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Leray’s Problem for the Nonstationary Micropolar Fluid Flow

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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.
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., [25]). Rigorous, mathematical justiﬁcation of various eﬀective models
describing ﬂow of a micropolar ﬂuid can be found in [611]. A detailed survey
of the mathematical theory underlying the micropolar ﬂuid model can be
found in the monograph [12].
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