NOTE ON A CLASS OF SOLUTIONS OF THE NAVIER-STOKES EQUATIONS REPRESENTING STEADY ROTATIONALLY-SYMMETRIC FLOW

TLDR

Large‑Reynolds‑number flows are interesting because the two bounding planes are separated by a region of essentially inviscid rigid‑body rotation and translation. The note presents one‑ and two‑parameter families of steady, rotationally‑symmetric viscous‑flow solutions. The solutions are obtained by reducing the Navier‑Stokes equations to ordinary differential equations in a single variable, yielding a one‑parameter family that includes von Kármán’s 1921 solution of rigid‑body rotation at infinity and over a plane through the origin, and a two‑parameter family describing rigid‑body rotation over two planes separated by a finite distance.

Abstract

This note describes one- and two-parameter families of solutions of steady rotationally-symmetric viscous flow. The solutions are such that the Navier-Stokes equations reduce to ordinary differential equations in a single position variable. The one-parameter family represents flow which is rigid-body rotation at infinity and over a plane through the origin; the solution given by von Kármán in 1921 is one member of this family. The two-parameter family represents flow which is rigid body rotation over each of two planes at a finite distance apart. The case of large Reynolds number is particularly interesting, since the two bounding planes are then separated by a region of rigid-body rotation and translation in which viscous effects are negligible.