# Approximate solutions for MHD squeezing fluid flow by a novel method

Mustafa Inc1 and Ali Akgül23*

Author Affiliations

1 Department of Mathematics, Science Faculty, Fırat University, Elazığ, 23119, Turkey

2 Department of Mathematics, Education Faculty, Dicle University, Diyarbakır, 21280, Turkey

3 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, USA

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Boundary Value Problems 2014, 2014:18  doi:10.1186/1687-2770-2014-18

 Received: 5 October 2013 Accepted: 10 December 2013 Published: 16 January 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, a steady axisymmetric MHD flow of two-dimensional incompressible fluids has been investigated. The reproducing kernel Hilbert space method (RKHSM) has been implemented to obtain a solution of the reduced fourth-order nonlinear boundary value problem. Numerical results have been compared with the results obtained by the Runge-Kutta method (RK-4) and optimal homotopy asymptotic method (OHAM).

MSC: 46E22, 35A24.

##### Keywords:
reproducing kernel method; series solutions; squeezing fluid flow; magnetohydrodynamics; reproducing kernel space

### 1 Introduction

Squeezing flows have many applications in food industry, principally in chemical engineering [1-4]. Some practical examples of squeezing flow include polymer processing, compression and injection molding. Grimm [5] studied numerically the thin Newtonian liquids films being squeezed between two plates. Squeezing flow coupled with magnetic field is widely applied to bearing with liquid-metal lubrication [2,6-8].

In this paper, we use RKHSM to study the squeezing MHD fluid flow between two infinite planar plates. This problem has been solved by RKHSM and for comparison it has been compared with the OHAM and numerically with the RK-4 by using Maple 16.

The RKHSM, which accurately computes the series solution, is of great interest to applied sciences. The method provides the solution in a rapidly convergent series with components that can be elegantly computed. The efficiency of the method was used by many authors to investigate several scientific applications. Geng and Cui [9] and Zhou et al.[10] applied the RKHSM to handle the second-order boundary value problems. Yao and Cui [11] and Wang et al.[12] investigated a class of singular boundary value problems by this method and the obtained results were good. Wang and Chao [13], Li and Cui [14], Zhou and Cui [15] independently employed the RKSHSM to variable-coefficient partial differential equations. Du and Cui [16] investigated the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and the RKM. Lv and Cui [17] presented a new algorithm to solve linear fifth-order boundary value problems. Cui and Du [18] obtained the representation of the exact solution for the nonlinear Volterra-Fredholm integral equations by using the RKHSM. Wu and Li [19] applied iterative RKHSM to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. For more details about RKHSM and the modified forms and its effectiveness, see [9-37] and the references therein.

The paper is organized as follows. We give the problem formulation in Section 2. Section 3 introduces several reproducing kernel spaces. A bounded linear operator is presented in Section 4. In Section 5, we provide the main results, the exact and approximate solutions. An iterative method is developed for the kind of problems in the reproducing kernel space. We prove that the approximate solution converges to the exact solution uniformly. Some numerical experiments are illustrated in Section 6. There are some conclusions in the last section.

### 2 Problem formulation

Consider a squeezing flow of an incompressible Newtonian fluid in the presence of a magnetic field of a constant density ρ and viscosity μ squeezed between two large planar parallel plates separated by a small distance 2H and the plates approaching each other with a low constant velocity V, as illustrated in Figure 1, and the flow can be assumed to quasi-steady [1-3,39]. The Navier-Stokes equations [3,4] governing such flow in the presence of magnetic field, when inertial terms are retained in the flow, are given as [38]

(2.1)

and

(2.2)

where u is the velocity vector, ∇ denotes the material time derivative, T is the Cauchy stress tensor,

and

J is the electric current density, B is the total magnetic field and

represents the imposed magnetic field and b denotes the induced magnetic field. In the absence of displacement currents, the modified Ohm law and Maxwell’s equations (see [40] and the references therein) are given by [38]

(2.3)

and

(2.4)

in which σ is the electrical conductivity, E is the electric field and is the magnetic permeability.

Figure 1. A steady squeezing axisymmetric fluid flow between two parallel plates[38].

The following assumptions are needed [38].

(a) The density ρ, magnetic permeability and electric field conductivity σ are assumed to be constant throughout the flow field region.

