Journal of Engineering Mathematics

, Volume 75, Issue 1, pp 63–79 | Cite as

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

  • Richard E. HewittEmail author
  • Iain Harrison


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.


Exact solutions Exponential asymptotics Stability Stagnation-point flows 


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  1. 1.
    Berman AS (1953) Laminar flow in channels with porous walls. J Appl Phys 24: 1232–1235MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Terrill RM (1966) Flow through a porous annulus. Appl Sci Res 17: 204–222CrossRefGoogle Scholar
  3. 3.
    von Kármán T (1921) Über laminare und turbulente Reibung. Z Angew Math 1: 233–252zbMATHGoogle Scholar
  4. 4.
    Brady JF, Acrivos A (1981) Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier–Stokes equations with reverse flow. J Fluid Mech 112: 127–150MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Riley N, Drazin PG (2006) The Navier–Stokes equations: a classification of flows and exact solutions. London Mathematical Society lecture note series 334. Cambridge University Press, CambridgeGoogle Scholar
  6. 6.
    Hall P, Papageorgiou DT (1999) The onset of chaos in a class of exact Navier–Stokes solutions. J Fluid Mech 393: 59–87MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Robinson WA (1976) The existence of multiple solutions for the laminar flow in a uniformly porous channel with suction at both walls. J Eng Math 10: 23–40zbMATHCrossRefGoogle Scholar
  8. 8.
    Cox SM, King AC (1997) On the asymptotic solution of a high-order nonlinear ordinary differential equation. Proc Roy Soc Lond A 453: 711–728MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Terrill RM (1973) On some exponentially small terms arising in flow through a porous pipe. QJMAM 26: 347–354zbMATHGoogle Scholar
  10. 10.
    Boyd JP (1999) The devil’s invention: asymptotic, superasymptotic and hyperasymptotic series. Acta Appl Math 56: 1–98MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gresho PM, Sani RL (1998) Incompressible flow and the finite element method. Wiley, ChichesterzbMATHGoogle Scholar
  12. 12.
    Heil M, Hazel AL (2006) oomph-lib: An object-oriented multi-physics finite-element library. In: Schafer M, Bungartz H-J (eds) Fluid–structure interaction. Springer, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.School of MathematicsUniversity of ManchesterManchesterUK

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