^{1,a)}and Yu. A. Polovinka

^{1}

### Abstract

The behavior of a single acoustically driven bubble tethered to a wire ring is considered. The method of restraining the bubble against rising by attaching it to a wire is a common procedure in conducting precision acoustic measurements. The dynamics of the tethered bubble differs from those of free bubble due to variation in inertial (or added) mass. The objective of this study is to obtain a closed-form, leading order solution for the volume oscillations, assuming smallness of the bubble radius R 0 in comparison with the acoustic wavelength λ. It was shown, by using the invariance of the Laplace equation to conformal transformations and the geometry of the problem, that the toroidal coordinates provide separation of variables and are most suitable for analysis of the oscillations of the tethered bubble. Thus, the dynamics of the bubble restraining by a wire loop in toroidal coordinates can be investigated by using analytical approach and by analogy to the dynamics of a free spherical bubble.

This study was supported by the RFBR, Project No. 11-05-00212a and FEBRAS, Project No. 12-III-A-07-124.

I. INTRODUCTION

II. BUBBLEDYNAMICS MODEL

III. BUBBLE KINEMATICS IN TOROIDAL COORDINATES

IV. BUBBLEDYNAMICS ON A WIRE

V. AXISYMMETRIC OSCILLATION MODES

VI. DISCUSSION

VII. CONCLUSIONS

### Key Topics

- Bubble dynamics
- 24.0
- Surface tension
- 14.0
- Boundary value problems
- 10.0
- Laplace equations
- 9.0
- Wetting
- 9.0

## Figures

Gas bubble tethered to a wire loop (a) and cross section view (b) illustrating wetting conditions. The center of the bubble is located at the point O. The diameter of the wire ring AB is denoted by L. The center of the wire constraint is located at ϑ c . Location of the bubble surface relative to the wire loop for the non wetting, partial wetting, and full wetting condition are shown by dashed-dotted line, solid line, and dotted line, respectively. The case of the partial wetting corresponds to the wetting angle equal to 90°.

Gas bubble tethered to a wire loop (a) and cross section view (b) illustrating wetting conditions. The center of the bubble is located at the point O. The diameter of the wire ring AB is denoted by L. The center of the wire constraint is located at ϑ c . Location of the bubble surface relative to the wire loop for the non wetting, partial wetting, and full wetting condition are shown by dashed-dotted line, solid line, and dotted line, respectively. The case of the partial wetting corresponds to the wetting angle equal to 90°.

Illustration of toroidal coordinates. The toroidal coordinates of any point are given by the intersection of a sphere, a torus, and an azimuthal plane (a). Surfaces of constant ϑ correspond to spheres of different radii that all pass through the focal ring but are not concentric. The surfaces of constant ξ are non-intersecting tori of different radii. The centers of the constant-ϑ spheres lie along the z-axis, whereas the constant-ξ tori are centered in the xy plane. The coordinate α is the azimuthal angle about the z axis. Circles of constant ξ and ϑ in the (ρ, z) plane of cylindrical polar coordinates (ρ2 = x 2 + y 2) are shown in (b).

Illustration of toroidal coordinates. The toroidal coordinates of any point are given by the intersection of a sphere, a torus, and an azimuthal plane (a). Surfaces of constant ϑ correspond to spheres of different radii that all pass through the focal ring but are not concentric. The surfaces of constant ξ are non-intersecting tori of different radii. The centers of the constant-ϑ spheres lie along the z-axis, whereas the constant-ξ tori are centered in the xy plane. The coordinate α is the azimuthal angle about the z axis. Circles of constant ξ and ϑ in the (ρ, z) plane of cylindrical polar coordinates (ρ2 = x 2 + y 2) are shown in (b).

Streamlines (solid lines) near the wire loop marked by the values of the dimensionless stream function: 1, 0.5, 0.1, –0.1, –0.5, –1. The dashed lines and the dashed-dotted line represent cross section of the bubble surface and the wire.

Streamlines (solid lines) near the wire loop marked by the values of the dimensionless stream function: 1, 0.5, 0.1, –0.1, –0.5, –1. The dashed lines and the dashed-dotted line represent cross section of the bubble surface and the wire.

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