Nonuniqueness of second-order fluids under steady radial flow in annuli

Abstract

We consider the axisymmetric plane radial and angular steady flow of incompressible second-order fluids within the region bounded by two fixed porous coaxial cylinders. The radial component of the velocity field has the same structure as that in the Navier-Stokes theory, and the angular velocity profile is shown to satisfy a second order ordinary differential equation. While the solution of this differential equation subject to homogeneous boundary conditions is zero for the Navier-Stokes theory, we discuss conditions under which it has eigenvalues and eigenfunctions in the present theory. Thus, with the aid of three theorems concerning the character of these eigenvalues and eigenfunctions we conclude, roughly, that for annuli with ‘large enough’ ratio of outer to inner radii and for ‘sufficiently large’ radial volumetric flow rates, the velocity field in the annulus can also possess a non-trivial steady component in the azimuthal direction without affecting a change in the boundary conditions. Elementary examples of the phenomenon are constructed for both radial flow directed toward and away from the axis of an annulus.