Collapse of a one-dimensional cavity

Abstract

Isentropic potential flows arising when a one-dimensional cavity collapses into an ideally polytropic continuous medium are examined. The analysis is carried out up to the instant of focussing or up to the instant infinite gradient arise in the flow. As a result of the investigation of the analytic solutions in special variables for the polytropy exponent 1 < γ < 3, it is proved that a free boundary separating the medium and the vacuum moves for some time with a constant velocity. Next, the solution is sought in physical space as a series in a neighborhood of the free boundary. When γ > 1 it is proved that the series converges and the free boundary's acceleration begins only from the instant of origin of an infinite gradient. An ordinary differential equation is obtained, governing the behavior of the gradient on the free boundary. The solutions of this equation are studied by numerical calculations and particular solutions are found. It turned out that the instant of origin of the singularity depends in an essential manner on the initial data. Exponents γ ∗ are introduced, such that when γ ⩽ γ ∗ there are no singularities on the free boundary up to the focussing instant if at the initial instant the medium was homogeneous and at rest. When γ > γ ∗ the function t ∗ = t ∗ (γ) , viz., the instant of origin of an infinite gradient on the free boundary, is obtained. Calculations carried out by a difference scheme showed that when γ ⩽ γ ∗ up to the focussing instant and when γ > γ ∗ up to instant t ∗ there are no large gradient inside the whole flow region. The exponents γ ∗ coincide with those found earlier papers ( ∗∗) in which the problem being investigated was investigated by means of constructing several terms of asymptotic series. It is concluded that when 1< γ < 3 the free boundary moves for some time t ∗ > 0 at a constant velocity, and when γ ⩽ γ ∗ = 1 + 2 i the time t ∗ coincides with the focussing instant t 1 = (γ −1) 2 and t ∗ < t 1 when γ > γ ∗ The complete construction of the asymptotic expansions and the exact estimates of these expansions was not carried out. Power series solving the original problem in the exact statement are constructed recurrently in this paper. All the facts obtained are proved on the basis of the study of the convergence domains of these series. When 1< γ < 3 they coincide with earlier derivations.