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Stability of step shocks. (English) Zbl 0111.38403

fluid mechanics
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##### References:
 [1] For a discussion of shock structure, see the article by W. D. Hayes inFundamentals of Gas Dynamics, edited by H. W. Emmons, Vol. III ofHigh Speed Aerodynamics and Jet Propulsion(Princeton University Press, Princeton, New Jersey, 1958), pp. 416–481. [2] H. A. Bethe, ”The Theory of Shock Waves for an Arbitrary Equation of State,” Office of Scientific Research and Development, Rept. 445 (1942). [3] Freeman, Proc. Roy. Soc. (London) A228 pp 341– (1955) [4] Freeman, J. Fluid Mech. 2 pp 397– (1957) [5] A. E. Roberts, ”Stability of a Steady Plane Shock,” Los Alamos Scientific Laboratory Rept. LA-299 (1945). [6] Richtmeyer, Communs. Pure and Appl. Math. 13 pp 297– (1960) [7] Erpenbeck, Phys. Fluids 5 pp 604– (1962) [8] D’yakov, Zhur. Eksptl. i Teoret. Fiz. 27 pp 288– (1954) [9] (translation: Atomic Energy Research Establishment AERE Lib.Trans. 648, 1956). [10] For mixtures in which no chemical transformations are induced by shocks, the composition could be regarded as an independent thermodynamic variable. The composition, though uniform in the unperturbed flow, could then be perturbed independently of S^ and {$$\upsilon$$}^. In the absence of diffusion, however, such perturbations merely propagate along streamlines without affecting the stability of the flow. [11] In I, {$$\theta$$} was defined through the inverse of the solution matrix of the homogeneous equation. The transpose of this inverse is, however, a solution matrix of the adjoint equation so that the definitions are equivalent. [12] See, for example, E. T. Copson,Theory of Functions of a Complex Variable(Oxford University Press, London, England, 1935), p. 119. [13] We, of course, must take care to remain to the right of the branch cut defining s({$$\omega$$}). This cut can be taken along the imaginary axis between {$$\omega$$} = {$$\pm$$}i[{$$\eta$$}(1-{$$\eta$$})]12. [14] D. V. Widder,The Laplace Transform(Princeton University Press, Princeton, New Jersey, 1946), p. 197. · Zbl 0060.24801 [15] See reference 2 for a discussion of the high-pressure form of the Hugoniot. It is to be noted, however, that the argument of Bethe whereby the shocked fluid approaches ideal-gas behavior is inconsistent with his Fig. 2B in which the vertical asymptote in the S-{$$\upsilon$$} (or p-{$$\upsilon$$}, since pS>0 for an ideal gas) plane is approached from the left. Since the ideal-gas Hugoniot, has negative slope throughout (see reference 15), this asymptote must be approached from the right, as assumed in our discussion. [16] The sonic character of the shock will always be with respect to the unperturbed state behind it. [17] R. Courant and K. O. Friedrichs,Supersonic Flow and Shock Waves(Interscience Publishers, Inc., New York, 1948), p. 138. · Zbl 0041.11302
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