Multiple solutions for flow between coaxial disks

Abstract

The problem of obtaining a numerical solution for the steady flow between two coaxial infinite disks, one fixed and porous, the other rotating, is reduced by von Kámán's hypothesis to solution of a system of nonlinear equations. A Newton-type iteration results in several solutions to these equations, as a number of authors have already indicated. Nevertheless, an interval in which only one solution is found exists for small values of the Reynolds number based on the angular velocity of the rotating disk, the distance between the disks and the kinematic viscosity of the fluid. At large values of this Reynolds number, two solutions appear and have been the subject of intense controversy.In this paper, both physical and numerical arguments are presented which support a Batchelor-type solution for the flow between infinite disks, in which part of the fluid rotates as a solid body. The other solution, following Stewartson, assumes that the velocity of the fluid outside the boundary layers is entirely axial. This only seems to be verified experimentally when the distance between the disks is large compared with the (finite) radius of the disks.