Numerical study of frequency mixing in a focused ultrasonic field in bubbly liquids from a dual-frequency spherical source

Abstract

The aim of this work is to study the nonlinear frequency mixing that comes out in the focus region in a bubbly liquid when a spherical transducer emits a dual-frequency ultrasonic wave of finite amplitude in a three-dimensional open domain with axial symmetry. The nonlinear interaction between the acoustic field and the bubble vibrations is modeled through a coupled differential system formed by the wave equation and a Rayleigh-Plesset equation. Simulations are performed by means of an ad-hoc numerical model developed on the basis of multi-dimensional finite-volume and finite-difference techniques. Results show the nonlinear response of the system by showing the space distribution of the fundamental and harmonics for a single-frequency excitation (for which a convergence study is also carried out) and the space distribution of the difference and sum-frequency components for a two-frequency excitation. In the former case a law is proposed for the evolution of each one of the second and third harmonics vs. amplitude. In the latter configuration a law is proposed for the evolution of each one of the difference and sum frequencies vs. amplitude. This work suggests that the combination of two frequencies emitted from a spherical transducer can allow the enhancement of the generation of the difference and sum-frequencies in a restricted volume of the bubbly liquid. This could be used to control the amplitude and space extent of these frequency components by modifying the shape of the spherical transducer, the primary frequencies, and the pressure amplitude. Our results can be useful for industrial and medical processes based on nonlinear frequency mixing. Keywords: Focused ultrasonic field, Numerical model, Nonlinear distortion, Nonlinear frequency mixing