The flow due to a rotating disc

TLDR

Kármán studied the steady motion of an incompressible viscous fluid induced by an infinite rotating plane lamina rotating with constant angular velocity ω about the axis r = 0 in cylindrical coordinates. The authors analyze the fluid on the side where z > 0, assuming the fluid is infinite and bounded only by the rotating plane, and derive velocity components u, v, w and pressure p that satisfy the equations of motion and continuity. 1.

Abstract

1. The steady motion of an incompressible viscous fluid, due to an infinite rotating plane lamina, has been considered by Kármán. If r , θ, z are cylindrical polar coordinates, the plane lamina is taken to be z = 0; it is rotating with constant angular velocity ω about the axis r = 0. We consider the motion of the fluid on the side of the plane for which z is positive; the fluid is infinite in extent and z = 0 is the only boundary. If u, v, w are the components of the velocity of the fluid in the directions of r , θ and z increasing, respectively, and p is the pressure, then Kármán shows that the equations of motion and continuity are satisfied by taking