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Related Experiment Video

Updated: May 2, 2026

Induction of Microstreaming by Nonspherical Bubble Oscillations in an Acoustic Levitation System

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Multiple steadily translating bubbles in a Hele-Shaw channel.

Christopher C Green¹, Giovani L Vasconcelos²

¹Department of Mathematics , Imperial College London , 180 Queen's Gate, London SW7 2AZ, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences

|March 11, 2014

Summary

Analytical solutions for multiple bubbles moving in Hele-Shaw channels were developed using conformal mapping. This generalized method accurately models bubble dynamics without assuming symmetrical arrangements.

Keywords:

Hele-Shaw flows Schottky–Klein prime function Schwarz–Christoffel formula bubble dynamics free boundary problems

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Area of Science:

Fluid dynamics
Mathematical physics
Complex analysis

Background:

Hele-Shaw flows are crucial for understanding fluid dynamics in confined geometries.
Modeling multiparticle systems, like bubble assemblies, presents significant analytical challenges.

Purpose of the Study:

To develop general analytical solutions for the steady motion of any finite number of bubbles in a Hele-Shaw channel.
To provide a framework for analyzing complex bubble interactions and configurations.

Main Methods:

Constructing conformal mappings from a multiply connected circular domain to the exterior of the bubbles.
Utilizing the generalized Schwarz-Christoffel formula and the Schottky-Klein prime function.
Decomposing the mapping into functions representing laboratory and co-moving frames.

Main Results:

Analytical solutions were derived for arbitrary configurations of bubbles.
The method accommodates any finite number of bubbles without symmetry constraints.
Demonstrated applicability through examples of various bubble arrangements.

Conclusions:

The developed conformal mapping technique offers a powerful and general approach to solving multiparticle Hele-Shaw flow problems.
This method advances the understanding of bubble dynamics in confined flows.
The solutions are applicable to diverse bubble configurations, enhancing predictive capabilities.