^{1}, Aaron Zakrzewski

^{1}, Christina Rossi

^{1}, Diane Dalecki

^{2}and Sheryl Gracewski

^{3,a)}

### Abstract

Motivated by various clinical applications of ultrasound contrast agents within blood vessels, the natural frequencies of two bubbles in a compliant tube are studied analytically, numerically, and experimentally. A lumped parameter model for a five degree of freedom system was developed, accounting for the compliance of the tube and coupled response of the two bubbles. The results were compared to those produced by two different simulation methods: (1) an axisymmetric coupled boundary element and finite element code previously used to investigate the response of a single bubble in a compliant tube and (2) finite element models developed in comsol Multiphysics. For the simplified case of two bubbles in a rigid tube, the lumped parameter model predicts two frequencies for in- and out-of-phase oscillations, in good agreement with both numerical simulation and experimental results. For two bubbles in a compliant tube, the lumped parameter model predicts four nonzero frequencies, each asymptotically converging to expected values in the rigid and compliant limits of the tube material.

The coupled boundary element and finite element code used in this work was written by Hongyu (Jacky) Miao. The authors gratefully acknowledge Sally Child and Chris Jensen for their assistance with the experiments. This work was supported by research Grants Nos. NIH R01HL69824 and NSF Grant No. CMI-0652947. Christina Rossi and Aaron Zakrzewski were supported by supplemental REUs on NSF Grant No. CMI-0652947.

I. INTRODUCTION

II. FIVE-DEGREE-OF-FREEDOM MODEL

III. SIMULATIONS: COUPLED BEM-FEM

IV. SIMULATIONS: COMSOL

V. EXPERIMENTS

VI. RESULTS AND DISCUSSION

VII. SUMMARY

### Key Topics

- Bubble dynamics
- 18.0
- Acoustic modeling
- 12.0
- Elasticity
- 10.0
- Surface tension
- 10.0
- Acoustical effects
- 8.0

## Figures

(a) Schematic diagram for two bubbles in a tube, showing bubble radii, *R* _{1}, *R* _{2}, bubble positions with respect to center of the tube, *z* _{1}, *z* _{2}, tube inner radius, *r* _{tube}, tube thickness, *t* _{tube}, and tube length, *L*. (b) Each spherical bubble is replaced with a cylindrical body of equal volume and radius equal to tube inner radius, so that the width of the *i*th cylindrical bubble is

(a) Schematic diagram for two bubbles in a tube, showing bubble radii, *R* _{1}, *R* _{2}, bubble positions with respect to center of the tube, *z* _{1}, *z* _{2}, tube inner radius, *r* _{tube}, tube thickness, *t* _{tube}, and tube length, *L*. (b) Each spherical bubble is replaced with a cylindrical body of equal volume and radius equal to tube inner radius, so that the width of the *i*th cylindrical bubble is

(a) Schematic diagram showing five degrees of freedom for the two bubbles in a compliant tube. (b) Schematic diagram of a tapered tube of angle θ with a single bubble positioned in the middle of the tube, as simulated with comsol Multiphysics.

(a) Schematic diagram showing five degrees of freedom for the two bubbles in a compliant tube. (b) Schematic diagram of a tapered tube of angle θ with a single bubble positioned in the middle of the tube, as simulated with comsol Multiphysics.

(Color online) A stainless steel cylindrical exposure chamber (25.5 cm in diameter and 35.5 cm in height) is filled with degassed, deionized water at room temperature, with a shaker, mounted on the bottom, controlled by a digital signal generator and a power amplifier. A tube was clamped horizontally inside the tank and located using a three-way positioner, such that the bubble centers were consistently 10 cm below the water surface. A hydrophone was placed in a hole at the center of the tube to measure the pressure in the vicinity of the bubble(s). The resonance frequencies were identified by the peaks in the pressure versus frequency plot.

(Color online) A stainless steel cylindrical exposure chamber (25.5 cm in diameter and 35.5 cm in height) is filled with degassed, deionized water at room temperature, with a shaker, mounted on the bottom, controlled by a digital signal generator and a power amplifier. A tube was clamped horizontally inside the tank and located using a three-way positioner, such that the bubble centers were consistently 10 cm below the water surface. A hydrophone was placed in a hole at the center of the tube to measure the pressure in the vicinity of the bubble(s). The resonance frequencies were identified by the peaks in the pressure versus frequency plot.

