The production of a stable population of gas bubbles

The production of a stable population of gas bubbles in a liquid in which acoustic waves are propagating is not an easy task. It is thus extremely difficult to obtain stable experimental measures in bubbly liquids. Therefore, an intense endeavor has been carried out lately in order to create numerical tools able to simulate the ultrasonic propagation in such media [17]. Sáez et al. characterize the ultrasonic field propagation in a 20kHz sonoreactor [18]. They use the finite lp-pla2 inhibitor method to solve the linear equation wave. Klíma et al. study the geometric optimization of a 20kHz sonoreactor by using the element finite method [19]. They show that an appropriate geometry can compensate the decrease in intensity of the acoustic signal when the distance from the source is increased. Sutkar et al. predict the cavitation activity in terms of pressure field [20]. They solve the wave equation by taking into account the damping effects (reflection, refraction and scattering), modifying the density and sound speed in the medium. They use a software based on the finite element method. Prosperetti develop a linear model based on the Caflisch equations [21,22]. They take into account the attenuation in the wave number. Dänke et al. simulate the three-dimensional distribution of a linear acoustic field in a liquid with an inhomogeneous distribution of bubbles by using a finite difference approach [23,24]. They assume a Gaussian distribution of bubble sizes and a wave number that considers attenuation. Nomura and Nakagawa analyze the pressure field inside an ultrasonic cleaning vessel [25]. They use a boundary element method to solve the wave equation considering the dissipation of the bubbles. Yasui et al. use the finite element method to calculate the pressure field by considering the vibration of the reactor walls and the attenuation of the bubbles [26]. Vanhille and Campos-Pozuelo characterize the acoustic field in a bubbly liquid by taking into account the nonlinearity, dispersion, and dissipation in one, two, and three dimensions [27–29].
Since the presence of bubbles in a liquid enhances the nonlinear behavior of ultrasound, bubbly liquids are interesting media for all the applications that require the nonlinear frequency mixing technique. Several studies have been carried out about this method in a bubbly liquid. Zabolotskaya and Soluyan [30] study the generation of harmonics and the creation of the sum and difference frequencies theoretically. Kobelev and Sutin [31] perform a theoretical and experimental study about the difference-frequency generation with a continuous size distribution of bubbles. Druzhinin et al. [32] analyze the low-frequency generation from two high-frequency waves in a resonant bubble layer analytically and numerically. Vanhille et al. [33] study the nonlinear frequency mixing in a resonant cavity numerically.
New developments are necessary to progress in this field and to get rid of the restrictions associate to the existing methods. In this paper we develop a new numerical tool based on the finite-volume method [34,35] and on the finite-difference method [35,36]. We consider a system that couples the acoustic pressure with the volume variation of the bubbles. We solve a set of differential equations formed by the wave equation and a Rayleigh-Plesset equation.
The objective of this work is to study the propagation of nonlinear ultrasonic waves in bubbly liquids by means of a new numerical tool able to describe the particular effects due to the oscillating bubbles on the acoustic field, and to analyze some aspects of the nonlinear frequency mixing. After describing the physical model and the development of the numerical model in Section 2, some numerical experiments are presented in Section 3. The results shown make Enhancer possible to validate the new model. A comparison between linear and nonlinear data obtained at infinitesimal and finite amplitudes, respectively, allows us to highlight the nonlinear effects due to the presence of the bubbles in the liquid and to study the distribution of the harmonic components of the pressure wave in the cavity. A law is proposed for the description of the second and third harmonics as a function of the source amplitude and the bubble density. A study is carried out by modifying the resonator length. The nonlinear mixing of two frequencies in the cavity shows the generation of the difference frequency component. The amplitude of this created frequency is studied versus the amplitude and the range of the two driving frequencies.