Axial Symmetric Navier Stokes Equations and the Beltrami /anti Beltrami spectrum in view of Physics Informed Neural Networks

TL;DR

<3-5 sentence high-level summary> The paper develops an axial-symmetric Navier–Stokes framework in a tube topology by constructing a complete L2_tube basis of harmonic 1-forms, split per energy shell into Beltrami, anti-Beltrami, and closed components, enabling an algebraic reduction of the steady NS equation to a quadratic system for expansion coefficients. It formulates the NS dynamics in geometric language using lattices, dual lattices, foliations, and contact geometry, linking Beltrami fields to Reeb flows and topological constraints. A hexagonal/diamond algebra is introduced to describe nonlinear mode interactions, and a plan is laid out to determine coefficients via Physics-Informed Neural Network optimization, offering a principled route toward Axial Beltrami spectra in practical simulations. The work provides explicit zero-modes, boundary-condition-consistent bases, and a structured pathway for future PN-network driven discovery of axisymmetric NS solutions with constant Bernoulli function, with potential implications for controlled mixing and flow design in tubular geometries.

Abstract

In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply revised and developed in 2022 with M. Trigiante. Instead of a compact $3$-torus $T^3=\mathbb{R}^3/Λ$ where $Λ$ is a crystallographic lattice, as done in previous work, here I considered flows confined in a cylinder with identified opposite bases. In this topology I considered axial symmetric flows and found a complete basis of axial symmetric harmonic $1$-forms that, for each energy level, decomposes into six components: two Beltrami, two anti-Beltrami and two closed forms. These objects, that are written in terms of trigonometric and Bessel functions, constitute a function basis for an $L^2$ space of axial symmetric flows. I have presented a general scheme for the search of axial symmetric solutions of Navier Stokes equation by reducing the latter to an hierachy of quadratic relations on the development coefficients of the flow in the above described functional basis. It is proposed that the coefficients can be determined by means of a Physics Informed like Neural Network optimization recursive algorithm. Indeed the present paper provides the theoretical foundations for such a algorithmic construction that is planned for a future publication.

Figures (17)

• Figure 1:
• Figure 2: Schematic vision of the standard contact structure in $\mathbb{R}^3$.
• Figure 3: Schematic vision of a cylinder hosting an axial symmetric flow. The cylinder is infinite in the $z$-direction but we are interested in a flow that is confined to the finite region of the cylinder comprised between the two disks, the red and the green one, that are separated by a horizontal distance $\ell$, evidenced by the black line. The most convenient way to describe such a flow is to impose periodic boundary conditions on the flow in the $z$-direction
• Figure 4: Behavior of the Bessel functions appearing in the expression of the harmonic $1$-form displayed in eq.(\ref{['armoniosa']}). As we see the second solution $Y_{1,0}\left(2 k \pi r \sqrt{\lambda ^2-1}\right)$ has to be excluded since it has a singularity for $r=0$ namely at the very center of the tube. Instead the first solution $J_{1,0}\left(2 k \pi r \sqrt{\lambda ^2-1}\right)$ has the perfect behavior to satisfy the desired boundary conditions.
• Figure 5: In this figure we show one streamline starting at $r=1/2,\phi=0,z=-1$ of the Beltrami flow ${\Omega}B_0[5]$. As one sees the radius never changes during the time evolution, while the longitudinal coordinate $z$ advances linearly in time as the angle $\phi$. The result is the helix shown in the picture.

Theorems & Definitions (3)

• Theorem 2.1
• Theorem 2.2
• Definition 6.1