Normalized natural frequencies for two equal-sized bubbles (*R* _{10} = *R* _{20} = 1 cm) as a function of normalized separation distance 2*z*/*L*, for bubbles (a) inside a rigid tube of *r* _{tube} = 1.27 cm and *L* = 20 cm and (b) in an open volume. Frequencies are normalized by *f* _{0}, the natural frequency of a single bubble of radius *R* _{0} = 1 cm in an open volume, represented by dashed line in (b). In (a), the solid lines are the results of the 3-degree-of-freedom model, and the circle markers are results from both simulations: coupled BEM-FEM and comsol acoustics model for two bubbles in a rigid tube. The dashed line represents the normalized frequency of a single bubble of radius *R* _{0} = 1 cm situated at the same axial position as one of the two bubbles. In (b), the solid lines are calculated using Eq. (22).

Normalized natural frequencies for two equal-sized bubbles (*R* _{10} = *R* _{20} = 1 cm) as a function of normalized separation distance 2*z*/*L*, for bubbles (a) inside a rigid tube of *r* _{tube} = 1.27 cm and *L* = 20 cm and (b) in an open volume. Frequencies are normalized by *f* _{0}, the natural frequency of a single bubble of radius *R* _{0} = 1 cm in an open volume, represented by dashed line in (b). In (a), the solid lines are the results of the 3-degree-of-freedom model, and the circle markers are results from both simulations: coupled BEM-FEM and comsol acoustics model for two bubbles in a rigid tube. The dashed line represents the normalized frequency of a single bubble of radius *R* _{0} = 1 cm situated at the same axial position as one of the two bubbles. In (b), the solid lines are calculated using Eq. (22).

Comparison of normalized frequencies f/f_{0} versus normalized separation distance 2*z*/*L* for two different sized bubbles equidistant from the center of the tube. Three sets of curves, corresponding to *R* _{2}/*R* _{1} = 0.5 (triangular markers), 0.75 (square markers), and 1.0 (circular marker), show both in-phase (lower) and out-of-phase (upper) frequencies. Lumped parameter model (solid line) and comsol acoustics model results for two bubbles in a rigid tube (markers) agree well when the bubbles are placed deep inside the tube.

Comparison of normalized frequencies f/f_{0} versus normalized separation distance 2*z*/*L* for two different sized bubbles equidistant from the center of the tube. Three sets of curves, corresponding to *R* _{2}/*R* _{1} = 0.5 (triangular markers), 0.75 (square markers), and 1.0 (circular marker), show both in-phase (lower) and out-of-phase (upper) frequencies. Lumped parameter model (solid line) and comsol acoustics model results for two bubbles in a rigid tube (markers) agree well when the bubbles are placed deep inside the tube.

Plot of the four natural frequencies f versus elastic modulus *E* for two bubbles (*R* _{10} = *R* _{20} = 1 cm) equidistant from the tube center (*z* _{1},*z* _{2} = ± 1.5 cm) in a compliant tube with *r* _{tube} = 1.27 cm, *t* _{tube} = 0.25 cm, and *L* = 20 cm. Lines show the analytical results from the 5- degree-of-freedom lumped parameter model and markers show the simulation results from comsol coupled fluid-structure interaction model for two bubbles in a flexible tube. The solid lines and filled markers correspond to in-phase frequencies. The dashed lines and open markers correspond to out-of-phase frequencies.

Plot of the four natural frequencies f versus elastic modulus *E* for two bubbles (*R* _{10} = *R* _{20} = 1 cm) equidistant from the tube center (*z* _{1},*z* _{2} = ± 1.5 cm) in a compliant tube with *r* _{tube} = 1.27 cm, *t* _{tube} = 0.25 cm, and *L* = 20 cm. Lines show the analytical results from the 5- degree-of-freedom lumped parameter model and markers show the simulation results from comsol coupled fluid-structure interaction model for two bubbles in a flexible tube. The solid lines and filled markers correspond to in-phase frequencies. The dashed lines and open markers correspond to out-of-phase frequencies.

