Contribución al estudio de la mezcla no lineal de ondas ultrasónicas en líquidos con burbujas mediante modelos numéricos

Abstract

Bubbly liquids are media with important potential in di erent kinds of medical and industrial applications. They present very interesting characteristics since a very small amount of gas bubbles (of the order of 0:001% in volume) is able to modify the acoustic behavior of the liquid drastically. Important acoustic properties such as sound speed, attenuation, compressibility or nonlinearity are modi ed up to several orders of magnitude. Very interesting nonlinear phenomena take place when an acoustic eld excites a bubbly liquid, like, for instance, the production of new frequencies. The experimental study of bubbly liquids is complex since it is not an easy task to control the amount, density, spatial distribution, and size of bubbles. This di culties lead to look for alternatives that allow us to analyze them by means of mathematical models able to describe the propagation of acoustic waves through this kind of media. The mutual nonlinear interaction between acoustic pressure and bubble vibrations is described by the coupled di erential system of equations formed by the wave equation and a Rayleigh-Plesset equation. Moreover, since this di erential system, due to its complexity, does not admit exact analytical solutions, the development of numerical models is necessary to obtain approximate solutions. The goal of this doctoral thesis is to develop numerical models that allow us to study the nonlinear phenomena occurring when an acoustic wave travels through a liquid with gas bubbles. Mainly, the objective of this work is the analysis of new frequencies produced during this nonlinear process, harmonic components when we work with a mono-frequency source and both sum and di erence frequency components when we work with a dual-frequency source. The following procedure is followed to achieve this objective. First, di erent physical models are proposed (in one and several dimensions). Then, their mathematical modelization is carried out through the corresponding systems of di erential equations and auxiliary conditions (boundary and initial conditions). Afterwards, we develop the numerical models that solve the di erential systems, based on nite-volume and nitedi erence techniques for the spatial coordinates and time domain respectively, before performing the implementation of the di erent computational codes. Finally, the corresponding simulations are carried out to analyze and study the nonlinear behaviors and to propose physical laws that describe them. This procedure allows us, from a complex problem, to obtain physical conclusions that simplify its comprehension. The results presented in this doctoral thesis show the usefulness and versatility of the numerical models we have developed. These models allow us to study thoroughly the behavior of this kind of media when they are excited by ultrasonic waves of nite amplitude in many con gurations: i) one dimension, ii) two dimensions, iii) three dimensions (axial symmetry), for both resonators and open- eld problems. The nonlinear behavior of harmonic production and frequency mixing is analyzed in detail according to the main elements that in uence its generation (bubble size and density, wave amplitude) to de ne how to enhance them. Moreover, the study provides us with the observation of e ects that suggest a sound speed change of the medium as a function of the amplitude of acoustic pressure.