Hostname: page-component-cb9f654ff-r5d9c Total loading time: 0 Render date: 2025-08-31T05:46:19.831Z Has data issue: false hasContentIssue false
  • English
  • Français

The motion generated by a slowly rising disk in an unbounded rotating fluid for arbitrary Taylor number

Published online by Cambridge University Press:  26 April 2006

D. Vedensky  and
M. Ungarish
M. Ungarish
Affiliation:
Department of Computer Science, Technion, Haifa 32000, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

The motion of a disk rising steadily parallel to the axis of rotation in a uniformly rotating unbounded liquid is considered. In the limit of zero Rossby number the linear viscous equations of motion are reduced to a system of dual integral equations which renders an ‘exact’ solution for arbitrary values of the Taylor number, Ta. The investigation is focused on the drag and the flow field. In the limits of small and large Ta the asymptotic results of the present formulation agree with – and extend – previous investigations by different approaches.

A particular novel feature, for large Ta, is the contribution of the Ekman-layer flux to the outer motion. New insight into the structure of the Taylor column is gained; in particular, it is shown that the main part of the column is a ‘bubble’ of recirculating fluid, detached from the body and not communicating with the Ekman layer. However, it turns out that the essential discrepancy in drag between experiments (Maxworthy 1970) and previous theories cannot be attributed to the Ekman-layer suction effect.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 262 , 3 October 1994 , pp. 1 - 26
Copyright
© 1994 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.)

Article purchase

Temporarily unavailable

References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Dover.
Barnard, B. J. S. & Pritchard, W. G. 1975 The motion generated by a body moving through a stratified fluid at large Richardson numbers. J. Fluid Mech. 71, 43–64.Google Scholar
Childress, S. 1964 The slow motion of a sphere in a rotating, viscous fluid. J. Fluid Mech. 20, 305–314.Google Scholar
Dennis, C. R., Ingham, D. B. & Singh, S. N. 1982 The slow translation of a sphere in a rotating viscous fluid. J. Fluid Mech. 117, 251–267.Google Scholar
Goldshtik, M. A. 1990 Viscous-flow paradoxes. Ann. Rev. Fluid Mech. 22, 441–472.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Hocking, L. M., Moore, D. W. & Walton, I. C. 1979 The drag on a sphere moving axially in a long rotating container. J. Fluid Mech. 90, 781–793.Google Scholar
Kantorovich, L. V. & Krylov, V. I. 1964 Approximate Methods of Higher Analysis. Interscience.
Maxworthy, T. 1965 An experimental determination of the slow motion of a sphere in a rotating, viscous fluid. J. Fluid Mech. 23, 373–384.Google Scholar
Maxworthy, T. 1970 The flow created by a sphere moving along the axis of a rotating, slightly-viscous fluid. J. Fluid Mech. 40, 453–479.Google Scholar
Miles, J. W. 1972 Axisymmetric rotating flow past a circular disk. J. Fluid Mech. 53, 689–700.Google Scholar
Moore, D. W. & Saffman, P. G. 1969 The structure of free vertical shear layers in a rotating fluid and the motion produced by a slowly rising body. Phil. Trans. R. Soc. Lond. A 264, 597–634.Google Scholar
Morrison, J. W. & Morgan, G. W. 1956 The slow motion of a disk along the axis of viscous, rotating liquid. Rep. 56207/ 8. Div. of Appl. Maths, Brown University.
Orloff, K. & Bossel, H. 1971 Laser-Doppler velocity measurements of swirling flows with upstream influence. Bull. Am. Phys. Soc. 16(2), 1331 (abstract only).Google Scholar
Ray, M. 1936 Application of Bessel functions in the solution of problems of motion of a circular disk in viscous liquid. Phil. Mag. (7) 21, 546–564.Google Scholar
Stewartson, K. 1952 On the slow motion of an ellipsoid in a rotating fluid. Q. J. Mech. Appl. Maths 6, 141–162.Google Scholar
Tanzosh, J. & Stone, H. A. 1994 Motion of a rigid particle in a rotating viscous flow: An integral equation approach. Submitted.
Taylor, G. I. 1922 The motion of a sphere in a rotating liquid. Proc. R. Soc. Lond. A 102, 180–189.Google Scholar
Tranter, C. J. 1951 Integral Transforms in Mathematical Physics. Methuen.
Ungarish, M. 1993 Hydrodynamics of Suspensions: Fundamentals of Centrifugal and Gravity Separation. Springer.
Watson, G. N. 1952 A Treatise on the Theory of Bessel Functions. Cambridge University Press.
Weisenborn, A. J. 1985 Drag on a sphere moving slowly in a rotating viscous fluid. J. Fluid Mech. 153, 215–227.Google Scholar