An exponential decay estimate for the stationary axisymmetric perturbation of Poiseuille flow in a circular pipe

We prove a decay estimate for the steady state incompressible Navier-Stokes equations. The estimate describes the exponential decay, in the axial direction of a semi-infinite circular tube, for an energy-type functional in terms of the axisymmetric perturbation of Poiseuille flow, provided that the Reynolds number does not exceed a critical value, for which we exhibit a lower and an upper bound. Since the motion is considered axisymmetric we use a stream function formulation, and the results are similar to those obtained by Horgan 8], for a two-dimensional channel flow problem. For the Stokes problem our estimate for the rate of decay is a lower bound to the actual rate of decay which is obtained from an asymptotic solution to the Stokes equations. Finally we describe a numerical approach to computing bounds to the energy functionalE(0).