Response curves for plane Poiseuille flow

Flow through a plane channel, in which the external force is the pressure gradient and the response function is the mass flux, is considered. The dimensionless response function relates the friction factors to the Reynolds numbers. The statistically stationary solutions of the Navier-Stokes equations for flow through a plane channel are studied in order to obtain the response function. The horizontal average of such solutions is found to be time independent. It is shown that this is true of the time-periodic motion which bifurcates from laminar Poiseuille flow, and it is assumed that other solutions observed as turbulence are statistically stationary. The concept of stable turbulence is developed in order to explain the response diagram.