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Leray’s Problem for the Nonstationary Micropolar Fluid Flow

Authors:
Michal Benes at Czech Technical University in Prague
Michal Benes
Igor Pazanin at University of Zagreb
Igor Pazanin
Marko Radulović at University of Zagreb
Marko Radulović
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Abstract

Motivated by the pipe network problems, in this paper, we consider the Leray’s problem for the nonstationary flow of a micropolar fluid. We prove that in an unbounded domain with cylindrical outlets to infinity, there exists a unique solution to the nonlinear micropolar equations which exponentially tends to the generalized nonstationary micropolar Poiseuille solution in each cylindrical outlet.
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Content available from Mediterranean Journal of Mathematics
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Mediterr. J. Math. (2020) 17:50
https://doi.org/10.1007/s00009-020-1493-9
1660-5446/20/020001-32
published online February 20, 2020
c
Springer Nature Switzerland AG 2020
Leray’s Problem for the Nonstationary
Micropolar Fluid Flow
Michal Beneˇs, Igor Paˇzanin and Marko Radulovi´c
Abstract. Motivated by the pipe network problems, in this paper, we
consider the Leray’s problem for the nonstationary flow of a micropolar
fluid. We prove that in an unbounded domain with cylindrical outlets
to infinity, there exists a unique solution to the nonlinear micropolar
equations which exponentially tends to the generalized nonstationary
micropolar Poiseuille solution in each cylindrical outlet.
Mathematics Subject Classification. 35Q35, 76A05, 76D03.
Keywords. Micropolar fluid, nonstationary flow, Leray’s problem,
solvability, pipe networks.
1. Introduction
The micropolar fluid model has been introduced in mid 60s by Eringen in his
famous paper [1]. The main advantage of the micropolar model, as compared
to the classical Navier–Stokes system, lies in the fact that it takes into account
the microstructure of the fluid. More precisely, to capture the effects such as
rotation and shrinking of the particles, the microrotation field is introduced
(along with the standard velocity and pressure fields) and, accordingly, a new
vector equation coming from the conservation of angular momentum. In this
way, numerous non-Newtonian fluids (such as liquid crystals, animal blood,
muddy fluids, certain polymeric fluids, and even water in models with small
scales) can be successfully described by the coupled system of micropolar
equations. This has been acknowledged by the engineering community and,
thus, micropolar fluid flows have been extensively investigated in the last
decade. In particular, one can find a vast amount of recent results concerning
engineering applications in biomedicine, mostly in blood flow modeling (see,
e.g., [2–5]). Rigorous, mathematical justification of various effective models
describing flow of a micropolar fluid can be found in [6–11]. A detailed survey
of the mathematical theory underlying the micropolar fluid model can be
found in the monograph [12].
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