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

Hewitt, R.E. and Harrison, I. (2012) Exponential sensitivity to symmetry imperfections in an exact Navier–Stokes solution. Journal of Engineering Mathematics, 75. pp. 63-79. ISSN 1573-2703

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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.

Item Type: Article
Uncontrolled Keywords: Asymptotics Exponential Self-similar Exact Solution Navier-Stokes
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 76 Fluid mechanics
PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 40 ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS > 47 Fluid dynamics
Depositing User: Dr Richard E. Hewitt
Date Deposited: 14 Nov 2013
Last Modified: 20 Oct 2017 14:13
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2042

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