Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T23:19:45.956Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on rotationally symmetric flow above an infinite rotating disc

Published online by Cambridge University Press:  26 February 2010

J. B. McLeod
J. B. McLeod
Affiliation:
Wadham College, Oxford.
Get access

Extract

The fluid mechanical problem with which we are concerned is the behaviour of fluid occupying the half-space x > 0 above a rotating disc which is coincident with the horizontal plane x = 0 and rotating about its axis which remains fixed. Studying rotationally symmetric solutions of this problem, von Kármán [1] (see also [2; p. 93]; [3; p. 133]) reduced it to the solution of two simultaneous equations in functions f(x), g(x) which may, with suitable normalisation, be written in the form

with the boundary conditions

Type
Research Article
Information
Mathematika , Volume 17 , Issue 2 , December 1970 , pp. 243 - 249
Copyright
Copyright © University College London 1970

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

1.von Kàrmàn, T., “Über laminare und turbulente Reibung”, Z. Ang. Math. Mech., 1 (1921), 233–252.CrossRefGoogle Scholar
2.Schlichting, H., Boundary layer theory (6th ed., McGraw-Hill, 1968).Google Scholar
3.Greenspan, H. P., The theory of rotating fluids (Cambridge, 1968).Google Scholar
4.Batchelor, G. K., “Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow”, Quart. J. Mech. Appl. Math., 4 (1951), 29–41.CrossRefGoogle Scholar
5.Stewartson, K., “On the flow between two rotating coaxial disks”, Proc. Cambridge Phil. Soc., 49 (1953), 333–341.CrossRefGoogle Scholar
6.Lance, G. N. and Rogers, M. H., “The axially symmetric flow of a viscous fluid between two infinite rotating disks”, Proc. Roy. Soc. A, 266 (1962), 109–121.Google Scholar
7.Pearson, C. E., “Numerical solutions for the time-dependent viscous flow between two rotating coaxial disks”, J. Fluid Mech., 21 (1965), 623–633.CrossRefGoogle Scholar
8.McLeod, J. B., “Von Karman“s swirling flow problem”, Arch. Rat. Mech. Anal., 33 (1969), 91–102.CrossRefGoogle Scholar
9.Iglisch, R., “Elementarer Existenzbeweis für die Strömung in der laminaren Grenzschicht zur Potentialströmung V = u 1x m mit m > 0 bei Absaugen und Ausblasen”, Z. Ang. Math. Mech., 33 (1953), 143–147.CrossRefGoogle Scholar
10.Coppel, W. A., “On a differential equation of boundary-layer theory”, Phil. Trans. Roy. Soc. A, 253 (1960), 101–136.Google Scholar
11.Ho, D. and Wilson, H. K., “On the existence of a similarity solution for a compressible boundary layer”, Arch. Rat. Mech. Anal., 27 (1967), 165–174.Google Scholar
12.McLeod, J. B. and Serrin, J., “The existence of similar solutions for some boundary layer problems”, Arch. Rat. Mech. Anal., 31 (1968), 288–303.CrossRefGoogle Scholar
13.Rogers, M. H. and Lance, G. N., “The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk “, J. Fluid Mech., 7 (1960), 617–631.CrossRefGoogle Scholar
14.McLeod, J. B., “The asymptotic form of solutions to von Kàrmàn's swirling flow problem” Quart. J. Math., Oxford (2), 20 (1969), 483–496.CrossRefGoogle Scholar
15.Evans, D. J., “The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disc with uniform suction”, Quart. J. Mech. Appl. Math., 22 (1969), 467–485.CrossRefGoogle Scholar
16.Moore, F. K., “Three-dimensional boundary-layer theory”, Adv. Appl. Mech., 4 (1956), 166–171.Google Scholar
17.Watson, J., “On the existence of solutions for a class of rotating disc flows and the convergence of a successive approximation scheme”, J. Inst. Math. App., 1 (1966), 348–371.CrossRefGoogle Scholar
18.McLeod, J. B., “The existence of axially symmetric flow above a rotating disc”, to appear.Google Scholar