(b) The electrical conductivity σ of the fluid is considered to be finite.

(c) Total magnetic field B is perpendicular to the velocity field V and the induced magnetic field b is negligible compared with the applied magnetic field so that the magnetic Reynolds number is small (see [40] and the references therein).

(d) We assume a situation where no energy is added or extracted from the fluid by the electric field, which implies that there is no electric field present in the fluid flow region.

Under these assumptions, the magnetohydrodynamic force involved in Eq. (2.2) can be put into the form

(2.5)

An axisymmetric flow in cylindrical coordinates r, θ, z with z-axis perpendicular to plates and at the plates. Since we have axial symmetry, u is represented by

when body forces are negligible, Navier-Stokes Eqs. (2.1)-(2.2) in cylindrical coordinates, where there is no tangential velocity (), are given as [38]

(2.6)

and

(2.7)

where p is the pressure, and the equation of continuity is given by [38]

(2.8)

The boundary conditions require

(2.9)

Let us introduce the axisymmetric Stokes stream function Ψ as

(2.10)

The continuity equation is satisfied using Eq. (2.10). Substituting Eqs. (2.3)-(2.5) and Eq. (2.10) into Eqs. (2.7)-(2.8), we obtain

(2.11)

and

(2.12)

Eliminating the pressure from Eqs. (2.11) and (2.12) by the integrability condition, we get the compatibility equation as [38]

(2.13)

where

The stream function can be expressed as [1,3]

(2.14)

In view of Eq. (2.14), the compatibility equation (2.13) and the boundary conditions (2.9) take the form

(2.15)

subject to

(2.16)

Non-dimensional parameters are given as [38]

For simplicity omitting the ∗, the boundary value problem (2.15)-(2.16) becomes [38]

(2.17)

with the boundary conditions

(2.18)

where Re is the Reynolds number and m is the Hartmann number.

### 3 Reproducing kernel spaces

In this section, we define some useful reproducing kernel spaces.

Definition 3.1 (Reproducing kernel)

Let E be a nonempty abstract set. A function is a reproducing kernel of the Hilbert space H if and only if

(3.1)

The last condition is called ‘the reproducing property’: the value of the function φ at the point t is reproduced by the inner product of φ with .

Definition 3.2 We define the space by

The fifth derivative of u exists almost everywhere since is absolutely continuous. The inner product and the norm in are defined respectively by

and

The space is a reproducing kernel space, i.e., for each fixed and any , there exists a function such that

Definition 3.3 We define the space by

The fourth derivative of u exists almost everywhere since is absolutely continuous. The inner product and the norm in are defined respectively by

and

The space is a reproducing kernel space, i.e., for each fixed and any , there exists a function such that

Theorem 3.1The spaceis a reproducing kernel Hilbert space whose reproducing kernel function is given by

whereandcan be obtained easily by using Maple 16 and the proof of Theorem 3.1 is given in Appendix.

Remark 3.1 The reproducing kernel function of is given as

This can be proved easily as the proof of Theorem 3.1.

### 4 Bounded linear operator in

In this section, the solution of Eq. (2.17) is given in the reproducing kernel space .

On defining the linear operator as

Model problem (2.17)-(2.18) changes the following problem:

(4.1)

where

and

Theorem 4.1The operatorLdefined by (4.1) is a bounded linear operator.

Proof We only need to prove

where P is a positive constant. By Definition 3.3, we have

and

By the reproducing property, we have

and

Therefore, by the Cauchy-Schwarz inequality, we get

Thus

Since

then

Therefore, we have

and

that is,

where is a positive constant. This completes the proof. □

### 5 Analysis of the solution of (2.17)-(2.18)

Let be any dense set in and , where is the adjoint operator of L and is given by Remark 3.1. Furthermore

Lemma 5.1is a complete system of.

Proof For , let

that is,

Note that is the dense set in . Therefore . Assume that (4.1) has a unique solution. Then L is one-to-one on and thus . This completes the proof. □

Lemma 5.2The following formula holds:

where the subscriptηof the operatorindicates that the operatorLapplies to a function ofη.

Proof

This completes the proof. □

Remark 5.1 The orthonormal system of can be derived from the Gram-Schmidt orthogonalization process of as

(5.1)

where are orthogonal coefficients.

