- Research
- Open Access
The solutions for the flow of micropolar fluid through an expanding or contracting channel with porous walls
- Xinhui Si^{1}Email author,
- Mingyang Pan^{1},
- Liancun Zheng^{1},
- Jianhui Zhou^{2} and
- Lin Li^{1}
- Received: 28 March 2016
- Accepted: 21 September 2016
- Published: 30 September 2016
Abstract
The unsteady, two-dimensional laminar flow of an incompressible micropolar fluid in a channel with expanding or contracting porous walls is investigated. The governing equations are transformed into a coupled nonlinear two-points boundary value problem by a suitable similarity transformation. Unlike the classic Berman problem (Berman in J. Appl. Phys. 24:1232-1235, 1953), three new solutions (totally six solutions) and no-solution interval, which is one of important characteristics for the laminar flow through porous pipe with stationary wall (Terrill and Thomas in Appl. Sci. Res. 21:37-67, 1969), are found numerically for the first time. The multiplicity of the solutions is strictly dependent on the expansion ratio. Furthermore, the asymptotic solutions are constructed by the Lighthill method, which eliminates the singularity of the similarity solution, for large injection and by the matching theorem for the suction Reynolds number, respectively. The analytical solutions also are compared with the numerical ones and the results agree well.
Keywords
- micropolar fluid
- expansion ratio
- multiple solutions
- Lighthill method
- matching theorem
- boundary value problem
1 Introduction
The studies of laminar flow in an expanding or contracting channel or pipe with permeable wall have received considerable attentions due to its application in biophysical flows in recent years. The pioneering work may have been done by Uchida and Aoki [3] to simulate the successive peristaltic motion of the artery, who examined the viscous flow inside an impermeable tube with a contracting cross section. Later, Goto and Uchida [4] analyzed the incompressible laminar flow in a semi-infinite porous pipe whose radius varied with time. In addition, further experiments carried out by Wilens [5], Evans et al. [6], and Michel [7] proved the endothelial walls to be permeable with ultra-microscopic pores, through which the mass transfer takes place between blood, air, and tissue. It has also been pointed out that the deposition of cholesterol and the dilated damaged and inflamed arterial walls may cause an increase in its permeability. Recently, Majdalani et al. [8–10] investigated the flow of the fluid through an expanding channel with porous walls by numerical or asymptotical methods and analyzed the influence of the permeability and the expansion ratio on the velocity and pressure distribution.
In general, blood flow is regarded as a non-Newtonian fluid through a pipe or channel. The rheological properties of the blood are mainly due to the suspension of red blood cells, white cells, and platelets in the blood plasma. These properties strongly affect the dynamics of blood flow and render blood a non-Newtonian fluid [11]. The theory of micopolar fluids proposed by Eringen [12, 13] is one of the best theories of fluids to describe the deformation of such materials. Physically these fluids may represent the fluids consisting of rigid randomly oriented particles suspended in a viscous medium undergoing both translational and rotational motion. Furthermore, experimental studies have revealed that the blood’s viscosity decreases with the increase in shear rate, and that blood has a small yield stress. Airman et al. [14, 15] made excellent reviews about the applications of micropolar fluids. Pazanin [16] presented the result as regards an asymptotic approximation of the micropolar fluid flow through a thin or long straight pipe with variable cross section and explicitly addressed the effects of the microstructure on the flow. Ziabakhsh and Domairry [17] and Joneidi et al. [18] discussed the micropolar fluid in a porous channel by a homotopy analysis method (HAM). Ramachandran et al. [19] studied the heat transfer of a micropolar fluid past a curved surface with suction and injection using Van Dyke’s singular perturbation technique. Si et al. [20] investigated the mass and heat transfer of the micropolar fluid through an expanding channel with porous walls by HAM and analyzed the effects of the heat dissipation on the velocity and temperature distribution.
