Instability of Hagen-Poiseuille flow for non-axisymmetric mode

An investigation is described for instability problem of flow through a pipe of circular cross-section. As a disturbance motion, we consider a general non-axisymmetric mode. An associated amplitude or modulation equation has been derived for this disturbance motion. This equation belongs to a diffusion type. The coefficient of it can be negative while Reynolds number increases, because of the complex interaction between molecular diffusion and convection. The negative diffusivity, when it occurs, causes a concentration and focussing of energy within decaying slugs, acting as a role of reversing natural decays.