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.
Effects of mass transfer on a resonance instability in the laminar boundary layer flow over a rotating-disk