Apart from the above work, solution multiplicity is a classic problem for the equations describing the flow in channels or pipes with porous walls. A similar problem has been studied by Berman [1] and Terrill [21], who assumed that the wall is stationary. The multiplicity of the solutions has been addressed by Terrill [21], Terrill and Thomas [2], Robinson [22], MacGillivray and Lu [23], Lu et al. [24], and Zaturska et al. [25]. More recently, a theoretical treatment was done by Xu et al. [26], who revisited the Newtonian fluid in the channel with porous orthogonally moving walls by homotopy analysis method (HAM), and the results have shown that there exist dual or triple solutions corresponding to the cases reported by Zaturska et al. [25]. Almost at the same time, Si et al. [27, 28] also obtained the dual solutions of the flow in the expanding pipe and channel with porous walls, where they consider the small expansion ratio and large suction Reynolds number by singular perturbation methods. In addition, some other similar numerical and experimental studies also have been reported on multiple solutions. For example, Luo and Pedley [29] investigated two-dimensional limitation flow in a collapsible channel and multiple steady solutions were found for a range of physical parameters. Siviglia and Toffolon [30] discussed one-dimensional flow through a collapsible tube and found that geometrical alterations or variations of the mechanical properties of the tube wall affected the occurrence of multiple flow states. Lanzerstorfer and Kuhlmann [31] studied numerically the global stability of multiple solutions for the flow in a sudden expansion plane. Putkaradze and Vorobieff [32] made some experiments to show the multiple solutions of the flow in an expanding channel.
Motivated by the above-mentioned studies, the aim of this paper is to investigate the flow of an incompressible micropolar fluid in a channel with deforming porous walls. The paper is organized as follows. In Section 2 we introduce the laminar flow equations and the similarity transformation which reduce the governing equations to a system of high order nonlinear ordinary differential ones. In Section 3 we discuss the characteristics of different types of solution obtained by Bvp4c, which is a collocation method equivalent to the fourth order monoimplicit-Runge-Kutta method. In Section 4 we construct the asymptotic solutions with large injection or suction velocity, which are compared with the numerical ones obtained in Section 3. In Section 5 the conclusion is drawn.
2 Governing equations
3 Numerical results and discussions
As mentioned before, multiple solutions associated with fluid mechanics have been attracted many researchers’ attention. Recently, Yao [34] claimed that the Reynolds number is insufficient to determine a flow field uniquely for a given geometry. The non-uniqueness of fluid flow means that the different frequencies of flows can exist on each stable bifurcation branch. Here the multiple solutions are obtained numerically by Bvp4c [35] and we set the maximum residual error as 10^{−4} during the computation.
- (i)
- (ii)
- (iii)
For some values of the positive expansion ratio (i.e. \(\alpha=5\)), the interval \((2.72,9.77)\) of no solution appears. This phenomenon is similar to the case of the Newtonian flow through a stationary pipe with porous wall.
- (iv)
4 Perturbation analysis for this problem
4.1 Solution for the large injection Reynolds number
The singularity is due to large blowing at each wall which pushes the shear layer to the core region [36]. In order to eliminate the singular behavior, we employ the Lighthill method to obtain the asymptotic solution which is valid to arbitrary order as regards the derivative of the axial velocity.
