Exact solutions of the Navier-Stokes equations with spiral or elliptical oscillation between two infinite planes

Exact solutions of the Navier-Stokes equations between two infinite planes are considered, where the velocity components parallel to the planes depend linearly on two spatial coordinates, and the third component depends only on the coordinate perpendicular to the planes. A class of unsteady exact solutions is found in this form with spiral or elliptical oscillation as an eigenmode of exponential time dependence, which can be arbitrarily superposed, while the pressure and boundary conditions remain unchanged. As a specific case, the flow between two infinite rotating disks is considered, and the corresponding eigenvalue problems are numerically investigated. Multiple solutions have been taken into account under axisymmetric and non-axisymmetric distributions of pressure. The eigenvalues, which are dependent on the Reynolds number, the rotation ratio, and the pressure parameter ratio, are calculated, and the phase diagrams containing neutral curves are presented. It is shown that some axisymmetric flows between two parallel rotating disks can be associated with an added periodic oscillation at low frequency proportional to the rotation ratio and with arbitrarily large amplitude.