Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe

Abstract

The nonsteady Navier–Stokes equations are considered in a thin infinite pipe with the small diameter ϵ in the case of the Reynolds number of order ϵ. The time-dependent flow rate is a given function. The complete asymptotic expansion is constructed and justified. The error estimate of order O(ϵ ^J ) for the difference of the exact solution and the J-th asymptotic approximation is proved for any real J.

Keywords:

• Navier--Stokes equations
• Poiseuille flow
• thin pipe
• asymptotics

AMS Subject Classifications::

• 76M45
• 35B27

Acknowledgements

This work is partially supported by the grant no. 02.270.11.5091 ‘Multiscale models in physics, biology and technologies: asymptotic and numerical analysis’ of Russian Federal Agency for Sciences and Innovations, by French-Russian PICS CNRS grant ‘Modelling of blood diseases’ and by French Research and Education Ministry grant MODMAD.

Notes

Note

1. Formulae (3.14) were derived by a student of Vilnius University V. Pilipauskaite during her summer practice supported by the Lithuanian Science Council Student Research Fellowship Award.