The influence of nanobubble size and stability on ultrasound enhanced drug delivery / D. V. B. Batchelor, F. J. Armistead, N. Ingram, S. A. Peyman, J. R. McLaughlan, P. L. Coletta, S. D. Evans // Langmuir. - 2022. - Vol. 38, no. 45. - P. 13943-13954.

Behnia, S. Nonlinear transitions of a spherical cavitation bubble / S. Behnia, A. Jafari, W. Soltanpoor, O. Jahanbakhsh // Chaos, Solitons & Fractals. - 2009. - Vol. 41, no. 2. - P. 818-828.

Bird, R. B. Transport phenomena / R. B. Bird, W. E. Stewart, E. N. Lightfoot. - N.Y.: John Wiley and Sons, Inc., 1960. - 780 p.

Blake, F. G. The onset of cavitation in liquids / F. G. Blake // Tech. Memo. No. 12, Acoustics Res. Lab., Harvard University. 1949.

Blake, J. R. Cavitation bubbles near boundaries / J. R. Blake, D. C. Gibson // Annu. Rev. Fluid Mech.

- 1987. - Vol. 19. - P. 99-123.

Brennen, C. E. Cavitation and Bubble Dynamics / C. E. Brennen. - N.Y.: Cambridge University Press, 2013. - 268 p.

Brenner, M. P. Mechanisms for Stable Single Bubble Sonoluminescence / M. P. Brenner, D. Lohse, D. Oxtoby, T. F. Dupont // Phys. Rev. Lett. - 1996. - Vol. 76, no. 7. - P. 1158-1161.

Brenner, M. P. Single-bubble sonoluminescence / M. P. Brenner, S. Hilgenfeldt, D. Lohse // Rev. Mod. Phys. - 2002. - Vol. 74, no. 2. - P. 425-484.

Caupin, F. Cavitation in water: a review / F. Caupin, E. Herbert // Comptes Rendus Physique. - 2006.

- Vol. 7, no. 9-10. - P. 1000-1017.

Caupin, F. Fluid inclusions in minerals: from geosciences to the physics of water and back / F. Caupin // Comptes Rendus Physique. - 2022. - Vol. 23, no. S2. - P. 71-87.

Chowdhury, S. MUltrasound and microbubble mediated therapeutic delivery: Underlying mechanisms and future outlook / S. M. Chowdhury, L. Abou-Elkacem, T. Lee, J. Dahl, A. M. Lutz // J. Control. Release. - 2020. - V. 326. - P. 75-90.

Church, C. C. The effects of an elastic solid surface layer on the radial pulsations of gas bubbles / C. C. Church // J. Acoust. Soc. Am. - 1995. - V. 97, no. 3. - P. 1510-1521.

Cochard, H. Cavitation in trees / H. Cochard // Comptes Rendus Physique. - 2006. - Vol. 7, no. 9-10.

- P.1018-1026.

Coussios, C. C. Applications of acoustics and cavitation to noninvasive therapy and drug delivery / C. C. Coussios, R. A. Roy // Annu. Rev. Fluid Mech. - 2008. - Vol. 40. - P. 395-420.

Crum, L. A. Sonochemistry and Sonoluminescence / L. A. Crum, T. J. Mason, J. L. Reisse, K. S. Suslick. - Springer Dordrecht, 1999. - 404 p.

Del Grosso, V. A. Speed of sound in pure water / V. A. Del Grosso, C. W. Mader // J. Acoust. Soc. Am. - 1972. - Vol. 52, no. 5B. - P. 1442-1446.

Deweber, K. Knuckle cracking and hand osteoarthritis / K. Deweber, M. Olszewski, R. Ortolano // J. Am. Board Fam. Med. - 2011. - Vol. 24, no. 2. - P. 169-174.

Doinikov, A. A. Model for the growth and the oscillation of a cavitation bubble in a spherical liquid-filled cavity enclosed in an elastic medium / A. A. Doinikov, B. Dollet, P. Marmottant // Phys. Rev. E.

