Hostname: page-component-758b78586c-t6tmf Total loading time: 0 Render date: 2023-11-28T17:13:34.980Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false

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

Published online by Cambridge University Press:  25 June 2000

Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK
Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK


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.

Research Article
© 2000 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)