On pulsating and cellular forms of hydrodynamic instability in liquid-propellant combustion

An extended Landau-Levich model of liquid-propellant combustion, one that allows for a local dependence of the burning rate on the (gas) pressure at the liquid-gas interface, exhibits not only the classival hydrodynamic cellular instability attributed to Landau but also a pulsating hydrodynamic instability associated with sufficiently negative pressure sensitivities. Exploiting the realistic limit of small values of the gas-to-liquid density ratio p, analytical formulas for both neutral sability boundaries may be obtained by expanding all quatities in appropriate powers of p in each of three distinguished wave-number regimes. In particular, composite analytical expressions are derived for the neutral stability boundaries A_p(k), where A_p is the pressure sensitivity of the burning rate and k is the wave number of the disturbance. For the cellular boundary, the results demonstrate explicitly the stabilizing effect of gravity on long-wave disturbances, the stabilizing effect of viscosity (both liquid and gas) and surface tension on short-wave perturbations, and the instability associated with intermediate wave numbers for negative values of A_p, which is characteristic of many hydroxylammoninum nitrate-based liquid propellants over certain pressure ranges. In contrast, the pulsating hydrodynamic stability boundary is insensitive to gravitational and surface-tension effects but is more sensitive to the effects of liquid viscosity because, for typical nonzero values of the latter, the pulsating boundary decreases to larger negative values of A_p as k increases through O(1) values. Thus, liquid-propellant combustion is predicted to be stable (that is, stealy and planar) only for a range of negative pressure sensitivities that lie below the cellular boundary that exists for sufficiently small negative values of A_p and above the pulsating boundary that exists for larger negative values of this parameter.