Mathematical Model Describes Collective Dynamics of Vortex Clusters in Confined Fluid Domains
Researchers developed an analytical framework using the Schottky-Klein prime function to describe how clusters of rotating vortices behave in confined, periodic fluid domains. The model reduces complex vortex interactions to a single quadrupole moment that governs both the cluster's rotation rate and its size oscillations. The findings, confirmed by numerical simulations, provide a simplified mathematical description applicable to various compact fluid systems.
A new analytical model explains the collective behavior of co-rotating vortex clusters in doubly periodic (toroidal) fluid domains by decomposing their dynamics into universal planar interactions and geometry-dependent corrections. The framework, based on exact mathematical representations using the Schottky-Klein prime function, reduces the two-vortex problem to a single complex degree of freedom and extends to larger clusters through a small-cluster expansion. The model reveals that the essential dynamics can be encoded in a single complex quadrupole moment: its real component determines corrections to the orbital rotation frequency, while its imaginary component controls the slow breathing (size oscillation) of the cluster. Direct numerical simulations quantitatively validate these theoretical predictions, establishing a reduced mathematical description that captures the essential physics of vortex clusters on flat tori and other compact fluid domains.
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The study does not discuss potential physical applications or experimental systems where this theoretical framework might be tested or applied.
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- arXiv physicsCenter
Collective Dynamics of Vortex Clusters in Compact Fluid Domains: From Pair Interactions to a Quadrupole Description
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