On the theory of a shock wave driven by a corrugated piston in a non-ideal fluid

Abstract

AbstractIn the context of an Eulerian fluid description, we investigate the dynamics of a shock wave that is driven by the steady impulsively initiated motion of a two-dimensional planar piston with small corrugations superimposed on its surface. This problem was originally solved by Freeman (Proc. Royal Soc. A, vol. 228, 1955, pp. 341–362), who showed that piston-driven shocks are unconditionally stable when the fluid medium through which they propagate is an ideal gas. Here, we generalize Freeman’s mathematical framework to account for a fluid characterized by an arbitrary equation of state. We find that a sufficient condition for shock stability is $\ensuremath{-} 1\lt h\lt {h}_{c} $, where $h$ is the D’yakov parameter and ${h}_{c} $ is a critical value less than unity. For values of $h$ within this range, linear perturbations imparted to the front by the piston at time $t= 0$ attenuate asymptotically as ${t}^{\ensuremath{-} 3/ 2} $. Outside of this range, the temporal behaviour of perturbations is more difficult to determine and further analysis is required to assess the stability of a shock front under such circumstances. As a benchmark of the main conclusions of this paper, we compare our generalized expression for the linearized shock-ripple amplitude with an independent Bessel-series solution derived by Zaidel’ (J. Appl. Math. Mech., vol. 24, 1960, pp. 316–327) and find excellent agreement.