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The motion of a rising disk in a rotating axially bounded fluid for large Taylor number

Published online by Cambridge University Press:  26 April 2006

Marius Ungarish  and
Dmitry Vedensky
Marius Ungarish
Affiliation:
Department of Computer Science, Technion, Haifa 32000, Israel
Dmitry Vedensky
Affiliation:
Department of Computer Science, Technion, Haifa 32000, Israel
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Abstract

The motion of a disk rising steadily along the axis in a rotating fluid between two infinite plates is considered. In the limit of zero Rossby number and with the disk in the middle position, the boundary value problem based on the linear, viscous equations of motion is reduced to a system of dual-integral equations which renders ‘exact’ solutions for arbitrary values of the Taylor number, Ta, and disk-to-wall distance, H (scaled by the radius of the disk). The investigation is focused on the drag and on the flow field when Ta is large (but finite) for various H. Comparisons with previous asymptotic results for ‘short’ and ‘long’ containers, and with the preceding unbounded-configuration ‘exact’ solution, provide both confirmation and novel insights.

In particular, it is shown that the ‘free’ Taylor column on the particle appears for H > 0.08 Ta and attains its fully developed features when H > 0.25 Ta (approximately). The present drag calculations improve the compatibility of the linear theory with Maxworthy's (1968) experiments in short containers, but for the long container the claimed discrepancy with experiments remains unexplained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 291 , 25 May 1995 , pp. 1 - 32
Copyright
© 1995 Cambridge University Press

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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
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. 1968 The observed motion of a sphere through a short, rotating cylinder of fluid. J. Fluid Mech. 31, 643–655.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
Moore, D. W. & Saffman, P. G. 1968 The rise of a body through a rotating fluid in a container of finite length. J. Fluid Mech. 31, 635–642.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
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. J. Fluid Mech. 275, 225–256.Google Scholar
Tranter, C. J. 1951 Integral Transforms in Mathematical Physics Methuen.
Ungarish, M. 1993 Hydrodynamics of Suspensions. Springer, Berlin.
Ungarish, M. 1995 Some shear-layer and inertial modifications of the geostrophic drag on a slowly rising particle or drop in a rotating fluid, in preparation.
Vedensky, D. & Ungarish, M. 1994 The motion generated by a slowly rising disk in an unbounded rotating fluid for arbitrary Taylor number. J. Fluid Mech. 262 1–26 (referred to herein as VU).Google Scholar
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