Abstract
We investigate an unsteady incompressible laminar micropolar flow in a semi-infinite porous pipe with large injection or suction through a deforming pipe wall. Using suitable similarity transformations, the governing partial differential are transformed into a coupled nonlinear singular boundary value problem. For large injection, the asymptotic solutions are constructed using the Lighthill method, which eliminates singularity of solution in the high order derivative. For large suction, a series expansion matching method is used. Analytical solutions are validated against the numerical solutions obtained by Bvp4c.
1 Introduction
Since Eringen [1, 2] proposed a mathematical model to describe the non-Newtonian behaviour of liquids such as polymers, colloidal suspensions, animal blood and liquid crystals, there has been interest in micropolar fluids. In particular, micropolar fluids flowing in porous channels or pipes have received more attention due to their relevance to a number of practical biological problems. For example, Mekheimer and Elkot [3] presented a micropolar model for axisymmetric blood flow through an axially nonsymmetric but radially symmetric mild-stenosis-tapered artery. Mekheimer et al. [4] investigated the effects of an induced magnetic field on peristaltic transport of an incompressible micropolar fluid in a symmetric channel. Furthermore, Subhardra Ramachandran et al. [5] used Van Dyke's singular perturbation technique to study the heat transfer of a micropolar fluid past a curved surface with suction and injection. Anwar Kamal and Hussain [6] examined the steady, incompressible and laminar flow of micropolar fluids inside an infinite channel where the flow was driven due to a surface velocity proportional to the streamwise coordinates. Joneidi et al. [7] obtained similarity equations for a micropolar fluid in a porous channel and used the homotopy analysis method (HAM) to discuss the velocity distribution. In addition, Ariman et al. [8] and Lukaszewicz [9] gave reviews of micropolar fluid mechanics and its applications.
The purpose of this paper is to extend previous investigations by presenting analytical solutions for the flow inside a deforming porous pipe with large injection or suction. Equations describing the unsteady flow of an incompressible Newtonian fluid in a porous expanding channel are presented by White [10] as one of the new exact Navier-Stokes solutions attributed to Dauenhauer and Majdalani [11] . In their work [11], they numerically discussed the influence of the expansion ratio and Reynolds number on the velocity and pressure distribution. Furthermore, Majdalani, Zhou and Dawson [12] also obtained an asymptotic solution for the flow in a porous channel, with slowly expanding or contracting walls, by considering the permeation Reynolds number and expansion ratio as two small parameters. Boutros et al. [13, 14] also discussed the flow through an expanding porous channel or pipe using the Lie group method and obtained the analytical solution with the perturbation method. Recently Si et al. [15] also investigated the micropolar fluid in a porous deforming channel and discussed the effects of the micropolar parameter and the expansion ratio on the velocity and microrotation distribution. As further research, Li, Lin and Si [16] numerically analyzed the flow of a micropolar fluid through a porous pipe with an expanding or contracting wall.
In this paper, asymptotic solutions are constructed for the flow of a micropolar fluid through an expanding or contracting porous pipe. For large injection, analytical solutions are constructed using the Lighthill method, which eliminates singularity of the solution in the high order derivative [17-19];a series expansion matching method is used for large suction. The accuracy of the analytical solutions for each case is compared with its numerical results.
2 Preliminaries
Consider a micropolar fluid flowing through a pipe with a vertical moving porous wall. Here we assume that one end of the pipe is closed by a complicated solid membrane and the wall of the pipe moves in the radial direction and expands or contracts uniformly at a time-dependent rate
Under these assumptions, the governing equations of the incompressible and homogeneous micropolar fluid flowing with no body force are expressed as follows:
where ρ and μ are the density and the dynamic viscosity, and j, y and κ are the micro-inertial coefficient, spin gradient viscosity and vortex viscosity, respectively. Here y is assumed to be
The corresponding boundary conditions are [11, 12, 15]
where A is the measure of the permeability. Here we also assume that there is a strong concentration of microelements, and the microelements close to the wall are unable to rotate [21, 22].
Introduce the stream function Ψ(r, z, t) such that
In this paper, the stream function Ψ and the microrotation velocity N are assumed as follows:
where
Similar to Dauenhauer and Majdalani [11], Uchida and Aoki [20], and Boutros et al. [13, 14], we substitute Eqs. (6) and (7) into governing equations and consider the similarity solutions with respect to space and time, then the following ordinary equations can be obtained:
where
The corresponding boundary conditions can be written as
3 Perturbation analysis for this problem
3.1 Solution for the large injection Reynolds number
For large injection Reynolds numbers, the asymptotic solution of (8) and (9), subject to the boundary conditions (10), is obtained by the Lighthill method. One treats
where λ is an integral constant. Firstly, we introduce a variable transformation of η
where the functions X1, X2 are unknown and will be determined in the following process. One assumes that the functions f, g and the constant λ are expanded as
Substituting (13)-(14) into (11)-(12) and collecting the same powers of ε, one can obtain the leading solution
and the first order solution
Here ˙ denotes the derivative with respect to ξ.