Normalized natural frequencies for two equal-sized bubbles (*R* _{1}∼*R* _{2}∼1.1 cm) and a single bubble (*R* _{0}∼1.1 cm) inside a stiff tube (*r* _{tube} = 1.75 cm, *t* _{tube} = 0.64 cm, *L* = 20 cm, *E* = 3200 MPa). The frequencies are normalized by the average of the two bubbles experimental free field natural frequencies, *f* _{0}. The solid lines correspond to the 3-degree-of- freedom model for two bubbles in a rigid tube, and the squares are from the experimental results for two bubbles in a tube. The dashed line represents the numerical prediction of the frequency of a single bubble at the same axial positions as one of the two bubbles, and the circles represent the single bubble experimental results.

Normalized natural frequencies for two equal-sized bubbles (*R* _{1}∼*R* _{2}∼1.1 cm) and a single bubble (*R* _{0}∼1.1 cm) inside a stiff tube (*r* _{tube} = 1.75 cm, *t* _{tube} = 0.64 cm, *L* = 20 cm, *E* = 3200 MPa). The frequencies are normalized by the average of the two bubbles experimental free field natural frequencies, *f* _{0}. The solid lines correspond to the 3-degree-of- freedom model for two bubbles in a rigid tube, and the squares are from the experimental results for two bubbles in a tube. The dashed line represents the numerical prediction of the frequency of a single bubble at the same axial positions as one of the two bubbles, and the circles represent the single bubble experimental results.

Normalized natural frequencies for one bubble in a compliant tube (solid lines and squares) and one bubble in a rigid tube (dashed line and circles) for *R* _{0} = 0.78 cm, *r* _{tube} = 1.27 cm, *t* _{tube} = 0.32 cm, *L* = 20 cm. The frequencies are normalized by the experimental free field natural frequency, *f* _{0}. The solid lines are from the 3-degree-of-freedom model for one bubble in a compliant tube and the squares are the corresponding experimental results. An elastic modulus of *E* = 0.6 MPa in the 3-degree-of-freedom model minimized the least squared error between the predicted and measured frequencies, so was used for this plot. The dashed line is the numerical prediction from the 1 -degree-of-freedom model for one bubble in a rigid tube and the circles are the corresponding experimental results.

Normalized natural frequencies for one bubble in a compliant tube (solid lines and squares) and one bubble in a rigid tube (dashed line and circles) for *R* _{0} = 0.78 cm, *r* _{tube} = 1.27 cm, *t* _{tube} = 0.32 cm, *L* = 20 cm. The frequencies are normalized by the experimental free field natural frequency, *f* _{0}. The solid lines are from the 3-degree-of-freedom model for one bubble in a compliant tube and the squares are the corresponding experimental results. An elastic modulus of *E* = 0.6 MPa in the 3-degree-of-freedom model minimized the least squared error between the predicted and measured frequencies, so was used for this plot. The dashed line is the numerical prediction from the 1 -degree-of-freedom model for one bubble in a rigid tube and the circles are the corresponding experimental results.

Normalized natural frequencies for two equal-sized bubbles (*R* _{10} = *R* _{20} = 4 µm) as a function of normalized separation distance 2*z*/*L*, for bubbles inside a rigid tube of *r* _{tube} = 5.08 µm, *L* = 80 µm, and *t* _{tube} = 1 µm. The open and closed markers correspond to comsol model results with and without surface tension (σ = 0.0643 N/m), respectively. The lines represent the lumped parameter model for two bubbles within a tube (solid) and one bubble within a tube (dotted).

Normalized natural frequencies for two equal-sized bubbles (*R* _{10} = *R* _{20} = 4 µm) as a function of normalized separation distance 2*z*/*L*, for bubbles inside a rigid tube of *r* _{tube} = 5.08 µm, *L* = 80 µm, and *t* _{tube} = 1 µm. The open and closed markers correspond to comsol model results with and without surface tension (σ = 0.0643 N/m), respectively. The lines represent the lumped parameter model for two bubbles within a tube (solid) and one bubble within a tube (dotted).

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