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 $x=\beta (t)r^2$ satisfies \[ \frac{\mathrm D x}{\mathrm D t} = 4νβ(t)\big[(s-1)-x\big], \] where $\beta (t)$ is the inverse squared radial scale, $\nu$ 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 $|{\omega }_z|$ attains its annular maximum on this material cylinder, ${\partial }_r{\omega }_z=0$ there, and the enclosed circulation remains exactly $\Gamma P(s,s-1)$ 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.

arXiv comment41 pages, 8 figures, 3 tables