On a Case of steady flow of a Gas in Two Dimensions
Published online by Cambridge University Press: 24 October 2008
Extract
The problem of finding general types of solution for the steady flow of gases in two dimensions, adiabatically and without friction, does not appear to have received a great deal of attention. An interesting and suggestive treatment of hydrodynamics applied to gases is given in Riemann-Weber's Partielle Differential-Gleichungen, but the application is to pressure propagation. Reference must also be made to a paper by the late Lord Rayleigh, who gave a general differential equation which must be satisfied by the velocity potential ø, but the problem of utilizing this equation does not appear easy.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 3 , 20 September 1924 , pp. 350 - 362
- Copyright
- Copyright © Cambridge Philosophical Society 1924
References
* 1901 Edn, Bd. 2, p. 469 et seq.
† Phil. Mag., 32 (1916)Google Scholar, and Scientific Papers, 6, 402. With the notation used in this paper, the equation referred to for adiabatic flow is (for two dimensions)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151030103152863-0984:S0305004100014250eqnU1.gif?pub-status=live)
which may also be written
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151030103152863-0984:S0305004100014250eqnU2.gif?pub-status=live)
which does not appear to be utilized easily.
For two-dimensional flow, a stream function exists, and it may readily be verified that it satisfies
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151030103152863-0984:S0305004100014250eqnU3.gif?pub-status=live)
This would not appear any more easy to handle than Rayleigh's equation for ø.
In the above equations, a 0 is the velocity of sound for regions where the gas may be considered at rest, whilst q is the absolute velocity at (x, y).
* is the velocity of sound at (x, y). See Lamb, Hydrodynamics, 1916 Edn, §§ 23, 25. See also § 6 of this paper.
* The elimination of one of the variables u and v may also be conducted in another manner, for which I am indebted to Prof. W. McF. Orr, F.R.S. We write u + iv=u 1, u − iv=v 1. Equations (12) and (13) then become and u 1v 1 = ψ1 (z) respectively. On eliminating v 1 (say) from these, we are ultimately led to a result agreeing with (23), but the method is neater than that used by the writer.
* Lines of constant pressure (and density) are straight in thé general case. See equation (27).
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