- A preview of this full-text is provided by Springer Nature.
- Learn more

Preview content only

Content available from Mediterranean Journal of Mathematics

This content is subject to copyright. Terms and conditions apply.

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].

Content courtesy of Springer Nature, terms of use apply. Rights reserved.