A Scaling Analysis of Sal’nikov’s Reaction, P→A→B, in the Presence of Natural Convection and the Diffusion of Heat and Matter

1 Sal'nikov's chemical reaction is very simple; it consists of two consecutive first-order steps, producing a product B from a precursor P via an active intermediate A, in P → A → B. The first step is assumed to be thermoneutral, with zero activation energy, whilst the second step is exothermic and has a positive activation energy. These properties make this mechanism one of the simplest to display thermokinetic oscillations, as seen in cool flames. We consider a pure gas, P, undergoing Sal'nikov's reaction in a closed spherical vessel, whose walls are held at a constant temperature. Natural convection becomes significant once the temperature is high enough for the Rayleigh number (Ra) to reach ~ 10 3. The subsequent behaviour of the system depends on the interaction between convection, diffusion of heat and mass, and chemical kinetics. By examining the governing equations, we develop and evaluate scales for the characteristic velocity, concentration of the intermediate A and the temperature rise during the progress of the reaction. These scales depend on the characteristic timescales for the interacting phenomena of chemical reaction, diffusion and natural convection. Our theoretical predictions are verified by full numerical simulation for the two limiting cases when transport is dominated, respectively, by diffusion and natural convection.