Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-18T21:16:50.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiple solutions for flow between coaxial disks

Published online by Cambridge University Press:  29 March 2006

N. D. Nguyen ,
J. P. Ribault  and
P. Florent
N. D. Nguyen
Affiliation:
Laboratoire de Mécanique des Fluides, Université Laval, Québec, Canada
J. P. Ribault
Affiliation:
Laboratoire de Mécanique des Fluides, Université Laval, Québec, Canada
P. Florent
Affiliation:
Laboratoire de Mécanique des Fluides, Université Laval, Québec, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of obtaining a numerical solution for the steady flow between two coaxial infinite disks, one fixed and porous, the other rotating, is reduced by von Kámán's hypothesis to solution of a system of nonlinear equations. A Newton-type iteration results in several solutions to these equations, as a number of authors have already indicated. Nevertheless, an interval in which only one solution is found exists for small values of the Reynolds number based on the angular velocity of the rotating disk, the distance between the disks and the kinematic viscosity of the fluid. At large values of this Reynolds number, two solutions appear and have been the subject of intense controversy.

In this paper, both physical and numerical arguments are presented which support a Batchelor-type solution for the flow between infinite disks, in which part of the fluid rotates as a solid body. The other solution, following Stewartson, assumes that the velocity of the fluid outside the boundary layers is entirely axial. This only seems to be verified experimentally when the distance between the disks is large compared with the (finite) radius of the disks.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 68 , Issue 2 , 25 March 1975 , pp. 369 - 388
Copyright
© 1975 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. 1951 Note on a class of solution of the Navier-Stokes equations representing rotationally symmetric flow. Quart. J. Mech. Appl. Math. 4, 29.Google Scholar
Bödewadt, U. T. 1940 Die Drehströmung uber festem Grunde. Z. angew. Math. Mech. 20, 241.Google Scholar
Cochran, W. G. 1934 The flow due to a rotating disc. Proc. Camb. Phil. Soc. 30, 365.Google Scholar
Dorfman, L. A. 1967 Flow of a viscous fluid between fixed and blown rotating discs. N.A.S.A. Tech. Rep. TTF-10, 931.Google Scholar
Florent, P. & Nguyen, N. D. 1971 Ecoulement instationnaire entre un disque fixe poreux et un disque tournant. Iutam Symp. Unsteady Boundary Layers, Laval University, Quebec, p. 1216.Google Scholar
Florent, P., Nguyen, N. D. & Vo, N. D. 1973 Ecoulement instationnaire entre disques coaxiaux. J. Mécanigue, 12, 555.Google Scholar
Florent, P. & Thiolet, O. 1969 Importance de l'orientation du support de sonde à fil chaud par rapport à une paroi sur la détermination des vitesses moyennes dans une couche limite turbulente. C. R. Acad. Sci. Paris, 269, 405.Google Scholar
Greenspan, D. 1972 Numerical studies of flow between rotating coaxial discs. J. Inst. Math. Appl. 9, 370.Google Scholar
Grohne, D. 1955 Uber die laminare Strömung in einer Kreiszylindrischen Dose mit rotierendem Deckel. Nachr. Akda. Wiss. Göttingen, Math. Phys. 1 (IIa), 263.Google Scholar
Kármán, T. VON 1921 Laminar und turbulente Reibung. Z. angew. Math. Mech. 1, 233.Google Scholar
Maxworthy, T. 1963 The flow between a rotating disc and a coaxial stationary disc. Space Prog. Summ. no. 37.27, vol. 4, 327. Jet Propulsion Laboratory, Pasadena, California.Google Scholar
Mellor, G. L., Chappel, P. J. & Stokes, V. K. 1968 On the flow between a rotating and a stationary disk. J. Fluid Mech. 31, 95.Google Scholar
Pearson, C. E. 1965a A computational method for viscous flow problems. J. Fluid Mech. 21, 611.Google Scholar
Pearson, C. E. 1965b Numerical solutions for the time-dependent viscous flow between rotating coaxial disks. J. Fluid Mech. 21, 623.Google Scholar
Peube, J. L. 1967 Ecoulement entre un disque fixe poreux et un disque tournant. Cong. Can. Mécanique Appl., Québec.Google Scholar
Rogers, M. H. & Lance, G. N. 1960 The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk. J. Fluid Mech. 7, 617.Google Scholar
Rott, N. & Lewellen, W. S. 1965 Examples of boundary layers in rotating flows. Agardogrqh, 97, 613.Google Scholar
Schultz-Grunow, F. 1935 Der Reibungswiderstand rotierender Scheiben in Gehäusen. Z. angew. Math. Mech. 15, 191.Google Scholar
Stewartson, K. 1953 On the flow between two rotating coaxial discs. Proc. Camb. Phil. Soc. 3, 333.Google Scholar
Stuart, J. T. 1954 On the effects of uniform suction on the steady flow due to a rotating disc. Quart. J. Mech. Appl. Math. 7, 466.Google Scholar