Exact analysis of a nonlinear partial differential equation of gas dynamics
Authors:
P. L. Sachdev, S. Dowerah, B. Mayil Vaganan and Varughese Philip
Journal:
Quart. Appl. Math. 55 (1997), 201-229
MSC:
Primary 35Q30; Secondary 35A25, 35C99, 76N15
DOI:
https://doi.org/10.1090/qam/1447575
MathSciNet review:
MR1447575
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Abstract: A new second-order nonlinear partial differential equation is derived from one-dimensional unsteady non-isentropic gas-dynamic equations through the introduction of three “potential” functions. Appropriate boundary conditions at the shock and at the piston in terms of the new functions are obtained. The nonlinear partial differential equation is analysed in great detail. Intermediate integrals and generalized Riemann invariants are discovered. Using the classical Lie group method, the direct similarity method due to Clarkson and Kruskal (1989), and equation-splitting etc., large families of new solutions are found. The direct similarity method is found to yield the most general results. Solutions with shocks (both finite and strong) are constructed to illustrate the applicability of the solutions.
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R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948
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, 171–184 (1956)
G. C. McVittie, Spherically symmetric solutions of the equations of gas dynamics, Proc. Roy. Soc. London Ser. A 220, 339–355 (1953)
M. H. Martin, The propagation of a plane shock into a quiet atmosphere, Canadian Journal of Mathematics 5, 37–39 (1953)
G. S. S. Ludford and M. H. Martin, One dimensional anisentropic flows, Comm. Pure Appl. Math. 7, 45–63 (1954)
J. A. Steketee, Unsteady rectilinear flows of a non-homentropic gas, Acta Astronaut. 6, 413–434 (1979)
J. A. Steketee, Transformations of the equations of motion for the unsteady rectilinear flow of a perfect gas, J. Engrg. Math. 10, 69–94 (1976)
P. Smith, Anisentropic rectilinear gas flows, Appl. Sci. Res. A12, 66–72 (1964)
H. Ardavan-Rhad, The decay of a plane shock wave, J. Fluid Mech. 43, 737–751 (1970)
M. D. Ustinov, Ideal gas flow behind a finite-amplitude shock wave, Izv. Akad. Nauk. SSSR, Mekh. Zhid. Gaza. 2, 88–90 (1967)
P. L. Sachdev and A. Venkataswamy Reddy, Some exact solutions describing unsteady plane gas flows with shocks, Quart. Appl. Math. 40, 249–272 (1982)
P. L. Sachdev, Neelam Gupta, and D. S. Ahluwalia, Exact analytic solutions describing unsteady plane gas flows with shocks of arbitrary strength, Quart. Appl. Math. 50, 677–726 (1992)
M. D. Ustinov, Approximate solution to nonself-similar problem of motion of a piston after an impact, Izv. Akad. Nauk. SSSR, Mekh. Zhid. Gaza. 6, 167–171 (1982)
M. D. Ustinov, Motion of a piston under the influence of gas pressure in the presence of an initial temperature gradient, Izv. Akad. Nauk. SSSR, Mekh. Zhid. Gaza. 2, 177–180 (1984)
M. D. Ustinov, Some one-dimensional unsteady adiabatic gas flows with plane symmetry, Izv. Akad. Nauk. SSSR, Mekh. Zhid. Gaza. 5, 96–104 (1986)
P. Clarkson and M. D. Kruskal, New similarity reductions of the Boussinesq equation, J. Math. Phys. 30, 2201–2213 (1989)
G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation 18, 1025–1042 (1969)
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Article copyright:
© Copyright 1997
American Mathematical Society