4.1.1 A. The transformed boundary condition at the wall
4.1.2 B. The transformed boundary condition at the core
The comparison between asymptotic and numerical values of \(\pmb{-f''(1)}\) for large injection Reynolds number as \(\pmb{K=0.1}\) , \(\pmb{N_{1}=0.1}\)
Re | α = −5 | α = −2 | α = 2 | α = 5 | ||||
---|---|---|---|---|---|---|---|---|
Asy. | Num. | Asy. | Num. | Asy. | Num. | Asy. | Num. | |
−100 | 2.6864 | 2.6398 | 2.5694 | 2.5479 | 2.4252 | 2.4347 | 2.3251 | 2.3538 |
−110 | 2.6648 | 2.6236 | 2.5597 | 2.5405 | 2.4291 | 2.4367 | 2.3377 | 2.3636 |
−125 | 2.6394 | 2.6043 | 2.5481 | 2.5317 | 2.4337 | 2.4403 | 2.3529 | 2.3755 |
−150 | 2.6090 | 2.5809 | 2.5342 | 2.5209 | 2.4393 | 2.4447 | 2.3716 | 2.3902 |
The asymptotic and numerical values of \(\pmb{g'(1)}\) for large injection Reynolds number at \(\pmb{K=0.1}\) , \(\pmb{N_{1}=0.1}\)
Re | α = −5 | α = −2 | α = 2 | α = 5 | ||||
---|---|---|---|---|---|---|---|---|
Asy. | Num. | Asy. | Num. | Asy. | Num. | Asy. | Num. | |
−100 | 0.002574 | 0.002764 | 0.002517 | 0.002593 | 0.002445 | 0.002387 | 0.002394 | 0.002248 |
−110 | 0.002330 | 0.002488 | 0.002284 | 0.002348 | 0.002225 | 0.002177 | 0.002183 | 0.002060 |
−125 | 0.002041 | 0.002163 | 0.002006 | 0.002055 | 0.001960 | 0.001922 | 0.001927 | 0.001830 |
−150 | 0.001691 | 0.001776 | 0.001667 | 0.001702 | 0.001635 | 0.001608 | 0.001612 | 0.001544 |
4.2 Solution for large suction Reynolds number
The comparison between asymptotic and numerical solutions of \(\pmb{-f''(1)}\) for large suction Reynolds number as \(\pmb{K=0.1}\) , \(\pmb{N_{1}=0.1}\)
Re | α = −5 | α = −2 | α = 2 | α = 5 | ||||
---|---|---|---|---|---|---|---|---|
Asy. | Num. | Asy. | Num. | Asy. | Num. | Asy. | Num. | |
100 | 94.4545 | 94.5315 | 91.7273 | 91.7123 | 88.0909 | 87.9436 | 85.1087 | 85.3636 |
110 | 103.5455 | 103.6155 | 100.8182 | 100.8048 | 97.1818 | 97.0492 | 94.4545 | 94.2260 |
125 | 117.1818 | 117.2435 | 114.4545 | 114.4429 | 110.8182 | 110.7029 | 108.0909 | 107.8930 |
150 | 139.9091 | 139.9607 | 137.1818 | 137.1727 | 133.5455 | 133.4510 | 130.8182 | 130.6565 |
The asymptotic and numerical values of \(\pmb{g'(1)}\) for large suction Reynolds number at \(\pmb{K=0.2}\) , \(\pmb{N_{1}=0.1}\)
Re | α = −5 | α = −2 | α = 2 | α = 5 | ||||
---|---|---|---|---|---|---|---|---|
Asy. | Num. | Asy. | Num. | Asy. | Num. | Asy. | Num. | |
100 | 0.09785 | 0.09775 | 0.09785 | 0.09795 | 0.09785 | 0.09825 | 0.09785 | 0.09853 |
110 | 0.09761 | 0.09753 | 0.09761 | 0.09769 | 0.09761 | 0.09793 | 0.09761 | 0.09817 |
125 | 0.09732 | 0.09726 | 0.09732 | 0.09739 | 0.09732 | 0.09757 | 0.09732 | 0.09774 |
150 | 0.09698 | 0.09693 | 0.09698 | 0.09702 | 0.09698 | 0.09714 | 0.09698 | 0.09726 |
5 Conclusions
- (i)
The multiplicity of the solutions is influenced by the Reynolds number and the expansion ratio of the channel. New types of solutions are captured when the walls are expanding. Moreover, the non-existence of solutions for a specified range of Reynolds numbers with a particular positive expanding ratio is discovered.
- (ii)
In order to eliminate the singularity of the solution for the case of large injection, the analytical solution is constructed by the Lighthill method.
- (iii)
For the case of a large suction, the matched asymptotic expansion is used to obtain the analytical solution by the matching of the inner solution with the outer solution.
In this paper, we only construct two solutions for large injection or suction Reynolds number by singular perturbation methods. Some other multiple asymptotic solutions may be constructed to complement the above results in the future, which will help us to better understanding the stability of the micropolar flow through a deforming channel with porous walls.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundations of China (No. 11302024), the Fundamental Research Funds for the Central Universities (No. FRF-TP-15-036A3), and the foundation of the China Scholarship Council in 2014 (No. 154201406465041).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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