A Zero-Flux Material Cylinder in the Unsteady Kiknadze--Krasnov Vortex
Abstract
In viscous flow, the circulation enclosed by a deterministic material loop generally changes through vorticity diffusion. It is shown that the source-bearing unsteady Kiknadze--Krasnov vortex possesses a distinguished cylindrical material surface on which this viscous circulation transfer vanishes identically. For a single distributed mode, the scaled squared radius x=β(t)r2 satisfies (D x)/(D t) = 4νβ(t)[(s−1)−x], where β(t) is the inverse squared radial scale, ν is the kinematic viscosity, and s is the incomplete-gamma shape parameter. For s>1, the cylinder x=s−1 is transported exactly by the radial flow and preserves the inside/outside ordering of fluid trajectories. The same factor (s−1)−x governs the radial vorticity gradient and the material circulation-transfer rate. Consequently, the magnitude of the distributed axial vorticity |ωz| attains its annular maximum on this material cylinder, ∂rωz=0 there, and the enclosed circulation remains exactly $ΓP(s,s-1)$ for every admissible prescribed strain history. Neighbouring material cylinders continue to exchange circulation viscously, while the distinguished moving cylinder carries a fixed fraction of the distributed circulation throughout the admissible evolution. In the steady source--strain limit, the surface reduces to the radial stagnation cylinder. The complete velocity and pressure fields, exact radial-scale evolution, admissible parameter ranges, and qualifications for multiple radial scales are also derived. The resulting solution-specific material-circulation invariant provides an exact benchmark for numerical material-surface tracking, vortex-core evolution, and viscous circulation transport, with the principal analytical results additionally supported by machine-checked formal verification.
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BibTeX
@misc{saqr2026zeroflux,
title = {{A Zero-Flux Material Cylinder in the Unsteady Kiknadze--Krasnov Vortex}},
author = {Khalid M. Saqr},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00082},
eprint = {2610.00082},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00082},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Zero-Flux Material Cylinder in the Unsteady Kiknadze--Krasnov Vortex
AU - Khalid M. Saqr
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.00082
UR - https://arxiv.org/abs/2610.00082
AB - In viscous flow, the circulation enclosed by a deterministic material loop generally changes through vorticity diffusion. It is shown that the source-bearing unsteady Kiknadze--Krasnov vortex possesses a distinguished cylindrical material surface on which this viscous circulation transfer vanishes identically. For a single distributed mode, the scaled squared radius x=β(t)r^2 satisfies (D x)/(D t) = 4νβ(t)[(s−1)−x], where β(t) is the inverse squared radial scale, ν is the kinematic viscosity, and s is the incomplete-gamma shape parameter. For s>1, the cylinder x=s−1 is transported exactly by the radial flow and preserves the inside/outside ordering of fluid trajectories. The same factor (s−1)−x governs the radial vorticity gradient and the material circulation-transfer rate. Consequently, the magnitude of the distributed axial vorticity |ω_z| attains its annular maximum on this material cylinder, ∂_rω_z=0 there, and the enclosed circulation remains exactly $ΓP(s,s-1)$ for every admissible prescribed strain history. Neighbouring material cylinders continue to exchange circulation viscously, while the distinguished moving cylinder carries a fixed fraction of the distributed circulation throughout the admissible evolution. In the steady source--strain limit, the surface reduces to the radial stagnation cylinder. The complete velocity and pressure fields, exact radial-scale evolution, admissible parameter ranges, and qualifications for multiple radial scales are also derived. The resulting solution-specific material-circulation invariant provides an exact benchmark for numerical material-surface tracking, vortex-core evolution, and viscous circulation transport, with the principal analytical results additionally supported by machine-checked formal verification.
ER -