Combustion waves in Sal’nikov’s reaction scheme in a spherically symmetric gas

We model a chemical reaction that proceeds according to Sal’nikov’s combustion scheme in a spherically symmetric cloud of gas. The gas density, velocity, temperature and the concentration of intermediate reactant are all assumed to be small perturbations about a steady-state equilibrium, and linearised solutions to the governing partial differential equations are derived. We demonstrate that the behaviour of the solutions occurs in two different ways, depending on the value of a derived parameter which is defined in terms of various properties of the occurring reaction. When this parameter is zero, the solutions present as travelling waves, and exact solutions are found in terms of the initial perturbations to the chemical system. When this parameter is non-zero, solutions are found as integrals over a basis of spherical Bessel functions. A brief stability analysis of these solutions is also presented, and it is shown that the linear stability is dependent only on the same derived parameter. We then conclude by comparing the linear results to a numerical solution of the full non-linear problem, which verifies the accuracy of the linearised results.