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Transverse motion of a disk through a rotating viscous fluid

Published online by Cambridge University Press:  26 April 2006

John P. Tanzosh
Affiliation:
Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA
H. A. Stone
Affiliation:
Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA

Abstract

A thin rigid disk translates edgewise perpendicular to the rotation axis of an unbounded fluid undergoing solid-body rotation with angular velocity Ω. The disk face, with radius a, is perpendicular to the rotation axis. For arbitrary values of the Taylor number, [Tscr ] = Ωa2/ν, and in the limit of zero Reynolds number [Rscr ]e, the linearized viscous equations reduce to a complex-valued set of dual integral equations. The solution of these dual equations yields an exact representation for the velocity and pressure fields generated by the translating disk.

For large rotation rates [Tscr ] [Gt ] 1, the O(1) disturbance velocity field is confined to a thin O([Tscr ]−1/2) boundary layer adjacent to the disk. Within this boundary layer, the flow field near the disk centre undergoes an Ekman spiral similar to that created by a nearly geostrophic flow adjacent to an infinite rigid plate. Additionally, flow within the boundary layer drives a weak O([Tscr ]−1/2) secondary flow which extends parallel to the rotation axis and into the far field. This flow consists of two counter-rotating columnar eddies, centred over the edge of the disk, which create a net in-plane flow at an angle of 45° to the translation direction of the disk. Fluid is transported axially toward/away from the disk within the core of these eddies. The hydrodynamic force (drag and lift) varies as O([Tscr ]1/2) for [Tscr ] [Gt ] 1; this scaling is consistent with the viscous stresses created in the Ekman boundary layer. Additionally, an approximate expression, suitable for all Taylor numbers, is given for the hydrodynamic force on a disk translating broadside along the rotation axis and edgewise transverse to the rotation axis.

Type
Research Article
Copyright
© 1995 Cambridge University Press

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