Non-axisymmetric rotating-disk flows: nonlinear travelling-wave states

Abstract

We consider the classical problem of the laminar flow of an incompressible rotating fluid above a rotating, impermeable, infinite disk. There is a well-known class of solutions to this configuration in the form of an exact axisymmetric solution to the Navier–Stokes equations. However, the radial self-similarity that leads to the ‘rotating- disk equations’ can also be used to obtain solutions that are non-axisymmetric in nature, although (in general) this requires a boundary-layer approximation. In this manner, we locate several new solution branches, which are non-axisymmetric travelling-wave states that satisfy axisymmetric boundary conditions at infinity and at the disk. These states are shown to appear as symmetry-breaking bifurcations of the well-known axisymmetric solution branches of the rotating-disk equations. Numerical results are presented, which suggest that an infinity of such travelling states exist in some parameter regimes. The numerical results are also presented in a manner that allows their application to the analogous flow in a conical geometry.

Two of the many states described are of particular interest. The first is an exact, nonlinear, non-axisymmetric, stationary state for a rotating disk in a counter-rotating fluid; this solution was first presented by Hewitt, Duck & Foster (1999) and here we provide further details. The second state corresponds to a new boundary-layer-type approximation to the Navier–Stokes equations in the form of azimuthally propagating waves in a rotating fluid above a stationary disk. This second state is a new non-axisymmetric alternative to the classical axisymmetric Bödewadt solution.