Towards a Unified Swirl Vortex Model

A review of the literature shows the existence of dozens of single-cell, axisymmetric, Newtonian-fluid vortices that, in one form or another, are solutions to the Navier-Stokes Equation. However, little research has been conducted in the literature to investigate common traits that are shared by these vortices. This research lays out a foundation of common mathematical traits, with a focus on the azimuthal velocity v. Notwithstanding their diverse mathematical and physical origin, it can be shown that these vortices are diverse manifestations of a single vortex family—they are united by a set of at least seven common mathematical traits. As a first common trait, we show that there are only four possible categories of single-cell, axisymmetric, Newtonian-fluid vortices. Next, by normalizing and overlaying the vortices into a single figure, three additional traits are noted, namely that v(r=0) = 0; they have similar azimuthal velocity profiles; and the azimuthal velocity is asymmetric about r. Thereafter, a fifth trait is noted by doing a series expansion for the azimuthal velocity of each vortex (as found in the literature), and expressing it as an alternating series that expands geometrically with odd exponents (truncated Laurent series with real coefficients). Finally, two more traits are noted by taking limit bounds for each series, thus showing that one bound is the Rankine (solid body) vortex, and the other is a Lamb-Oseen sine-like bound. In brief, we note that the vortices have seven traits in common, and hereby propose that these vortices are essentially diverse manifestations of a single vortex family.