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On the shearing motion of fluid past a projection

Published online by Cambridge University Press:  24 October 2008

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

1. In this paper, a continuation of an earlier paper(1), we consider the two-dimensional motion of incompressible viscous liquid past a projection, the motion being one of uniform shear apart from the disturbance caused by the projection. A special form is assumed for the boundary, so that the area in the z-plane (Fig. 1) can be represented conformally on a circle in the ζ-plane by a rational function of ζ; this function contains a parameter a (0 < a ≤ 1), and by varying a the shape of the projection can be varied. Since a rational function is concerned in the conformal transformation a method lately developed by N. Muschelišvili(2) can be used in solving the biharmonic equation for the stream function, though the method actually used differs in some points of detail from that originally proposed by Muschelišvili and appears to be somewhat simpler.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 40 , Issue 1 , March 1944 , pp. 19 - 36
Copyright
Copyright © Cambridge Philosophical Society 1944

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References

REFERENCES

(1)Dean, W. R.Proc. Cambridge Phil. Soc. 36 (1940), 300.CrossRefGoogle Scholar
(2)Muschelišvili, N. Z.angew. Math. Mech. 13 (1933), 264.CrossRefGoogle Scholar
(3)Prandtl, L.The physics of solids and fluids, Part ii (London, 1936), pp. 199, 261.Google Scholar
(4)Modern developments in fluid dynamics (Oxford, 1938), pp. 63–65, Plates 7 and 8.Google Scholar