GAS-FLOWS WITH SHOCK-WAVES WHICH DIVERGE FROM AN AXIS OR CENTER OF SYMMETRY WITH FINITE VELOCITY

Cylindrically and spherically symmetric steady hows of an ideal gas ar e investigated. The Cauchy problem with data for zeta = 0 is considere d in the space of independent variables zeta = t/r, chi = r. The exist ence and uniqueness of the analytic solution of this problem is proved . The Cauchy problem with initial data on different surfaces is consid ered in the space of the t, r variables: for r = 0 the zero velocity o f the gas is specified, and the Hugoniot conditions are satisfied on t he unknown shock wave front, which diverges with finite velocity from an axis or centre of symmetry. The existence and uniqueness of the ana lytic solution of this problem is also proved, and the law of motion o f the diverging shock wave is determined uniquely. Two problems on the focusing of a gas and on its subsequent reflection with a finite velo city of the shock wave are solved, namely, (1) the compression wave du e to smooth motion of a piston in a gas at rest is focused, and (2) th e rarefaction wave that arises when a one-dimensional cavity collapses is focused. The solutions of these problems represent an extension of Sedov's self-similar solutions to the case of two independent variabl es [1-3]. Moreover, the solution of the second problem extends the mat hematical investigation of the process of one-dimensional cavity colla pse [4, 5]. Copyright (C) 1996 Elsevier Science Ltd.