A Fluid-Dynamic Hypergeometric Series: Multiple Convergence Regimes and the Pavesi Flow

We introduce a new class of series, called Pavesi fluid series,whose terms are velocity vectors of a particle immersed in a viscousfluid. The series is constructed from a hypergeometric kernel anddepends on a physical parameter \(α\) (the reciprocal Reynoldsnumber). For different ranges of \(α\), the sequence of partialsums converges, in appropriate topologies, to distinct limits:a single point (laminar regime), a compact set (turbulent regime),and a fractal measure (transition regime). This behaviour provides aunified mathematical framework for the onset of turbulence and linksthe Ramanujan–Sato theory of \(1/π\) series with the Navier–Stokesequations. {equation}{eq:main_abstract}{\,S_N(α) \;=\;∑ₙ₌₀N\,{({13})_n\,({23})_n\,({12})_n}{(n!)^3}\,(Aα\,n + Bα)\,q(α)^n\,e_n\,}.{equation}Here \(e_n\) are unit vectors in a Hilbert space, and\(Aα, Bα, q(α)\) are algebraic functions of thefluid viscosity.