The resonance behavior of a bubble near a boundary

Abstract

We present an analytical model for the frequency response of a gas microbubble oscillating near a spherical inclusion of arbitrary size and mechanical nature (rigid, fluid, or viscoelastic) immersed in a viscous compressible fluid. The model considers both radial and nonspherical oscillations in the linear regime and predicts how their resonance frequencies and oscillation amplitudes are altered by the bubble size, material properties, and distance to the nearby sphere. As a key application, we demonstrate that scanning the frequency response of a bubble near a viscoelastic object, such as an erythrocyte-like particle mimicking a biological cell, offers a way to recover its mechanical properties through inverse modeling, opening new possibilities for high-resolution elastography at the microscale.

Figures (12)

• Figure 1: Coordinate systems used in the theoretical model.
• Figure 2: Frequency response of the radial mode for a bubble of radius $R_{10} = \qty{10}{\um}$ driven at $P_{\mathrm{ac}} = \qty{1}{\kilo\Pa}$ in an unbounded liquid. Comparison between the linearized Rayleigh–Plesset model [Eq. (\ref{['eq:57']})] and the present model [Eq. (\ref{['eq:59']})]. Here, $f_0=\omega_0 / 2\pi$ is the resonance frequency of the bubble in an unbounded liquid, calculated by Eq. (\ref{['eq:63']}).
• Figure 3: The resonance frequencies of shape modes ($n \geq 2$) for a bubble of radius $R_{10}=\qty{10}{\um}$ in an unbounded liquid. Comparison between the classical Lamb formulation [Eq. (\ref{['eq:60']})] and the results provided by the current model.
• Figure 4: (a) Frequency response of a gas bubble with the equilibrium radius $R_{10} = \qty{10}{\um}$, driven at $P_{\mathrm{ac}} = \qty{1}{\kilo\Pa}$, for various distances $h$ from the surface of a rigid sphere of radius $R_{20} = 40R_{10}$. (b) Normalized resonance frequency of mode $0$ for the same bubble placed at distances $h \in \{1.5R_{10}, \dots, 15R_{10}\}$ from rigid spheres of different radii $R_{20}$.
• Figure 5: Normalized resonance frequency of shape modes (a) $n = 2$, (b) $n = 3$, and (c) $n = 4$ for a gas bubble with the equilibrium radius $R_{10} = \qty{10}{\micro\meter}$, placed at distances $h \in \{1.2R_{10}, \dots, 5R_{10}\}$ from rigid spheres of varying radius $R_{20}$. The inset in panel (a) shows the normalized resonance frequency of mode $n = 2$ as a function of the sphere size ratio $R_{20}/R_{10}$.