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Exact solutions and numerical analysis of the problem of an intense explosion in certain ideal compressible media

Journal of Applied Mathematics and Mechanics
Publication Date
DOI: 10.1016/0021-8928(89)90065-8
  • Mathematics


Abstract The well-known selfsimilar problem of an intense explosion in an ideal compressible medium possessing a certain arbitrariness in the form of the internal energy is considered. The problem was formulated by Sedov. The existence of the first two integrals reduces the problem to the study of the integrability of a single, first-order differential equation. We will show that even in the simplest case when the problem has planar symmetry, and the equation reduces, in the general case, to an Abel equation with functional coefficients which is not integrable in quadratures. A special case of its integrability is found, which enables us to write out the analytic solutions of the problem for a certain family of media including real and dust-containing gases (under the assumption that the phase parameters are in equilibrium). The results generalize the results obtained earlier /1–4/. All solutions obtained can be continued to the plane of symmetry, and their asymptotic behaviour near it is investigated. A numerical analysis of the problem is carried out for the same family of media for the cylindrical and spherical cases. Two new effects are found for disperse media such as a liquid with bubbles and a dusty gas (previously studied numerically in /5, 6/), namely the non-monotonic form of the velocity behind the shock wave, and the effect of incompressibility when the mixture contains a fairly small amount of gas. In the spherical case the limit solution of the problem, when the amount of gas is reduced, is represented by the well-known solution of the problem of an intense explosion in an incompressible fluid.

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