Asymptotic solutions of stationary patterns in convection-reaction-diffusion systems

We study and map the possible stationary patterns that emerge in a convection-reaction-diffusion (CRD) system using a learning polynomial kinetics. We classify the patterns according to the kinetic model (oscillatory, bistable, or intermediate), the instability nature of the bounded system (convective or absolute), the applied boundary conditions and the system length. This analysis presents a unifying approach to various pattern-inducing mechanisms such as DIFICI (differential flow induced chemical instability), which predicts moving patterns in systems with wide difference of convective rates, and differential capacity patterns, which predicts stationary patterns in cross-flow reactors with a large heat capacity. Previous studies of CRD systems have considered only oscillatory kinetics. Nonlinear analysis, which follows the front motion by approximating its velocity, accounts for the stability of the stationary, whether spatially periodic or other, patterns. The most dominant state is the large-amplitude stationary spatially periodic pattern. With oscillatory kinetics these emerge in the convectively unstable domain above the amplification threshold. The domain of absolute instability, which is determined analytically for unbounded systems, is divided in the bounded system into two subdomains with moving DIFICI waves or stationary patterns. With bistable kinetics the large-amplitude stationary patterns can be sustained only within a narrow subdomain but other stationary patterns, that incorporate several fronts upstream and an "almost homogeneous" tail downstream, can be sustained as well. With intermediate kinetics the large-amplitude axisymmetric stationary patterns may coexist with small-amplitude stationary nonaxisymmetric patterns.