Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-22T12:27:27.113Z Has data issue: false hasContentIssue false
  • English
  • Français

Transverse motion of a disk through a rotating viscous fluid

Published online by Cambridge University Press:  26 April 2006

John P. Tanzosh  and
H. A. Stone
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
Get access Rights & Permissions [Opens in a new window]

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
Information
Journal of Fluid Mechanics , Volume 301 , 25 October 1995 , pp. 295 - 324
Copyright
© 1995 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

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bush, J. W. M., Stone, H. A. & Tanzosh, J. 1995 Particle motion in rotating viscous fluids: Historical survey and recent developments. To appear in Current Topics in the Physics of Fluids. Research Trends, Trivandrum, India.
Childress, S. 1964 The slow motion of a sphere in a rotating, viscous fluid. J. Fluid Mech. 20, 305–314.Google Scholar
Davies, P. A. 1972 Experiments on Taylor columns in rotating stratified fluids. J. Fluid Mech. 54, 691–717.Google Scholar
Davis, A. M. J. 1991 Slow viscous flow due to motion of an annular disk; pressure-driven extrusion through an annular hole in a wall. J. Fluid Mech. 231, 51–71.Google Scholar
Davis, A. M. J. 1992 Drag modification for a sphere in a rotational motion at small, non-zero Reynolds and Taylor numbers; wake interference and possible Coriolis effects. J. Fluid Mech. 237, 13–22.Google Scholar
Davis, A. M. J. 1993 Some asymmetric Stokes flows that are structurally similar. Phys. Fluids A 5, 2086–2094.Google Scholar
Davis, A. M. J. & Brenner, H. 1986 Steady rotation of a tethered sphere at small, non-zero Reynolds and Taylor numbers: wake interference effects on drag. J. Fluid Mech. 168, 151–167.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1965 Table of Integrals, Series and Products. Academic Press.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics. Noordhoff, Groningen.
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
Hide, R. 1966 On the dynamics of rotating fluids and related topics in geophysical fluid dynamics. Bull. Am. Met. Soc. 47, 873–885.Google Scholar
Hide, R. & Ibbetson, A. 1966 An experimental study of Taylor columns. Icarus 5, 279–290.Google Scholar
Hide, R. & Ibbetson, A. 1968 On slow transverse flow past obstacles in a rapidly rotating fluid. J. Fluid Mech. 32, 251–272.Google Scholar
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
Hughes, B. D., Pailthorpe, B. A. & White, L. R. 1981 The translational and rotational drag on a cylinder moving in a membrane J. Fluid Mech. 110, 349–372.Google Scholar
IMSL Math/Library V2.0 1991 User's Manual. IMSL, Houston.
Ingersoll, A. P. 1969 Inertial Taylor columns and Jupitor's Great Red Spot. J. Atmos. Sci. 26, 744–752.Google Scholar
Jacobs, S. J. 1964 The Taylor column problem. J. Fluid Mech. 20, 581–591.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
Lighthill, M. J. 1966 Dynamics of rotating fluids: a survey. J. Fluid Mech. 28, 411–431.Google Scholar
Lucas, S. K. 1995 Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Maths to appear.
Magnus, W., Oberhettinger, F. & Soni, R. P. 1966 Formulas and Theorems for the Special Functions of Mathematical Physics. Springer.
Mason, P. J. 1975 Forces on bodies moving transversely through a rotating fluid. J. Fluid Mech. 71, 577–599.Google Scholar
Maxworthy, T. 1969 Some experimental observations. Appendix to D. W. Moore & P. G. Saffman. J. Fluid Mech. 39, 831–847.Google Scholar
Moore, D. W. & Saffman, P. G.1969a The flow induced by the transverse motion of a thin disk in its own plane through a contained rapidly rotating viscous liquid. J. Fluid Mech. 39, 831–847.Google Scholar
Moore, D. W. & Saffman, P. G. 1969b The structure of vertical free 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 to the solution of problems of motion of a circular disk in viscous liquid. Phil. Mag. 21, 546–564.Google Scholar
Saffman, P. G. 1976 Brownian motion in thin sheets of viscous fluid. J. Fluid Mech 73, 593–602.Google Scholar
Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. John Wiley & Sons.
Stewartson, K. 1953 On the slow motion of an ellipsoid in a rotating fluid. Q. J. Mech. Appl. Maths 21, 353–373.Google Scholar
Stewartson, K. 1967 On the slow transverse motion of a sphere through a rotating fluid. J. Fluid Mech. 30, 357–369.Google Scholar
Tanzosh, J. P. 1994 Integral equation formulations of the linearized Navier-Stokes equation: applications to particle motions in rotating viscous flows. PhD dissertation, Harvard University.
Tanzosh, J. P. & 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
Tanzosh, J. P. & Stone, H. A. 1995 A general approach to analyzing the motion of a disk in a Stokes flow. Chem. Engng Commun. to appear.
Taylor, G. I. 1923 Experiments on the motion of solid bodies in rotating fluids. Proc. R. Soc. Lond. A 104, 213–218.Google Scholar
Tranter, C. J. 1966 Integral Transforms in Mathematical Physics. John Wiley & Co.
Ungarish, M. & Vedensky, D. 1995 The motion of a rising disk in a rotating axially bounded fluid for large Taylor number. J. Fluid Mech. 291, 1–32.Google Scholar
Vaziri, A. & Boyer, D. L. 1971 Rotating flow over shallow topographies. J. Fluid Mech. 50, 79–95.Google Scholar
Vedensky, D. & Ungarish, M. 1994 The motion generated by a slowly rising disk in a rotating fluid for arbitrary Taylor number. J. Fluid Mech. 262, 1–26.Google Scholar