A vortex in an infinite viscous fluid

TLDR: In this paper, a solution for a viscous vortex in an infinite liquid is given, where similarity arguments lead to a reduction of the equations of motion to a set of ordinary differential equations.

Abstract: A solution is given for a viscous vortex in an infinite liquid. Similarity arguments lead to a reduction of the equations of motion to a set of ordinary differential equations. These are integrated numerically. A uniform feature is the constant circulation K outside the vortex core, which is also a viscous boundary layer. The circulation decreases monotonically towards the axis. The axial velocity profiles and the radial velocity profiles have several characteristic shapes, depending on the value of the non-dimensional momentum transfer M. The solution has a singular point on the axis of the vortex. The radius of the core increases linearly with distance along the axis from the singularity, and, at a given distance, is proportional to the coefficient of viscosity and inversely proportional to K.Finally, a discussion is given to indicate that intense vortices above a plate, like the confined experimental vortex, or above the ground, like the atmospheric tornado and dust whirl, will not resemble the theoretical vortex except, possibly, far above the plate.