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Compressible rotational flows gnerated by the substitution principle. II. (English) Zbl 0661.76113

The substitution principle is a transformation of the dependent variables that leaves invariant the steady Euler equations. In Part I [the authors, ibid. 31, No.5, 1058-1063 (1988; Zbl 0643.76111)] the principle is applied to several well-known compressible flows. In this communication, the most general forms of the transformation are derived, including a generalization of the vorticity transformation. From a thermodynamic analysis and the energy shock wave jump condition, we conclude that a thermally perfect gas is usually the only assumptions compatible with the principle. The nonlinear equations for rotational flow of a perfect gas are derived in the hodograph plane. By means of the substitution principle, the solutions of the linear, irrotational hodograph equations can be transformed into solutions of the new rotational equations. As a result of the invariance of the physical coordinates under the substitution principle, the transformation does not alter the form of the solutions. A cylindrical source flow is used to illustrate the hodograph theory.

MSC:

76U05 General theory of rotating fluids
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations
76M99 Basic methods in fluid mechanics

Citations:

Zbl 0643.76111
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References:

[1] DOI: 10.1073/pnas.33.5.137 · doi:10.1073/pnas.33.5.137
[2] DOI: 10.1017/S002211206000092X · Zbl 0094.21203 · doi:10.1017/S002211206000092X
[3] DOI: 10.1063/1.866786 · Zbl 0643.76111 · doi:10.1063/1.866786
[4] Prim R. C., J. Rat. Mech. Anal. 1 pp 425– (1952)
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