Propagation of a reaction front in a narrow sample of energetic material with heat losses: chaotic regimes, extinction and intermittency

Abstract

The influence of heat-losses on the flame dynamics in narrow samples of energetic material is investigated numerically. The model is reduced to a one-dimensional form with the flame-sheet approximation applied for the reaction rate. Both the steady-state solutions and its linear stability analysis are treated analytically. A typical C-shaped response curve is found for the dependence of the flame-propagation velocity on the heat-loss parameter, with solutions along the lower branch of slower flames being always unstable. It is found that a part of the upper branch of the C-shaped response curve is also unstable and the Poincaré–Andronov–Hopf bifurcation takes place at a certain value of heat-loss intensity even if the steady state solution is stable under the corresponding adiabatic conditions. The numerical simulations show that an increase in heat-losses induces, for sufficiently high Zel’dovich numbers, the Feigenbaum’s cascade of period doubling bifurcations after which a chaotic dynamics is setting in. The chaotic dynamics precedes the flame extinction occurring for the further increase of the heat-loss parameter which, nevertheless, remains significantly lower than the steady extinction limit dictated by the C-shaped response curve. Apparently, the parametric dependence of the extinction time in these cases is also irregular with appreciable disparities in magnitude. Finally, the intermittency effect is detected slightly below the extinction limit with irregular dynamics alternating by apparently periodic stages. These results may be important for the flammability limits theory and practical fire safety applications