Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$

Abstract

Consider the reaction front formed when two initially separated reactants A and B are brought into contact and react at a rate proportional to $A^n B^m$ when the concentrations $A$ and $B$ are positive. Further, suppose that both $n$ and $m$ are less than unity. Then the leading order large time asymptotic reaction rate has compact support, i.e. the reaction zone where the reaction takes place has a finite width and the reaction rate is identically zero outside of this region. In the large time asymptotic limit an analytical approximate solution to the reactant concentrations is constructed in the vicinity of the reaction zone. The approximate solution is found to be in good agreement with numerically obtained solutions. For $n \ne m$ the location of the maximum reaction rate does not coincide with the centre of mass of the reaction, and further for $n>m$ this local maximum is shifted slightly closer to the zone that initially contained species A, with the reverse holding when $m>n$. The three limits $m\rightarrow 0$, $n\rightarrow 1$ and $m,n\rightarrow 1$ are given special attention.