Effective Flow of Incompressible Micropolar Fluid Through a System of Thin Pipes
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Effective Flow of Incompressible Micropolar Fluid Through a System of Thin Pipes Michal Beneš1 · Igor Pažanin2
Received: 30 March 2015 / Accepted: 9 October 2015 © Springer Science+Business Media Dordrecht 2015
Abstract In this paper, we consider the incompressible micropolar fluid flowing through a multiple pipe system via asymptotic analysis. Introducing the ratio between pipes thickness and its length as a small parameter ε, we propose an approach leading to a macroscopic model describing the effective flow. In the interior of each pipe (far from the junction), we deduce that the fluid behavior is different depending on the magnitude of viscosity coefficients with respect to ε. In particular, we prove the solvability of the critical case characterized by the strong coupling between velocity and microrotation. In the vicinity of junction, an interior layer is observed so we correct our asymptotic approximation by solving an appropriate micropolar Leray’s problem. The error estimates are also derived providing the rigorous mathematical justification of the constructed approximation. We believe that the obtained result could be instrumental for understanding the microstructure effects on the fluid flow in pipe networks. Keywords Micropolar fluid · Junction of thin pipes · Strong coupling · Micropolar Leray problem · Asymptotic analysis
1 Introduction The theory of micropolar fluids was introduced by Eringen [1] in 60’s and since then it has attracted much attention both in engineering and mathematical community. That is due to the fact that micropolar fluid model successfully describes some physical phenomena which can not be captured by the classical Navier-Stokes equations ignoring the fluid’s internal
B I. Pažanin
[email protected] M. Beneš [email protected]
1
Department of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic
2
Department of Mathematics, Faculty of Science, University of Zagreb, Bijeniˇcka 30, 10000 Zagreb, Croatia
M. Beneš, I. Pažanin Fig. 1 The domain considered (for m = 3)
structure. Indeed, if we consider fluids whose particles have complex shapes (e.g. liquid crystals, muddy fluids, animal blood, certain polymeric fluids even water in models with small scales), fluid particles can exhibit some microscopical effects (e.g. rotation, shrinking) that should definitely be taken into account. From the physical point of view, micropolar fluids consist of rigid, spherical particles suspended in a viscous medium characterized by the following main property: the individual particles can rotate, independently of the movement of the fluid as whole. The rotation is being described by a new vector field, the angular velocity field of rotation of particles called microrotation. Correspondingly, one new vector equation (coming from the conservation of the angular momentum) is coupled with the Navier-Stokes equations with four new viscosities introduced. As a result, a non-Newtonian mathematical model is obtained describi
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