Some exact solutions describing unsteady plane gas flows with shocks
Authors:
P. L. Sachdev and A. Venkataswamy Reddy
Journal:
Quart. Appl. Math. 40 (1982), 249-272
MSC:
Primary 76L05; Secondary 76N15
DOI:
https://doi.org/10.1090/qam/678197
MathSciNet review:
678197
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Abstract: A new class of exact solutions of plane gasdynamic equations is found which describes piston-driven shocks into non-uniform media. The governing equations of these flows are taken in the coordinate system used earlier by Ustinov, and their similarity form is determined by the method of infinitesimal transformations. The solutions give shocks with velocities which either decay or grown in a finite or infinite time depending on the density distribution in the ambient medium, although their strength remains constant. The results of the present study are related to earlier investigations describing the propagation of shocks of constant strength into non-uniform media.
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M. D. Ustinov, Transformation and some solutions of the equations of motion of an ideal gas, Izv. AN SSSR Mekh. Zhid, I Gaza 3, 68–74 (1966)
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Ya. B. Zel’dovich and Yu. P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena, Academic Press, New York and London, 1967
L. I. Sedov, Similarity and dimensional methods in mechanics, Academic Press, New York, 1959
J. B. Keller, Spherical, cylindrical and one-dimensional gas flows, Quart. Appl. Math. 14, 2, 171–184 (1956)
G. C. McVittie, Spherically symmetric solutions of the equations of gas dynamics, Proc. Roy. Soc. (London) 220, 339–355 (1953)
M. D. Ustinov, Ideal gas flow behind a finite-amplitude shock wave, Izv. AN SSSR Mekh. Zhid. I Gaza (Fluid dynamics) 2, 1, 88–90 (1967)
S. P. Castell and C. Rogers, Applications of invariant transformations in one-dimensional non-steady gas dynamics, Quart. Appl. Math. 32, 241–251 (1974)
M. D. Ustinov, Transformation and some solutions of the equations of motion of an ideal gas, Izv. AN SSSR Mekh. Zhid, I Gaza 3, 68–74 (1966)
G. W. Bluman and J. D. Cole, Similarity methods for differential equations, Springer-Verlag, New York, 1974
L. V. Ovsjannikov, Group properties of differential equations, translation by G. W. Bluman of Gruppovye Svoysta Differentsialny Uravneni, Novosibirsk, U.S.S.R., 1962
J. D. Logan and J. D. J. Perez, Similarity solutions for reactive shock hydrodynamics, SIAM J. Appl. Math. 39, 512–527 (1980)
Pierre A. Carrus, Phyllis A. Fox, Feliz Haas, and Zdenek Kopal, The propagation of shock waves in a stellar model with continuous density distribution, AP. J. 113, 496–518 (1951)
J. D. Murray, Perturbation effects on the decay of discontinuous solutions of nonlinear first order wave equations, SIAM J. Appl. Math. 19, 2, 135–160 (1970)
Ya. B. Zel’dovich and Yu. P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena, Academic Press, New York and London, 1967
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Article copyright:
© Copyright 1982
American Mathematical Society
