Effective flow of a viscous liquid through a helical pipe

Eduard Marušić-Paloka; Igor Pažanin

doi:10.1016/j.crme.2004.09.005

Effective flow of a viscous liquid through a helical pipe

Presented by: Évariste Sanchez-Palencia

Eduard Marušić-Paloka ¹; Igor Pažanin ¹

¹ Department of Applied Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia

Comptes Rendus. Mécanique, Volume 332 (2004) no. 12, pp. 973-978.

Abstract
Résumé

We study the flow of a viscous fluid through a pipe with helical shape parameterized with $r_{ɛ} (x_{1}) = (x_{1}, a \cos \frac{x_{1}}{ɛ}, a \sin \frac{x_{1}}{ɛ})$ , where the small parameter ɛ stands for the distance between two coils of the helix. The pipe has small cross-section of size ɛ. Using the asymptotic analysis of the microscopic flow described by the Navier–Stokes system, with respect to the small parameter ɛ that tends to zero, we find the effective fluid flow described by an explicit formula of the Poisseuile type including a small distorsion due to the particular geometry of the pipe.

On considère un écoulement dans un tube de section circulaire et de forme hélicoïdale paramétré par $r_{ɛ} (x_{1}) = (x_{1}, a \cos \frac{x_{1}}{ɛ}, a \sin \frac{x_{1}}{ɛ})$ , où ɛ est la distance entre deux tours de la spirale. Le rayon de la section du tube est lui aussi supposé égal à ɛ. A partir de l'écoulement microscopique décrit par le système de Navier–Stokes et en utilisant l'analyse asymptotique par rapport à ce petit paramètre ɛ on obtient l'écoulemment effectif décrit par une formule explicite de type Poiseuille associée à une petite déviation due à la géometrie du tube.

Received: 2004-07-21
Accepted: 2004-09-21
Published online: 2004-11-02

DOI: 10.1016/j.crme.2004.09.005

Keywords: Fluid mechanics, Helical pipe, Navier–Stokes equations
Mot clés : Mécanique des fluides, Tube hélicoïdale, Équations de Navier–Stokes

Author's affiliations:

Eduard Marušić-Paloka ¹; Igor Pažanin ¹

¹ Department of Applied Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia

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Eduard Marušić-Paloka; Igor Pažanin. Effective flow of a viscous liquid through a helical pipe. Comptes Rendus. Mécanique, Volume 332 (2004) no. 12, pp. 973-978. doi : 10.1016/j.crme.2004.09.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2004.09.005/

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[3] G. Panasenko Asymptotic expansion of the solution of the Navier–Stokes equation in a tube structure, C. R. Acad. Sci. Paris, Ser. IIb, Volume 326 (1998), pp. 867-872

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[5] E. Marušić-Paloka Rigorous justification of the Kirchhoff law for junction of thin pipes filled with viscous fluid, Asymptotic Analysis, Volume 33 (2003) no. 1, pp. 51-66

[6] E. Marušić-Paloka Incompressible newtonian flow through thin pipes, Dubrovnik 2001, Kluwer (2003), pp. 123-142

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