Self-similar swirl between contracting porous walls: the GD1998 exact Navier-Stokes solution revisited in the similarity variables of the OpenAI 2026 forced blow-up construction
Abstract
The finite-time blow-up construction for the forced Navier-Stokes equations released by OpenAI in Sep. 2026 (OpenAI 2026) has as its object an axisymmetric swirl core in anisotropic similarity variables. Its geometry is that of an exact steady swirl between rotating porous cylinders found by Gumerov and Duraiswami in 1998 (GD1998). We find that the GD1998 boundary value problem does not recast into the similarity variables and that its generalization is a two-dimensional profile problem between porous walls held at fixed similarity radii, second order in the radial variable X and first order in the time-like axial variable η. We solve it by tensor Chebyshev collocation with a pinned pressure gauge, complex-step Newton and arclength continuation, verify it three ways, and sweep the radial Reynolds number, the wall swirl and a symmetry-breaking datum. Below inflow 9 the solution is one smooth branch; above it the symmetric branch folds at a critical swirl F*=0.41V02. The Dirichlet axis problem has no resolution-stable solution; the core must be a Cauchy problem from the axis. Its moment identities cannot be met by a core symmetric about the dividing plane: the core is an axial through-flow, as its authors chose it; a non-symmetric core meets them to 0.2%. The cone condition is Rayleigh's criterion with axial shear and requires radii of order 1020. Dynamic rescaling shows the profile to be an attractor at weak inflow and toward the fold, a saddle-node; at moderate inflow and strong swirl the spectrum does not converge in the axial resolution and is left open. A real fluid cavitates (water) or shocks (air) first; the GD1998 chamber is proposed as the experiment. Nothing found suggests the mechanism is reachable in a flow one computes or builds; the forced theorem, and this study, leave the unforced equations of practice as they were. Code, tests and log accompany the paper.
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BibTeX
@misc{duraiswami2026selfsimilar,
title = {{Self-similar swirl between contracting porous walls: the GD1998 exact Navier-Stokes solution revisited in the similarity variables of the OpenAI 2026 forced blow-up construction}},
author = {Ramani Duraiswami},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17642},
eprint = {2609.17642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17642},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Self-similar swirl between contracting porous walls: the GD1998 exact Navier-Stokes solution revisited in the similarity variables of the OpenAI 2026 forced blow-up construction
AU - Ramani Duraiswami
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17642
UR - https://arxiv.org/abs/2609.17642
AB - The finite-time blow-up construction for the forced Navier-Stokes equations released by OpenAI in Sep. 2026 (OpenAI 2026) has as its object an axisymmetric swirl core in anisotropic similarity variables. Its geometry is that of an exact steady swirl between rotating porous cylinders found by Gumerov and Duraiswami in 1998 (GD1998). We find that the GD1998 boundary value problem does not recast into the similarity variables and that its generalization is a two-dimensional profile problem between porous walls held at fixed similarity radii, second order in the radial variable X and first order in the time-like axial variable η. We solve it by tensor Chebyshev collocation with a pinned pressure gauge, complex-step Newton and arclength continuation, verify it three ways, and sweep the radial Reynolds number, the wall swirl and a symmetry-breaking datum. Below inflow 9 the solution is one smooth branch; above it the symmetric branch folds at a critical swirl F^*=0.41V_0^2. The Dirichlet axis problem has no resolution-stable solution; the core must be a Cauchy problem from the axis. Its moment identities cannot be met by a core symmetric about the dividing plane: the core is an axial through-flow, as its authors chose it; a non-symmetric core meets them to 0.2%. The cone condition is Rayleigh's criterion with axial shear and requires radii of order 10^20. Dynamic rescaling shows the profile to be an attractor at weak inflow and toward the fold, a saddle-node; at moderate inflow and strong swirl the spectrum does not converge in the axial resolution and is left open. A real fluid cavitates (water) or shocks (air) first; the GD1998 chamber is proposed as the experiment. Nothing found suggests the mechanism is reachable in a flow one computes or builds; the forced theorem, and this study, leave the unforced equations of practice as they were. Code, tests and log accompany the paper.
ER -