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Note on the slow motion of fluid

Published online by Cambridge University Press:  24 October 2008

W. R. Dean
W. R. Dean
Affiliation:
Trinity CollegeCambridge
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Extract

In the first part of the paper a slow two-dimensional motion of viscous fluid is considered which approximates to a motion of uniform shear past an infinite fixed plane, and differs from this motion because there is a gap in the plane (Fig. 1). A simple expression in finite terms is found for the stream function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 35 , Issue 1 , January 1939 , pp. 27 - 43
Copyright
Copyright © Cambridge Philosophical Society 1939

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References

* The area in the z-plane can accordingly be represented on a circle in the ζ-plane by a rational function of ζ. A method for the solution of the biharmonic equation in such a case has been developed by Muschelišvili, N., Z. angew. Math. 13 (1933), 264–82CrossRefGoogle Scholar, but with the particular rational function that occurs here a simpler alternative is available. It may also be noted that if the area is bounded by two intersecting circular arcs the Green's function has been found by Dixon, A. C., Proc. London Math. Soc. (2), 19 (1920), 373–86Google Scholar; we consider here a particular case of such an area, but it was not found convenient to use the Green's function, which is not in finite terms.

* Proc. Cambridge Phil. Soc. 32 (1936), 598–613.Google Scholar

† Aeronautical Research Committee, F.M. 101 (1933), 506.

* The negative direction is taken because − ∂ψ/∂y, ∂ψ/∂x are the x, y components of velocity if (20) is taken as the equation for the stream function.