One‐dimensional stability of self‐similar converging flows

A necessary condition for the asymptotic approach of symmetric converging flows to a self‐ similar form is the stability of the one‐dimensional partial differential equations when linearized about the appropriate similarity solutions. For the converging shock problem, and (for sufficiently large adiabatic exponent γ) also for the collapsing free‐surface problem, it is found that (1) the standard, analytic similarity solution is positively stable with respect to symmetric (one‐ dimensional) perturbations, and (2) that all other similarity solutions are positively unstable. For the free‐surface problem when γ is small, there seem to be three regimes: neutral stability, instability of all solutions, and stability of a degenerate solution.