Steady boundary-layer solutions for a swirling stratified fluid in a rotating cone

We consider a set of nonlinear boundary-layer equations that have been deri ved by Duck, Foster & Hewitt (1997a, DFH), for the swirling flow of a linea rly stratified fluid in a conical container. In contrast to the unsteady an alysis of DFH, we restrict attention to steady solutions and extend the pre vious discussion further by allowing the container to both co-rotate and co unter-rotate relative to the contained swirling fluid. The system is govern ed by three parameters, which are essentially nondimensional measures of th e rotation, stratification and a Schmidt number. Some of the properties of this system are related (in some cases rather subtly) to those found in the swirling flow of a homogeneous fluid above an infinite rotating disk; howe ver, the introduction of buoyancy effects with a sloping boundary leads to other (new) behaviours. A general description of the steady solutions to th is system proves to be rather complicated and shows many interesting featur es, including non-uniqueness, singular solutions and bifurcation phenomena. We present a broad description of the steady states with particular emphasi s on boundaries in parameter space beyond which steady states cannot be con tinued. A natural extension of this work (motivated by recent experimental results) is to investigate the possibility of solution branches corresponding to no n-axisymmetric boundary-layer states appearing as bifurcations of the axisy mmetric solutions. In an Appendix we give details of an exact, non-axisymme tric solution to the Navier-Stokes equations (with axisymmetric boundary co nditions) corresponding to the flow of homogeneous fluid above a rotating d isk.