- 2018. - Vol. 97, no. 1. - P. 013108.

Doinikov, A. A. Natural oscillations of a gas bubble in a liquid-filled cavity located in a viscoelastic medium / A. A. Doinikov, P. Marmottant // J. Sound Vib. - 2018. - V. 420. - P. 61-72.

Dollet, B. Bubble dynamics in soft and biological matter / B. Dollet, P. Marmottant, V. Garbin // Annu. Rev. Fluid Mech. - 2019. - Vol. 51. - P. 331-355.

Eller, A. Rectified diffusion during nonlinear pulsations of cavitation bubbles / A. Eller, H. G. Flynn // J. Acoust. Soc. Am. - 1965. - Vol. 37, no. 3. - P. 493-503.

Fourest, T. Confined Rayleigh-Plesset equation for Hydrodynamic Ram analysis in thin-walled containers under ballistic impacts / T. Fourest, J. M. Laurens, E. Deletombe, J. Dupas, M. Arrigoni // Thin-Walled Struct. - 2015. - Vol. 86. - P. 67-72.

Fraters, A. Inkjet nozzle failure by heterogeneous nucleation: bubble entrainment, cavitation, and diffusive growth / A. Fraters, M. van den Berg, Y. de Loore, H. Reinten, H. Wijshoff, D. Lohse, M. Versluis, T. Segers // Phys. Rev. Appl. - 2019. - Vol. 12, no. 6. - P. 064019.

Fyrillas, M. M. Dissolution or growth of soluble spherical oscillating bubbles / M. M. Fyrillas, A. J. Szeri // J. Fluid Mech. - 1994. - Vol. 277. - P. 381-407.

Fyrillas, M. M. Dissolution or growth of soluble spherical oscillating bubbles: the effect of surfactants / M. M. Fyrillas, A. J. Szeri // J. Fluid Mech. - 1995. - Vol. 289. - P. 295-314.

Gaitan, D. F. Sonoluminescence and bubble dynamics for a single, stable, cavitation bubble / D. F. Gaitan, L. A. Crum, C. C. Church, R. A. Roy // J. Acoust. Soc. Am. - 1992. - Vol. 91, no. 6. - P. 3166-3183.

Gedupudi, S. Confined bubble growth during flow boiling in a mini/micro-channel of rectangular cross-section. Part I: Experiments and 1-D modeling / S. Gedupudi, Y. Q. Zu, T. G. Karayiannis, D. B. R. Kenning, Y. Y. Yan // Intl. J. Therm. Sci. - 2011. - Vol. 50, no. 3. - P. 250-266.

Goldstein, H. Classical Mechanics, 3rd ed. / H. Goldstein, C. Poole, J. Safko, S. R. Addison. -Addison-Wesley, 2002.

Gondrexon, N. Degassing effect and gas-liquid transfer in a high frequency sonochemical reactor / N. Gondrexon, V. Renaudin, P. Boldo, Y. Gonthier, A. Bernis, C. Pettier // J. Chem. Eng. - 1997. - Vol. 66, no. 1. - P. 21-26.

Hilgenfeldt, S. Analysis of Rayleigh-Plesset dynamics for sonoluminescing bubbles / S. Hilgenfeldt, M. P. Brenner, S. Grossmann, D. Lohse // J. Fluid Mech. - 1998. - Vol. 365. - P. 171-204.

Hsieh, D. Theory of rectified diffusion of mass into gas bubbles / D. Hsieh, M. S. Plesset // J. Acoust. Soc. Am. - 1961. - Vol. 33, no. 2. - P. 206-215.

Kawchuk, G. N. Real-time visualization of joint cavitation / G. N. Kawchuk, J. Fryer, J. L. Jaremko, H. Zeng, L. Rowe, R. Thompson // PLoS One. - 2015. - Vol. 10, no. 4. - P. e0119470.

Kell, G. S. Density, thermal expansivity, and compressibility of liquid water from 0° to 150°C: Correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale / G. S. Kell // J. Chem. Eng. Data. - 1975. - Vol. 20, no. 1. - P. 97-105.

