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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
published online February 20, 2020
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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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 (Formula presented.), 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 (Formula presented.). 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.
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We study the junction of m pipes that are either thin or long (i.e., they have small ratio between the cross-section and the length, denoted by l). Pipes are filled with an incompressible Newtonian fluid and the values of the pressure π at the end of each pipe are prescribed. By rigorous asymptotic analysis, we justify the analog of the Kirchhoff law for computing the junction pressure. In the interior of each pipe the effective flow is the Poiseuille flow governed by the pressure drop between the end of the pipe and the junction point. The pressure at the junction point is equal to a weighted mean value of the prescribed π ’s (Kirchhoff law). In the vicinity of the junction there is an interior layer, with thickness llog(1/l). To get a better approximation and to control the velocity gradient in the vicinity of the junction, a first order asymptotic approximation has to be corrected by solving an appropriate Leray problem. We prove the asymptotic error estimate for the approximation.
In this paper we study a time-dependent flow of an incompressible micropolar fluid through a pipe with arbitrary cross-section. The effective behavior of the flow is found by means of rigorous asymptotic analysis with respect to the small parameter representing the pipe's thickness. The complete asymptotic expansion (up to an arbitrary order) of the solution is constructed with detailed analysis of the boundary layers provided. The convergence of the expansion is proved justifying the usage of the formally derived asymptotic model.
In this study a mathematical model for two-dimensional pulsatile blood flow through overlapping constricted tapered vessels is presented. In order to establish resemblance to the in vivo conditions, an improved shape of the time-variant overlapping stenosis in the elastic tapered artery subject to pulsatile pressure gradient is considered. Because it contains a suspension of all erythrocytes, the flowing blood is represented by micropolar fluid. By applying a suitable coordinate transformation, tapered cosine-shaped artery turned into non-tapered rectangular and a rigid artery. The governing nonlinear partial differential equations under the imposed realistic boundary conditions are solved using the finite difference method. The effects of vessel tapering on flow characteristics considering their dependencies with time are investigated. The results show that by increasing the taper angle the axial velocity and volumetric flow rate increase and the microrotational velocity and resistive impedance reduce. It has been shown that the results are in agreement with similar data from the literature.
We consider an initial boundary-value problem for the system of equations describing nonstationary flows of asymmetric fluids. In an earlier article [ibid. 12, No.1, 83-97 (1988; Zbl 0668.76045)] we established the existence of a weak solutio of the problem in the interval (O,T), where T is an arbitrary positive real. In the present paper we prove the existence of a unique global in time solution, provided the viscosity ν is greater than some ν * >0 and the data is “small” in comparison with ν. We show also that when no external forces act then the solution decays to zero when t→∞.
The steady motion of a micropolar fluid through a wavy tube with the dimensions depending on a small parameter is studied. An asymptotic expansion is proposed and error estimates are proved by using a boundary layer method. We apply the method of partial asymptotic decomposition of domain and we prove that the solution of the partially decomposed problem represents a good approximation for the solution of the considered problem.
In this paper, the flow of blood through catheterized artery with mild constriction at the outer wall is considered. The closed form solutions are obtained for velocity and microrotation components. The impedance (resistance to the flow) and wall shear stress are calculated. The effects of catheterization, coupling number, micropolar parameter, and height of the stenosis on impedance and wall shear stresses are discussed.