Asymptotic Analysis of a Viscous Flow in a Curved Pipe with Elastic Walls

  • Gonzalo CastiñeiraEmail author
  • José M. Rodríguez
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 8)


This communication is devoted to the presentation of our recent results regarding the asymptotic analysis of a viscous flow in a tube with elastic walls. This study can be applied, for example, to the blood flow in an artery. With this aim, we consider the dynamic problem of the incompressible flow of a viscous fluid through a curved pipe with a smooth central curve. Our analysis leads to the obtention of an one dimensional model via singular perturbation of the Navier-Stokes system as \(\varepsilon\), a non dimensional parameter related to the radius of cross-section of the tube, tends to zero. We allow the radius depend on tangential direction and time, so a coupling with an elastic or viscoelastic law on the wall of the pipe is possible. To perform the asymptotic analysis, we take a change of variable to a reference domain where we assume the existence of asymptotic expansions on \(\varepsilon\) for both velocity and pressure which, upon substitution on Navier-Stokes equations, leads to the characterization of various terms of the expansion. This allows us to obtain an approximation of the solution of the Navier-Stokes equations.


Asymptotic Expansion Asymptotic Analysis Pipe Wall Transversal Velocity Rigid Wall 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This research was partially supported by Ministerio de Economía y Competitividad under grant MTM2012-36452-C02-01 with the participation of FEDER.


  1. 1.
    Castiñeira, G., Rodríguez, J.M.: Asymptotic analysis of a viscous flow in a curved pipe with moving walls. arXiv:1602.06121 (
  2. 2.
    Formaggia, L., Lamponi, D., Quarteroni, A.: One-dimensional models for blood flow in arteries. J. Eng. Math. 47, 251–276 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gammack, D., Hydon, P.E.: Flow in pipes with non-uniform curvature and torsion. J. Fluid Mech. 433, 357–382 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lyne, W.H.: Unsteady viscous flow in a curved pipe. J. Fluid. Mech. 45, 13–31 (1970)CrossRefzbMATHGoogle Scholar
  5. 5.
    Marušić-Paloka, E.: The effects of flexion and torsion on a fluid flow through a curved pipe. Appl. Math. Optim. 44, 245–272 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Marušić-Paloka, E., Pažanin, I.: Fluid flow through a helical pipe. Z. Angew. Math. Phys. 58, 81–89 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Panasenko, G.P., Pileckas, K.: Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary-layer-in-time. Nonlinear Anal. 122, 125–168 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Panasenko, G.P., Pileckas, K.: Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. II. General case. Nonlinear Anal. 125, 582–607 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Panasenko, G.P., Stavre, R.: Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall. J. Math. Pures Appl. 85, 558–579 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pedley, T.J.: Mathematical modelling of arterial fluid dynamics. J. Eng. Math. 47, 419–444 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Riley, N.: Unsteady fully-developed flow in a curved pipe. J. Eng. Math. 34, 131–141, (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Smith, F.T.: Fluid flow into a curved pipe. Proc. R. Soc. Lond. A 351, 71–87 (1976)CrossRefzbMATHGoogle Scholar
  13. 13.
    Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, New York (2000)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Facultad de Matemáticas, Departamento de Matemática AplicadaUniv. de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Departamento de Métodos Matemáticos y de Representación, E.T.S. ArquitecturaUniversidade da CoruñaA CoruñaSpain

Personalised recommendations