Kim, W. Mechanism of particle removal by megasonic waves / W. Kim, T.-H. Kim, J. Choi, H.-Y. Kim // Appl. Phys. Lett. - 2009. - Vol. 94, no. 8. - P. 081908.

Landau, L. D. Theory of Elasticity / L. D. Landau, E. M. Lifshitz. - N.Y.: Pergamon Press, 1970. Landau, L. D. Mechanics / L. D. Landau, E. M. Lifshitz. - Butterworth-Heinemann, 1976. Landau, L. D. Fluid Mechanics / L. D. Landau, E. M. Lifshitz. - N.Y.: Pergamon Press, 1987. Lauterborn, W. Numerical investigation of nonlinear oscillations of gas bubbles in liquids / W.

Lauterborn // J. Acoust. Soc. Am. - 1976. - Vol. 59, no. 2. - P. 283-293.

Lauterborn, W. Experimental and theoretical bubble dynamics / W. Lauterborn, T. Kurz, R. Mettin,

C.-D. Ohl // Adv. Chem. Phys. - 1999. - Vol. 110. - P. 295-380.

Lauterborn, W. Physics of bubble oscillations / W. Lauterborn, T. Kurz // Rep. Prog. Phys. - 2010. -Vol. 73. - P. 106501.

Lohse, D. Fundamental fluid dynamics challenges in inkjet printing / D. Lohse // Annu. Rev. Fluid Mech. - 2022. - Vol. 54. - P. 349-382.

Louisnard, O. Acoustic cavitation. In: Feng H., Barbosa-Canovas G., Weiss J. (eds) Ultrasound Technologies for Food and Bioprocessing. Food Engineering Series / O. Louisnard, J. González-García. - N.Y.: Springer, 2011. P. 13-64.

Marmottant, P. Controlled vesicle deformation and lysis by single oscillating bubbles / P. Marmottant, S. Hilgenfeldt // Nature. - 2003. - Vol. 423. - P. 153-156.

Marti, D. The effect of surface tension on liquid-gas equilibria in isochoric systems and its application to fluid inclusions / D. Marti, Y. Krüger, D. Fleitmann, M. Frenz, J. Ricka // Fluid Phase Equilib. -2012. - Vol. 314. - P. 13-21.

Minnaert, M. XVI. On musical air-bubbles and the sounds of running water / M. Minnaert // The London, Edinburgh, and Dublin Philos. Mag. J. Sci. - 1933. - Vol. 16, no. 104. - P. 235-248.

Moreno Soto, A. Diffusive growth of successive bubbles in confinement / Á. Moreno Soto, D. Lohse,

D. Van der Meer // J. Fluid Mech. - 2020. - Vol. 882. - P. A6.

Nakoryakov, V. E. Wave Mechanics of the Gas and Vapor-Liquid Systems / V. E. Nakoryakov, B. G. Pokusaev, I. R. Shraiber. - M.: Energoatomizdat, 1990. (in Russian)

Nigmatulin, R I. Dynamics of Multiphase Media / R. I. Nigmatulin. - N.Y.: Hemisphere Publishing Corporation, 1991. Vols. 1,2.

Obreschkow, D. Cavitation bubble dynamics inside liquid drops in microgravity / D. Obreschkow, P. Kobel, N. Dorsaz, A. de Bosset, C. Nicollier, M. Farhat // Phys. Rev. Lett. - 2006. - Vol. 97, no. 9. -P. 094502.

Parlitz, U. Bifurcation structure of bubble oscillators / U. Parlitz, V. Englisch, C. Scheffczyk, W. Lauterborn // J. Acoust. Soc. Am. - 1990. - Vol. 88, no. 2. - P. 1061-1077.

Petrov, A. G. Analytical Fluid Dynamics / A. G. Petrov. - M.: Fizmatlit, 2010. (in Russian)

Petzold, L. Automatic selection of methods for solving stiff and nonstiff systems of ordinary

differential equations / L. Petzold // SIAM J. Sci. Stat. Comput. - 1983. - Vol. 4, no. 1. - P. 136-148.

