The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. II. An initial-value problem with an immobilized or nearly immobilized autocatalyst

Abstract

Abstract We study the isothermal autocatalytic reaction schemes, A + B -> 2B (quadratic autocatalysis), and A + 2B → 3B (cubic autocatalysis), where A is a reactant and B is an autocatalyst. We consider the situation when a quantity of B is introduced locally into a uniform expanse of A, in one-dimensional slab geometry. In addition, we allow the chemical species A and B to have unequal diffusion rates DA and DB respectively, and study the two closely related cases, (DB/ DA) = 0 and 0 < (DB/ DA) < 1. When (Db/Da) = 0 a spike forms in the concentration of B, which grows indefinitely, and we can obtain both large and small time asymptotic solutions. For 0 < (DB/ DA) < 1 there is a long induction period during which a large spike forms in the concentration of B, before a minimum speed travelling wave is generated. We can relate the results for case (DB/DA) = 0 to the solution when 0 < (DB/ DA) < 1 to obtain detailed information about its behaviour.