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Rectilinear plastic flow of a Bingham solid

III. A more general discussion of steady flow

Published online by Cambridge University Press:  24 October 2008

J. G. Oldroyd
J. G. Oldroyd
Affiliation:
Courtaulds Ltd., Research LaboratoryMaidenhead, Berks
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Extract

It is apparent from the two previous papers of the same main title (1, 2) that velocity distributions in steady rectilinear plastic flow of a Bingham solid between moving boundaries are not easy to determine, even when the boundaries are of simple shape. In the familiar case of a Newtonian liquid (which can be regarded as a special case of a Bingham solid with zero yield value) the velocity ω is a harmonic function and can be obtained by a conformal transformation of the region of flow of the type

In order to generalize and include materials of finite yield value, in which ω is not harmonic, one must first regard the Newtonian case from a slightly different point of view. The transformation

must be regarded as defining a change of coordinates, from Cartesian coordinates x, y to the natural curvilinear coordinates for a particular problem ω, W, one of which is the velocity. For a Bingham solid in general, it is shown in the present paper that natural coordinates ω, W exist, but are not obtainable by a conformal transformation from Cartesian coordinates, except in the limit when the yield value tends to zero. In practice it may be difficult to determine the natural coordinate system in a given problem, but when it is found the velocity distribution is automatically known.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 44 , Issue 2 , April 1948 , pp. 200 - 213
Copyright
Copyright © Cambridge Philosophical Society 1948

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References

REFERENCES

(1)Oldroyd, J. G.Proc. Cambridge Phil. Soc. 43 (1947), 396.CrossRefGoogle Scholar
(2)Oldroyd, J. G.Proc. Cambridge Phil. Soc. 43 (1947), 521.Google Scholar
(3)Oldroyd, J. G.Proc. Cambridge Phil. Soc. 43 (1947), 100.Google Scholar
(4)Oldroyd, J. G.Proc. Cambridge Phil. Soc. 43 (1947), 383.Google Scholar
(5)Lamé, G.Leçons sur les coordonnées curvilignes et leurs diverses applications (Paris, 1859), p. 73.Google Scholar
(6)See, for example, Copson, E. T.An Introduction to the Theory of Functions of a Complex Variable (Oxford, 1935), p. 184.Google Scholar