Plesset, M. S. The dynamics of cavitation bubbles / M. S. Plesset // J. Appl. Mech. - 1949. - Vol. 16, no. 3. - P. 277-282.

Plesset, M. S. A nonsteady heat diffusion problem with spherical symmetry / M. S. Plesset, S. A. Zwick // J. Appl. Phys. - 1952. - Vol. 23, no. 1. - P. 95-98.

Qin, S. Ultrasound contrast microbubbles in imaging and therapy: physical principles and engineering / S. Qin, C. F. Caskey, K. W. Ferrara // Phys. Med. Biol. - 2009. - Vol. 54, no. 6. - P. R27-R57.

Radhakrishnan, K. Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations / K. Radhakrishnan, A. C. Hindmarsh // NASA Reference Publication, Lawrence Livermore National Laboratory Report. 1993.

Rayleigh, L. On the pressure developed in a liquid during the collapse of a spherical cavity / L. Rayleigh // The London, Edinburgh, and Dublin Philos. Mag. J. Sci. - 1917. - Vol. 34, no. 200. - P. 94-98.

Reinten, H. Resonance behavior of a compliant piezo-driven inkjet channel with an entrained microbubble / H. Reinten, Y. Jethani, A. Fraters, R. Jeurissen, D. Lohse, M. Versluis, T. Segers // J. Acoust. Soc. Am. - 2022. - Vol. 151, no. 4. - P. 2545-2557.

Reuter, F. Cavitation erosion by shockwave self-focusing of a single bubble / F. Reuter, C. Deiter, CD. Ohl // Ultrason. Sonochem. - 2022. - Vol. 90. - P. 106131.

Roedder, E. Geologic pressure determinations from fluid inclusion studies / E. Roedder, R. J. Bodnar // Annu. Rev. Earth Planet. Sci. - 1980. - Vol. 8. - P. 263-301.

Rooze, J. Dissolved gas and ultrasonic cavitation - A review / J. Rooze, E. V. Rebrov, J. C. Schouten, J. T. F. Keurentjes // Ultrason. Sonochem. - 2013. - Vol. 20, no. 1. - P. 1-11.

Rosner, D. E. Effects of interface kinetics, capillarity and solute diffusion on bubble growth rates in highly supersaturated liquids / D. E. Rosner, M. Epstein // Chem. Eng. Sci. - 1972. - Vol. 27, no. 1. -P. 69-88.

Sander, R. Compilation of Henry's law constants (version 4.0) for water as solvent / R. Sander // Atmos. Chem. Phys. - 2015. - Vol. 15, no. 8. - P. 4399-4981.

Scherer, G. W. Cavitation during drying of a gel / G. W. Scherer, D. M. Smith // J. Non-Cryst. Solids.

- 1995. - Vol. 189, no. 3. - P. 197-211.

Strasberg, M. Onset of ultrasonic cavitation in tap water / M. Strasberg // J. Acoust. Soc. Am. - 1959.

- Vol. 31, no. 2. - P. 163-176.

Strasberg, M. Rectified diffusion: Comments on a paper of Hsieh and Plesset / M. Strasberg // J. Acoust. Soc. Am. - 1961. - Vol. 33, no. 3. - P. 359.

Stride, E. Nucleation, mapping and control of cavitation for drug delivery / E. Stride, C. Coussios // Nat. Rev. Phys. - 2019. - Vol. 1. - P. 495-509.

Stroock, A. D. The physicochemical hydrodynamics of vascular plants / A. D. Stroock, V. V. Pagay, M. A. Zwieniecki, N. Michele Holbrook // Annu. Rev. Fluid Mech. - 2014. - Vol. 46. - P. 615-642.

Tamidi, A. M. A review of recent development in numerical simulation of ultrasonic-assisted gasliquid mass transfer process / A. M. Tamidi, K. K. Lau, S. H. Khalit // Comput. Chem. Eng. - 2021. -Vol. 155. - P. 107498.

