On the influence of boundary condition on stability of Hagen–Poiseuille flow

Abstract

We analyze the influence of choice of boundary condition (no-slip and Navier’s slip boundary conditions) on linear stability of Hagen–Poiseuille flow. Several heuristic arguments based on detailed analysis of spectrum of the Stokes operator are given, and it is concluded that Navier’s slip boundary condition should have a destabilizing effect on the flow. Finally the linear stability problem is solved by numerical means, and quantitative results confirming the heuristic prediction are obtained. It is shown that the destabilization is not strong enough to maintain an unstable disturbance, and that the significant destabilization effects of Navier’s slip boundary condition are restricted to small values of the Reynolds number. As a byproduct we obtain explicit formulas for eigenfunctions of the Stokes operator subject to Navier’s slip boundary condition.