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The stability of viscous fluid flow between rotating cylinders

Published online by Cambridge University Press:  24 October 2008

S. Goldstein
S. Goldstein
Affiliation:
St John's College
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Extract

The stability of the motion of viscous incompressible fluid, of density ρ and kinematic viscosity ν, between two infinitely long coaxial circular cylinders, of radii a and a + d, where d/a is small, is investigated mathematically by the method of small oscillations. The inner cylinder is rotating with angular velocity ω and the outer one with angular velocity αω, and there is a constant pressure gradient parallel to the axis. The fluid therefore has a component velocity W parallel to the axis, in addition to the velocity round the axis. A disturbance is assumed which is symmetrical about the axis and periodic along it. The critical disturbance, which neither increases nor decreases with the time, is periodic with respect to the time (except when W = 0, when the critical disturbance is a steady motion). As Reynolds number of the flow we take | | d/ν, where is the average value of W across the annulus, and we denote by l the wave-length of the disturbance along the axis, by σ/2π the time period of the critical flow, by c the wavelength of the critical flow, by ωc the critical value of ω, and we put

approximately, if α is not nearly equal to 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 33 , Issue 1 , March 1937 , pp. 41 - 61
Copyright
Copyright © Cambridge Philosophical Society 1937

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References

* Phil. Trans. A, 223 (1923), 289–343.Google Scholar

* Phil. Trans. A, 223 (1922), 289–343.Google Scholar See also Dean, , Proc. Roy. Soc. A, 121 (1928), 405–10;Google ScholarFaxen, , Arkiv för Mat. Astr. och Fysik, A, 21 (1929),Google Scholar No. 26 (for a discussion of convergence); and Jeffreys, , Proc. Roy. Soc. A, 118 (1928), 195–208CrossRefGoogle Scholar (for some mathematically similar problems).

* Proc. Roy. Soc. A, 140 (1933), 227–40.Google Scholar My thanks are due to Mr Cornish for the opportunity of discussion of these experiments and their results.