Vanovskiy, V. V. Resonance mechanism of gas bubble crushing in a liquid / V. V. Vanovskiy // PhD Dissertation, Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, 2019. - 95 p. (in Russian)

Vincent, O. Dynamique de bulles de cavitation dans de l'eau micro-confinée sous tension. Application à l'étude de l'embolie dans les arbres (Dynamics of cavitation bubbles in a micro-confined water volume undergoing tension. Application to the study of embolism in trees) / O. Vincent // PhD Dissertation, University of Joseph Fourier, Grenoble, France, 2012. - 255 p. (in French)

Vincent, O. Birth and growth of cavitation bubbles within water under tension confined in a simple synthetic tree / O. Vincent, P. Marmottant, P. A. Quinto-Su, C.-D. Ohl // Phys. Rev. Lett. - 2012. -Vol. 108, no. 18. - P. 184502.

Vincent, O. The fast dynamics of cavitation bubbles within water confined in elastic solids / O. Vincent, P. Marmottant, S. R. Gonzalez-Avila, K. Ando, C.-D. Ohl // Soft Matter. - 2014a. - Vol. 10. - P. 1455-1461.

Vincent, O. Drying by cavitation and poroelastic relaxations in porous media with macroscopic pores connected by nanoscale throats / O. Vincent, D. A. Sessoms, E. J. Huber, J. Guioth, A. D. Stroock // Phys. Rev. Lett. - 2014b. - V. 113, no. 13. - P. 134501.

Vincent, O. On the statics and dynamics of fully confined bubbles / O. Vincent, P. Marmottant // J. Fluid Mech. - 2017. - Vol. 827. - P. 194-224.

Wang, S. S. Acoustically induced bubbles in a microfluidic channel for mixing enhancement / S. S. Wang, Z. J. Jiao, X. Y. Huang, C. Yang, N. T. Nguyen // Microfluid. Nanofluid. - 2009. - Vol. 6. - P. 847-852.

Wang, Q. X. Oscillation of a bubble in a liquid confined in an elastic solid / Q. X. Wang // Phys. Fluids. - 2017. - V. 29, no. 7. - P. 072101.

Wang, S.-P. Experimental observations of the behaviour of a bubble inside a circular rigid tube / S.-P. Wang, Q. Wang, A.-M. Zhang, E. Stride // Int. J. Multiph. Flow. - 2019. - Vol. 121. - P. 103096.

Wheeler, T. D. The transpiration of water at negative pressures in a synthetic tree / T. D. Wheeler, A. D. Stroock // Nature. - 2008. - Vol. 455. - P. 208-212.

Zu, Y. Q. Confined bubble growth during flow boiling in a mini-/micro-channel of rectangular cross-section. Part II. Approximate 3-D numerical simulation / Y. Q. Zu, Y. Y. Yan, S. Gedupudi, T. G. Karayiannis, D. B. R. Kenning // Intl. J. Therm. Sci. - 2011. - Vol. 50, no. 3. - P. 267-273.

Publications of the Author on the Subject of the Dissertation

Leonov, K. Towards a theory of dynamics of a single cavitation bubble in a rigid micro-confinement / K. Leonov, I. Akhatov // Int. J. Multiph. Flow. - 2020. - Vol. 130. - P. 103369.

Leonov, K. Dynamics of an externally driven cavitation bubble in an elastic microconfinement / K. Leonov, I. Akhatov // Phys. Rev. E. - 2021. - Vol. 104, no. 1. - P. 015105.

Leonov, K. The influence of dissolved gas on dynamics of a cavitation bubble in an elastic microconfinement / K. Leonov, I. Akhatov // Int. J. Heat Mass Transf. - 2022. - Vol. 196. - P. 123295.

Leonov, K. V. The diffusion stability of an externally driven cavitation bubble in micro-confinement / K. V. Leonov, I. Sh. Akhatov // Fluid Dynamics. - 2024. - Vol. 59, no. 1. - P. 60-73.