Article

On the Filtration of Micropolar Fluid Through a Thin Pipe

Authors:
To read the full-text of this research, you can request a copy directly from the author.

Abstract

This paper reports the analytical results on the incompressible micropolar fluid flowing through a thin (or long) cylindrical pipe filled with porous medium. We start from the Brinkman-type system governing the filtration of the micropolar flow and perform the asymptotic analysis in the critical case characterized by the strong coupling between the velocity and microrotation. The error estimates are also derived providing the rigorous justification of the proposed effective model.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the author.

ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Inspired by the lubrication framework, in this paper we consider a micropolar fluid flow through a rough thin domain, whose thickness is considered as the small parameter ε while the roughness at the bottom is defined by a periodical function with period of order ε ℓ and amplitude ε δ , with δ> ℓ >1. Assuming nonzero boundary conditions on the rough bottom and by means of a version of the unfolding method, we identify a critical case δ = 3/2 ℓ − 1/2 and obtain three macroscopic models coupling the effects of the rough bottom and the nonzero boundary conditions. In every case we provide the corresponding micropolar Reynolds equation. We apply these results to carry out a numerical study of a model of squeeze-film bearing lubricated with a micropolar fluid. Our simulations reveal the impact of the roughness coupled with the nonzero boundary conditions on the performance of the bearing, and suggest that the introduction of a rough geometry may contribute to enhancing the mechanical properties of the device.
Article
Full-text available
This paper considers membranes of globular structure in the framework of the cell model technique. The flow of a micropolar fluid through a spherical cell consisting of a solid core, porous layer and liquid envelope is modeled using coupled micropolar and Brinkman-type equations. The solution is obtained in analytical form. Boundary value problems with different conditions on the hypothetical cell surface are considered and compared. The hydrodynamic permeability of the membrane is investigated as a function of micropolar and porous media characteristics.
Article
Full-text available
The lubrication theory is mostly concerned with the behaviour of a lubricant flowing through a narrow gap. Motivated by the experimental findings from the tribology literature, we take the lubricant to be a micropolar fluid and study its behaviour in a thin domain with rough boundary. Instead of considering (commonly used) simple zero boundary conditions, we impose physically relevant (nonzero) boundary conditions for the microrotation and perform the asymptotic analysis of the corresponding 3D boundary value problem. We formally derive a simplified mathematical model acknowledging the roughness-induced effects and the effects of the nonzero boundary conditions on the macroscopic flow. Using the obtained asymptotic model, we study numerically the influence of the roughness on the performance of a linear slider bearing. The numerical results clearly indicate that the use of the rough surfaces may contribute to enhance the mechanical performance of such device.
Article
Full-text available
Metallic nanoparticles effect on magnetohydrodynamicsc (MHD) micropolar blood flow through a vertical artery with six different stenosis is investigated. Conservation of mass, momentum, and energy governing partial differential equations are transformed into ordinary differential equations by means of mild stenosis assumptions. Solutions for velocity, microrotation, stream function, temperature, resistance impedance, and wall shear stress are calculated and expressed through graphs against various emerging physical parameters. It is observed that as the nanofluid volume fraction ϕ increases, the velocity and wall shear stress increase, while resistance impedance has an inverse trend. It is also found that as the nanofluid volume fraction ϕ increases, the temperature decreases. Moreover, the trapping phenomena in the stenosed region are introduced through graphs. These findings illustrate that nanoparticles’ technique could be a promising therapeutic strategy against arterial diseases.
Article
Full-text available
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.
Article
Full-text available
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.
Article
Full-text available
This paper considers the stationary flow of incompressible micropolar fluid through a thin cylindrical pipe governed by the pressure drop between pipe's ends. Its goal is to investigate the influence of the viscosity coefficients on the effective flow. Depending on the magnitude of viscosity coefficients with respect to the pipe's thickness, it derives different asymptotic models and discusses their properties.
Article
Full-text available
The aim of this paper is to present the result about asymptotic approximation of the micropolar fluid flow through a thin (or long) straight pipe with variable cross section. We assume that the flow is governed by the prescribed pressure drop between pipe's ends. Such model has relevance to some important industrial and engineering applications. The asymptotic behavior of the flow is investigated via rigorous asymptotic analysis with respect to the small parameter, being the ratio between pipe's thickness and its length. In the case of circular pipe, we obtain the explicit formulae for the approximation showing explicitly the effects of microstructure on the flow. We prove the corresponding error estimate justifying the obtained asymptotic model.
Article
Full-text available
Field equations governing the steady flow of an incompressible micro-polar fluid through isotropic porous sediments are derived using intrinsic volume averaging. The model equations might be of applicability to the study of lubrication problems in configurations involving porous linings, and to the study of polymer and oil flow through porous structures.
Article
Full-text available
