Exponential sensitivity to symmetry imperfections in an exact Navier–Stokes solution

Abstract

We consider the (radial) stretching flow of an incompressible viscous fluid between two parallel plates. For infinite plates, a well-known self-similar solution reduces the Navier–Stokes equations to a simple nonlinear boundary-value problem. We demonstrate that, for large Reynolds numbers, a naïve matched asymptotic description of the self-similar flow yields a continuum of solutions. To describe which of the continuum of states is realised requires the inclusion of terms that are beyond all orders in the asymptotic description. Sensitivity to exponentially small terms in the asymptotic description has practical significance in that (i) exponentially small symmetry imperfections in the boundary conditions have a leading-order effect, and (ii) linearised perturbations are seen to decay only on exponentially long space/time scales owing to the presence of eigenmodes that are exponentially near neutral. The results of axisymmetric Navier–Stokes computations are presented to show that the asymptotic description of the self-similar states (and their stability) is of practical relevance to finite-domain solutions.