Geometric Vortex Collapse Model via Alternating Series Converging to 2/3 and Its Application to the Navier-Stokes Problem

Abstract:This work presents a geometric and visual model describing the limit behavior of a fluid under extreme vorticity, addressing the singularity and energy collapse issues in the Navier-Stokes equations. Instead of relying solely on abstract analytical treatments, the model uses a spatial discretization based on concatenated circular arcs that decrease in scale and invert flow direction, formalizing the exact mechanism by which energy accumulates and dampens. The resulting alternating geometric series converges exactly to a factor of 2/3, acting as a structural limit that prevents infinite energy divergence. Priority and Authorship Note:I, the undersigned Matteo Milo D. Rienzo, declare that I conceived and first presented this geometric and conceptual intuition, publicly documenting it on November 19, 2023, on my YouTube channel (Matteo Milo D. Rienzo) (with initial previews shared on social media platforms in the preceding months), originally using it as a geometric demonstration and interpretive key for the Navier-Stokes problem. Acknowledgments:I thank my family, especially my life, Milo.