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Rectilinear flow of non-Bingham plastic solids and non-Newtonian viscous liquids. I

Published online by Cambridge University Press:  24 October 2008

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

In a previous paper on steady rectilinear plastic flow of a Bingham solid (1), the point of view was adopted that the study of all possible velocity distributions was equivalent to a study of the geometry of velocity contours in a section normal to the flow. This led to an analysis of the differential geometry of velocity contours in rectilinear plastic flow under zero pressure gradient, and the problem of finding all possible velocity distributions was eventually reduced to the problem of finding the general integral of a single linear second-order partial differential equation with two independent variables. But the form of this differential equation (equation (25) of (1)) was such that only a very limited number of solutions could be found by standard methods. The particular solutions which were obtained corresponded to those flow patterns in which the velocity contours could be identified with a family of parametric curves in a curvilinear coordinate system obtainable by conformal transformation of a cartesian frame of reference. In effect, this means that the results so far achieved by using the new approach to the problem could have been obtained by more direct methods.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 45 , Issue 4 , October 1949 , pp. 595 - 611
Copyright
Copyright © Cambridge Philosophical Society 1949

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References

REFERENCES

(1)Oldroyd, J. G.Proc. Cambridge Phil. Soc. 44 (1948), 200.CrossRefGoogle Scholar
(2)Oldroyd, J. G.Proc. Cambridge Phil. Soc. 43 (1947), 100.CrossRefGoogle Scholar
(3)Forsyth, A. R.A treatise on differential equations, 3rd ed. (London, 1903), chap. x, §§ 265–70.Google Scholar
(4)Forsyth, A. R.Theory of differential equations, 6 (Cambridge, 1906), chap. xvii.Google Scholar
(5)Forsyth, A. R.A treatise on differential equations, chap. ix, §§ 184–9.Google Scholar