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

Hewitt, R.E. and Duck, P.W. (2000) Non-axisymmetric rotating-disk flows: nonlinear travelling-wave states. Journal of Fluid Mechanics, 413. pp. 287-316. ISSN 1469-7645

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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 Bodewadt solution.

Item Type:	Article
Additional Information:	© 2000 Cambridge University Press
Uncontrolled Keywords:	Exact Navier-Stokes solutions self-similar non-axisymmetric rotating disk nonlinear
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 76 Fluid mechanics PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 40 ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS > 47 Fluid dynamics
Depositing User:	Dr Richard E. Hewitt
Date Deposited:	14 Nov 2013
Last Modified:	20 Oct 2017 14:13
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2051

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