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The flow due to a rotating disc

Published online by Cambridge University Press:  24 October 2008

W. G. Cochran
W. G. Cochran
Affiliation:
St John's College
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Extract

1. The steady motion of an incompressible viscous fluid, due to an infinite rotating plane lamina, has been considered by Kármán. If r, θ, z are cylindrical polar coordinates, the plane lamina is taken to be z = 0; it is rotating with constant angular velocity ω about the axis r = 0. We consider the motion of the fluid on the side of the plane for which z is positive; the fluid is infinite in extent and z = 0 is the only boundary. If u, v, w are the components of the velocity of the fluid in the directions of r, θ and z increasing, respectively, and p is the pressure, then Kármán shows that the equations of motion and continuity are satisfied by taking

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 3 , July 1934 , pp. 365 - 375
Copyright
Copyright © Cambridge Philosophical Society 1934

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References

* Zeitschrift für angewandte Mathematik u. Mechanik, 1 (1921), 244–7.Google Scholar

* These results are given, as Kármán, stated them, in the Handbuch der Physik, 7 (1927), 158, 159Google Scholar; the Handbuch der Experimental-Physik, 4 (1931), Part I, 255–7Google Scholar; Bulletin No. 84 of the National Research Council; Lamb, , Hydrodynamics (1932), 280–2Google Scholar; Müller, , Einführung in die Theorie der zähen Flüssigkeiten (1932), 226–9Google Scholar. A comparison with experiment was given by Kempf, , Vorträge aus dem Gebiete der Hydro- und Aerodynamik (Innsbruck, 1922Google Scholar) edited by Kármán, and Levi-Civitá, (1924), 168–70Google Scholar. If ωa 2/ν is too large (greater than about 5 × 105), the motion is turbulent.

* Cf. Whittaker, and Robinson, , Calculus of Observations, 2nd ed., 363–7.Google Scholar