A micropolar model for axisymmetric blood flow through an axially nonsymmetreic but radially symmetric mild stenosis tapered artery is presented. To estimate the effect of the stenosis shape, a suitable geometry has been considered such that the axial shape of the stenosis can be changed easily just by varying a parameter (referred to as the shape parameter). The model is also used to study the effect of the taper angle f{\phi} . Flow parameters such as the velocity, the resistance to flow (the resistance impedance), the wall shear stress distribution in the stenotic region and its magnitude at the maximum height of the stenosis (stenosis throat) have been computed for different values of the shape parameter n, the taper angle f{\phi} , the coupling number N and the micropolar parameter m. It is shown that the resistance to flow decreases with increasing the shape parameter n and the micropolar parameter m while it increases with increasing the coupling number N. So, the magnitude of the resistance impedance is higher for a micropolar fluid than that for a Newtonian fluid model. Finally, the velocity profile, the wall shear stress distribution in the stenotic region and its magnitude at the maximum height of the stenosis are discussed for different values of the parameters involved on the problem.
Article
Full-text available
In this paper we study the flow of incompressible Newtonian fluid through a helical pipe with prescribed pressures at its ends. Pipe’s thickness and the helix step are considered as the small parameter ɛ. By rigorous asymptotic analysis, as ɛ→ 0 , the effective behaviour of the flow is found. The error estimate for the approximation is proved.
Article
The present paper considers the flow of micropolar fluid through a membrane modeled as a swarm of solid cylindrical particles with porous layer using the cell model technique. Traditional boundary conditions on hypothetical cell surface were added with an additional condition: the no spin condition / no couple stress condition. Expressions for velocity and microrotation vector components have been obtained analytically. Effect of various parameters such as particle volume fraction, permeability parameter, micropolarity number etc. on hydrodynamic permeability of membrane has been discussed.
Article
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.
Article
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.
Article
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.
Article
A thin micropolar fluid with new boundary conditions at the fluid-solid interface, linking the velocity and the microrotation by introducing a so-called "boundary viscosity" is presented. The existence and uniqueness of the solution is proved and, by way of asymptotic analysis, a generalized micropolar Reynolds equation is derived. Numerical results show the influence of the new boundary conditions for the load and the friction coefficient. Comparisons are made with other works retaining a no slip boundary condition.
Article
It is known that the asymptotics of slow fluid flow past an array of fixed obstacles is described in several situations by Brinkman's law or Darcy's law (with or without interaction between different obstacles). We show that there exists a continuous transition between the asymptotic structures corresponding to the different situations. Consequently, the asymptotic structure for concentrations of particles 0(1) gives by a limit process the asymptotic schemes for small concentration. Some results of existence, uniqueness and symmetry of the translation tensor are given for flow in 2 dimensions.
Article
Slow flow of an incompressible viscous fluid is studied in an array of a great number of small fixed solid particles. The particles size ϵ and the distance η between two neighbouring solids are such that ϵ⪡η⪡1. Using perturbation methods it is proved that Brinkman's law occurs really for a critical size of particles; for larger particles the fluid filtration is governed by the Darcy's law and smaller solids do not influence the flow. The 3 and 2-dimensional cases are studied.
Article
Equations of motion, constitutive equations and boundary conditions are derived for a class of fluids named micropolar fluids. These fluids respond to micro-rotational motions and spin inertia and therefore can support couple stress and distributed body couples. Thermodynamical restrictions are studied in detail and field equations are obtained for the density, velocity vector and micro-rotation vector. The system is solved for a channel flow exhibiting certain interesting phenomena.
Article
This paper is concerned with an asymptotic approach for a micropolar flow through a thin curvilinear channel. A priori estimates (which we obtain together with the existence and the uniqueness of the solution) are used to establish the error between the exact solution and the asymptotic one and to justify the asymptotic analysis. We obtain the expression of an expansion of order K and we study the general problems for the boundary layer functions. Under some additional assumptions on the data we obtain satisfactory error estimates.
Article
A calculation is given of the viscous force, exerted by a flowing fluid on a dense swarm of particles. The model underlying these calculations is that of a spherical particle embedded in a porous mass. The flow through this porous mass is decribed by a modification of Darcy's equation. Such a modification was necessary in order to obtain consistent boundary conditions. A relation between permeability and particle size and density is obtained. Our results are compared with an experimental relation due to Carman.
Article
We study the stationary motion of a micropolar fluid in a thin (or long) curved pipe via rigorous asymptotic analysis. An asymptotic solution is found, showing explicitly the effects of pipe’s distortion and microstructure on the effective behavior of the flow. We justify the obtained model by proving the corresponding error estimate. KeywordsMicropolar fluid–Curved pipe–Curvilinear coordinates–Asymptotic expansion
Article
This paper treats the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. (For example, in the three-dimensional case, the obstacles have a size of 3 and are located at the nodes of a regular mesh of size .) A suitable extension of the pressure is used to prove the convergence of the homogenization process to a Brinkman-type law (in which a linear zero-order term for the velocity is added to a Stokes or Navier-Stokes equation).
Article
Different laws are used for modeling flows in porous media. In this paper, we focus on Brinkman and Darcy law. We derive them from microscopic equations by upscaling, compare them and estimate the error made by their application. Our results justify the use of Brinkman law.
Fundamental equations of the theory of elastic media with rotationally interacting particles
  • E L Aero
  • E V Kuvshinsky
Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations
  • J G Heywood
  • R Rannacher
  • S Turek