A mathematical analogy between certain hollow vortex and bubble flows - existence, uniqueness, and identical boundary shapes
Abstract
We study two physically distinct steady free-boundary problems in a two-dimensional inviscid fluid: a hollow vortex in a constant-vorticity flow subject to simple shear, third order and linear strain at infinity, without surface tension, and an irrotational flow past a constant-pressure bubble subject to surface tension, linear strain, and circulation at infinity. Building on the mathematical vortex-bubble correspondence recently developed by Crowdy and Tanveer, we show that both problems are linked by a common mathematical formulation. For the vortex problem, we extend the sequence-space fixed-point approach of Crowdy and Tanveer to obtain new local existence, uniqueness, and analyticity results. For the bubble problem, we develop a separate existence theory, reducing it to a nonlinear singular integral operator fixed-point problem and deriving a scalar solvability condition that defines a realanalytic surface in the physical parameter space. On this surface we obtain locally unique bubble solutions with real-analytic boundaries and holomorphically extendible velocity fields. Finally, we prove that the scalar solvability condition is equivalent to a functional conformal mapping identity. The scalar condition between the parameters is enough for the vortex and bubble problems to have the same free-boundary shape, despite their different exterior flows.
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BibTeX
@misc{kunz2026mathematical,
title = {{A mathematical analogy between certain hollow vortex and bubble flows - existence, uniqueness, and identical boundary shapes}},
author = {Valentin D. Kunz and Saleh Tanveer and Darren G. Crowdy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07709},
eprint = {2610.07709},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07709},
}
RIS
TY - UNPB
TI - A mathematical analogy between certain hollow vortex and bubble flows - existence, uniqueness, and identical boundary shapes
AU - Valentin D. Kunz
AU - Saleh Tanveer
AU - Darren G. Crowdy
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.07709
UR - https://arxiv.org/abs/2610.07709
AB - We study two physically distinct steady free-boundary problems in a two-dimensional inviscid fluid: a hollow vortex in a constant-vorticity flow subject to simple shear, third order and linear strain at infinity, without surface tension, and an irrotational flow past a constant-pressure bubble subject to surface tension, linear strain, and circulation at infinity. Building on the mathematical vortex-bubble correspondence recently developed by Crowdy and Tanveer, we show that both problems are linked by a common mathematical formulation. For the vortex problem, we extend the sequence-space fixed-point approach of Crowdy and Tanveer to obtain new local existence, uniqueness, and analyticity results. For the bubble problem, we develop a separate existence theory, reducing it to a nonlinear singular integral operator fixed-point problem and deriving a scalar solvability condition that defines a realanalytic surface in the physical parameter space. On this surface we obtain locally unique bubble solutions with real-analytic boundaries and holomorphically extendible velocity fields. Finally, we prove that the scalar solvability condition is equivalent to a functional conformal mapping identity. The scalar condition between the parameters is enough for the vortex and bubble problems to have the same free-boundary shape, despite their different exterior flows.
ER -