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On a Case of steady flow of a Gas in Two Dimensions

Published online by Cambridge University Press:  24 October 2008

S. Lees
S. Lees
Affiliation:
St John's College
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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

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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)

which may also be written

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

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).