3.1.1 A. the transformed boundary conditions at the wall of the pipe
We assume
thus the conditions at the wall can be obtained
Hence, the boundary conditions of fi and gi at η = 1 are
3.1.2 B. the transformed boundary conditions at the center of the pipe
One supposes that
thus we can induce
Hence, the boundary conditions of fi at η = 0 are
Using (22)-(24) and (27), the solution for (15) can be obtained
and then
and
Here it should be noted that direct use of the method of variation of parameters will cause a singularity in the third-order derivative of f1 [17-19]. In order to eliminate the singularity and to simplify the equation of f1, we can set
Then we have
where
The solution of Eq. (33) is
where
Finally, one obtains the asymptotic solutions of f, g in terms of ξ
where
Fig. 2 shows the profiles of f′(η) and g(η) against η for the asymptotic and numerical results. Tables 1 and 2 give asymptotic and numerical values of f″(1) and g′(1) for some values of large injection Reynolds number and expansion ratio α, respectively. The results agree well.
f″(1) | g′(1) | |||
---|---|---|---|---|
Re | Numerical | Asymptotic | Numerical | Asymptotic |
50 | -2.6641619556581 | -2.7339624379749 | -0.2128340193109 | -0.2077811452861 |
100 | -2.5700001017056 | -2.5955567129755 | -0.1036184958045 | -0.1012267118060 |
150 | -2.5366158007420 | -2.5517506351567 | -0.0681380895087 | -0.0669125722108 |
200 | -2.5195961015861 | -2.5302623043972 | -0.0507066428440 | -0.0499726805118 |
500 | -2.4884675811385 | -2.4922614078153 | -0.0199682862609 | -0.0198384008062 |
f″(1) | g′(1) | |||
---|---|---|---|---|
α | Numerical | Asymptotic | Numerical | Asymptotic |
-5 | -2.569310686399770 | -2.595556712975505 | -0.155025023543228 | -0.151840067709067 |
-2 | -2.507901592906879 | -2.517499337080236 | -0.147379010820407 | -0.149539460622566 |
2 | -2.430977747070986 | -2.418783550899264 | -0.138027490934619 | -0.146578283184495 |
5 | -2.376702021744101 | -2.348507888421025 | -0.131584147099143 | -0.144433235137893 |
3.2 Solution for large suction Reynolds number
The boundary layer happens near the wall not only for the velocity but also for the microrotation when there is large suction. The solutions of (8) and (9), subject to the boundary conditions (10), can be obtained for large suction by using the method of matched asymptotic expansion. One treats
where k is a constant of integration,
and
where the coefficients δi, σi and ωi (i = 0,1,2, …) are constants determined by matching with the inner solution.
We assume that the forms of the outer solutions are
Substituting (41)-(42) into (38)-(39), and equating the same power of the coefficient ε, one obtains
…….
The corresponding boundary conditions for the outer solutions are
According to the boundary conditions (51), the solutions for (43) and (44) can be obtained
where
Then the solution of (53) that satisfies the boundary conditions (51) is
where
whose solution is
where
Thus one obtains
where
Then one obtains
Similarly,
where
In order to obtain the inner solution in the viscous layer, we introduce a stretching transformation τ = (1 - η)/ε. Substituting into Eqs. (11) and (12) yields
Here ˙ denotes the derivative with respect to τ. According to the boundary conditions (10), we assume the inner solutions near the wall to be
Substituting (65) into (63) and (64) yields the following equations
…….
The boundary conditions corresponding to the inner solution are
The solution of (66) satisfying the boundary conditions (70) is
where
The outer solution of f, expressed in terms of the inner variable τ, is
As τ → ∞, matching the inner solution (72) with (73) gives
The solution of (67) satisfying the boundary conditions (70) is
where
The solution of (68) satisfying the boundary conditions (70) is
where D3 is an integral constant. Then the inner solution of f can be expressed as
As τ → ∞, matching the inner solution with outer solution, one obtains
similarly, one can obtain
Hence, the complete solutions of (8) and (9) satisfying the boundary conditions (10) for large suction can be obtained as follows:
and
Fig. 3 shows the profiles of f′(η) and g(η) against η for the asymptotic and numerical results. Tables 3 and 4 give asymptotic and numerical values of f″(1) and g′(1) for some values of large suction Reynolds number and expansion ratio, respectively.
f″(1) | g′(1) | |||
---|---|---|---|---|
Re | Numerical | Asymptotic | Numerical | Asymptotic |
-60 | -26.029180587850 | -26.083333333333 | -1.9697087541999 | -1.9620553359684 |
-90 | -38.553222227512 | -38.583333333333 | -1.9168665563568 | -1.9140974967062 |
-100 | -42.723767379927 | -42.750000000000 | -1.9066699865053 | -1.9045059288538 |
-125 | -53.146883680302 | -53.166666666667 | -1.8885407560495 | -1.8872411067194 |
-150 | -63.567422760150 | -63.583333333333 | -1.8766007832311 | -1.8757312252964 |
f″(1) | g′(1) | |||
---|---|---|---|---|
α | Numerical | Asymptotic | Numerical | Asymptotic |
-5 | -42.7237673799268 | -42.7500000000000 | -1.90666998651 | -1.90450592885 |
-2 | -41.4054335101753 | -41.5000000000000 | -1.91280714642 | -1.90450592885 |
2 | -39.6395810279868 | -39.8333333333333 | -1.92210393045 | -1.90450592885 |
5 | -38.3082775719552 | -38.5833333333333 | -1.93007245250 | -1.90450592885 |
4 Conclusions
In this paper, we have proposed a model for the flow of micropolar flow through an expanding porous pipe. Using suitable similarity transformations, the governing equations are transformed into a coupled nonlinear singular boundary value problem, and the analytical solutions are compared with the numerical ones, showing good agreement. Some conclusions can be drawn:
Analytical solutions can be obtained for large injection or suction using the Lighthill method and series-expansion matching method;
The microrotation velocity also exists at the boundary layer for large suction;
The Lighthill method can also be used to solve similar problems.
Acknowledgement
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), Beijing Higher Education Young Elite Teacher Project(No.YETP0387), and the Foundation of the China Scholarship Council in 2014 (No.154201406465041).
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