Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-18T17:16:24.109Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady rotation of a tethered sphere at small, non-zero Reynolds and Taylor numbers: wake interference effects on drag

Published online by Cambridge University Press:  21 April 2006

A. M. J. Davis  and
H. Brenner
A. M. J. Davis
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT Present address: Department of Mathematics, University of Alabama, P.O. Box 1416, University, AL 35486, USA.
H. Brenner
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Matched asymptotic expansion methods are used to solve the title problem. First-order Taylor number corrections to both the Stokes-law drag and Kirchhoff's-law couple on the sphere are obtained for Rossby numbers of order unity. This calculation fills a gap between the Proudman-Pearson (1957) rectilinear trajectory analysis, which includes Reynolds-number effects but does not address Taylor-number effects arising from the curvilinear trajectory, and the Herron, Davis & Bretherton (1975) curvilinear-trajectory analysis, which incorporates Taylor-number effects but ignores those arising from the Reynolds number. At the same Reynolds number, the drag on the sphere is found to be greater or less than the classical Oseen (1927)-Proudman & Pearson (1957) value, depending upon the magnitude of a certain dimensionless length parameter B measuring the tether radius to the sphere radius. This drag difference is attributed, in part, to the fact that the sphere runs into the disturbance created by its own wake.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 168 , July 1986 , pp. 151 - 167
Copyright
© 1986 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.)

References

Batchelor G. K.1967 An Introduction to Fluid Dynamics, pp. 555–567. Cambridge University Press.
Childress S.1964 The slow motion of a sphere in a rotating, viscous fluid. J. Fluid Mech. 20, 305–314.Google Scholar
Cox R. G.1965 The steady motion of a particle of arbitrary shape at small Reynolds numbers. J. Fluid Mech. 23, 625–643.Google Scholar
Dennis S. 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
Dill, L. H. & Brenner H.1983 Taylor dispersion in systems of sedimenting nonspherical Brownian particles III. Time-periodic forces. J. Colloid Interface Sci. 94, 430–450.Google Scholar
Drew D. A.1978 The force on a small sphere in slow viscous flow. J. Fluid Mech. 88, 393–400.Google Scholar
Eiffel G.1907 Récherches expérimentales sur la résistance de l'air exécutées à la tour Eiffel. Paris.
Eiffel G.1910 Récherches de l'air et l'aviation. Paris.
Gradshteyn, I. S. & Ryzhik I. M.1980 Table of Integrals, Series, and Products (corrected and enlarged edition). Academic.
Greenspan H. P.1968 The Theory of Rotating Fluids. Cambridge University Press.
Happel, J. & Brenner H.1983 Low Reynolds Number Hydrodynamics. Nijhoff.
Happel, J. & Pfeffer R.1960 The motion of two spheres following each other in a viscous fluid. Am. Inst. Chem. Engrs J. 6, 129–133.Google Scholar
Herron I. H., Davis, S. H. & Bretherton F. P.1975 On the sedimentation of a sphere in a centrifuge. J. Fluid Mech. 68, 209–234.Google Scholar
Karanfilian, S. K. & Kotas T. J.1981 Motion of a spherical particle in a liquid rotating as a solid body Proc. R. Soc. Lond. A 376, 525–544.Google Scholar
Lamb H.1932 Hydrodynamics. Cambridge University Press.
Lilienthal O.1889 Der Vogelflug alds Grundage der Fliegekunst. (The Flight of Birds as the Foundation of the Art of Flying.) Berlin.
Nadim A., Cox, R. G. & Brenner H.1985 Transport of sedimenting Brownian particles in a rotating Poiseuille flow. Phys. Fluids 28, 3457–3466.Google Scholar
Oseen C. W.1927 Neure Methoden und Ergebnisse in der Hydrodynamik. Leipzig: Akademische.
Prandtl, L. & Tietens O. G.1934 Applied Hydro- and Aeromechanics. pp. 247–254. McGraw-Hill.
Proudman, I. & Pearson J. R. A.1957 Expansions at small Reynolds number for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237–262.Google Scholar
Rouse, H. & Ince S.1957 History of Hydraulics, pp. 116, 122–125. Iowa Institute of Hydraulic Research, State University of Iowa.
Rubinow, S. I. & Keller J. B.1961 Transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech. 11, 447–459.Google Scholar
Singh S. N.1975a The flow past a fixed sphere in a slowly rotating viscous fluid. Z. angew. Math. Phys. 26, 415–426.Google Scholar
Singh S. N.1975b The flow past a spinning sphere in a slowly rotating fluid at small Reynolds number. Intl J. Engng Sci. 13, 1085–1089.Google Scholar
Sneddon I. N.1972 The Use of Integral Transforms. McGraw-Hill.
Von Kármán T.1954 Aerodynamics: Selected Topics in the Light of Their Historical Development, pp. 11–13. Cornell 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