Global solutions describing the collapse of a spherical or cylindrical cavity (Q1201307)

The paper studies isentropic symmetric flow into a vaccum of a gas initially at rest with speed of sound \(c_ 0= 1\). The flow has cylindrical or spherical symmetry with \(\nu =1\) or 2 when the gas is outside a circular cylindrical or spherical gas/vacuum interface with initial radius \(R(t_ 0)= 1\). The isentropic flow field is bounded by the symmetric surface moving with speed \(c_ 0\) with zero radial velocity and the gas/vacuum interface where \(c= 0\). \textit{H. P. Greenspan} and \textit{D. S. Butler} [J. Fluid Mech. 13, 101 ff. (1962)] formulated the governing equations and constructed the solution for the expansion of the gas into a vacuum outside of the gas/vacuum interface. Numerical solution of the case considered here was carried out by \textit{L. P. Thomas}, \textit{V. Pais}, \textit{R. Grahlon} and \textit{J. Diez} [Phys. Fluids 21, 676 ff. (1986)] and they found that the gas/vacuum interface moves at constant speed \(2/(\gamma- 1)\) as in the two-dimensional case until the cavity collapses, \(R(t)= 0\), when the specific heat ratio, \(\gamma\) is less than a critical number \(\gamma< \gamma_ p= 1+ 2/(1+\nu)\). For \(\gamma>\gamma_ p\) the solution near the collapsing interface, \(1\gg r\geq R\), and the velocity of the interface is given by a similarity solution found by \textit{C. Hunter} [J. Fluid Mech. 15, 289 ff. (1962)]. \textit{R. B. Lazarus} [Phys. Fluids 25, 1146-1155 (1982; Zbl 0489.76079)] showed that similarity solution is unstable for \(\gamma< \gamma_ p\). The present paper applies the power series solution introduced by Greenspan and Butler to the current problem and reconfirms the conclusions of Thomas et al, Hunter and Lazarus.