Pattern formation induced by a differential Poiseuille flow

Differential advection, where a reactant is advected while another one is immobilized, leads to instabilities in reaction-advection-diffusion systems. In particular, a homogeneous steady state looses stability for strong enough flows, leading to chemical patterns moving in the direction of the flow. In this paper we study the effects of differential advection due to a two-dimensional Poiseuille flow. We carry out a linear stability analysis on a homogeneous state using an activator-inhibitor reaction. We find that shear dispersion induced by the Poiseuille flow may lead to instabilities at slower flow rates. We find that contrary to the one-dimensional system, the instability depends on which substance is advected. We find a critical average flow speed for instability depending on tube size. Numerical solutions of the nonlinear reaction-advection-diffusion result in patterns of constant shape propagating along the tube.