Nonlinear instability of flow in a rectangular pipe with large aspect ratio

The stability of flow in a rectangular pipe with large aspect ratio can be studied by regarding the flow as a perturbation of plane Poiseuille flow. In the central region of the pipe, the disturbance has the form of a slowly modulated wave. The amplitude equation for this wave at slightly supercritical Reynolds numbers is identical with that for the unbounded flow. The regions in the vicinity of the side walls are studied and the boundary conditions for the amplitude equation at the walls are obtained. It is shown that the critical Reynolds number is only increased by a term proportional to (depth/width)², so that even moderate aspect ratios should be suitable for studying unbounded plane Poiseuille flow instability.