Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-12T17:11:57.411Z Has data issue: false hasContentIssue false
  • English
  • Français

One-dimensional unsteady motion of a gas initially at rest and the dam-break problem

Published online by Cambridge University Press:  24 October 2008

A. G. Mackie
A. G. Mackie
Affiliation:
Mathematics DepartmentUniversity of Manchester*
Get access Rights & Permissions [Opens in a new window]

Abstract

The object of this paper is to discuss the one-dimensional unsteady adiabatic motion of a gas which is initially at rest with a prescribed density distribution such that the specific entropy is uniform. The contour integral methods which Copson developed recently for even analytic functions are extended to apply to general analytic initial conditions. The solution is valid in the range 1 < Υ > 3, where y is the adiabatic index of the gas. Of particular interest, in view of the hydraulic analogy, is the case Υ = 2 for which real variable methods cannot readily be adapted. The motion of the front of a water column Sowing into a dry, horizontal stream bed is discussed. A curious type of solution, corresponding to a particular choice of initial distribution, which was established by Pack for a, countable sequence of values of Υ, is verified to hold over the whole range and is interpreted in terms of the dam-break problem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 1 , January 1954 , pp. 131 - 138
Copyright
Copyright © Cambridge Philosophical Society 1954

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Copson, E. T.Mon. Not. B. astr. Soc. 110 (1950), 238–46.CrossRefGoogle Scholar
(2)Copson, E. T.Proc. roy. Soc. A, 216 (1953), 539–47.Google Scholar
(3)Dressler, R. F.Bur. Stand. J. Res., Wash., 49 (1952), 217–25.CrossRefGoogle Scholar
(4)Mackie, A. G.Proc. roy. Soc. Edin. (submitted for publication).Google Scholar
(5)Martin, M. H.Amer. J. Math. 55 (1943), 391–407.CrossRefGoogle Scholar
(6)Pack, D. C.Mon. Not. R. astr. Soc. 113 (1953), 43–51.CrossRefGoogle Scholar
(7)Pack, D. C.Proc. Camb. phil. Soc. 49 (1953), 493–7.CrossRefGoogle Scholar
(8)Stoker, J. J.Commun. appl. Math. 1 (1948), 1–87.CrossRefGoogle Scholar