In the following, we give the representation of the exact solution of Eq. (2.17) in the reproducing kernel space .

Theorem 5.1Ifuis the exact solution of (4.1), then

whereis a dense set in.

Proof From (5.1) and the uniqueness of solution of (4.1), we have

This completes the proof. □

Now the approximate solution can be obtained by truncating the n-term of the exact solution u as

Lemma 5.3 ([30])

Assume thatuis the solution of (4.1) andis the error between the approximate solutionand the exact solutionu. Then the error sequenceis monotone decreasing in the sense ofand.

### 6 Numerical results

In this section, comparisons of results are made through different Reynolds numbers Re and magnetic field effect m. All computations are performed by Maple 16. Figure 5.7 shows comparisons of for a fixed Reynolds number with increasing magnetic field effect . From this figure, the velocity decreases due to an increase in m. Figure 5.8 shows comparisons of for a fixed magnetic field with increasing Reynolds numbers . It is observed that much increase in Reynolds numbers affects the results. The RKHSM does not require discretization of the variables, i.e., time and space, it is not affected by computation round of errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKHSM for the MHD squeezing fluid flow is controllable and absolute errors are small with present choice of x (see Tables 1-6 and Figures 2-7). The numerical results we obtained justify the advantage of this methodology. Generally it is not possible to find the exact solution of these problems.

Figure 2. Comparison RKHSM, OHAM and RK-4 solutions for.

Figure 3. Comparison RKHSM, OHAM and RK-4 solutions forand.

Figure 4. AE and RE forand.

Figure 5. AE and RE forand.

Figure 6. Comparison of squeezing flow for a fixed Reynolds numberand increasing magnetic field effect.

Figure 7. Comparison of squeezing flow for a fixed magnetic field effectand increasing Reynolds numbers.

Table 1. Numerical results atand

Table 2. Numerical results atand

Table 3. Numerical results atand

Table 4. Numerical results atand

Table 5. Numerical results atand

Table 6. Numerical results atand

### 7 Conclusion

In this paper, we introduced an algorithm for solving the MHD squeezing fluid flow. We applied a new powerful method RKHSM to the reduced nonlinear boundary value problem. The approximate solution obtained by the present method is uniformly convergent. Clearly, the series solution methodology can be applied to much more complicated nonlinear differential equations and boundary value problems. However, if the problem becomes nonlinear, then the RKHSM does not require discretization or perturbation and it does not make closure approximation. Results of numerical examples show that the present method is an accurate and reliable analytical method for this problem.

### Appendix

Proof of Theorem 3.1 Let . By Definition 3.2 we have

(A.1)

Through several integrations by parts for (A.1), we have

(A.2)

Note the property of the reproducing kernel

Now, if

(A.3)

then (A.2) implies that

when

and therefore

Since

we have

(A.4)

and

(A.5)

Since , it follows that

(A.6)

From (A.3)-(A.6), the unknown coefficients and () can be obtained. This completes the proof. □

### Competing interests

The authors declare that they do not have any competing or conflict of interests.

### Authors’ contributions

Both authors contributed equally to this paper.

### Acknowledgements

We presented this paper in the International Symposium on Biomathematics and Ecology Education Research in 2013. We would like to thank the organizers of this conference and the reviewers for their kind and helpful comments on this paper. Ali Akgül gratefully acknowledge that this paper was partially supported by the Dicle University and the Firat University. This paper is a part of PhD thesis of Ali Akgül.

### References

1. Papanastasiou, TC, Georgiou, GC, Alexandrou, AN: Viscous Fluid Flow, CRC Press, Boca Raton (1994)

2. Stefa Hughes, WF, Elco, RA: Magnetohydrodynamic lubrication flow between parallel rotating disks. J. Fluid Mech.. 13, 21–32 (1962). Publisher Full Text

3. Ghori, QK, Ahmed, M, Siddiqui, AM: Application of homotopy perturbation method to squeezing flow of a Newtonian fluid. Int. J. Nonlinear Sci. Numer. Simul.. 8, 179–184 (2007)

4. Ran, XJ, Zhu, QY, Li, Y: An explicit series solution of the squeezing flow between two infinite plates by means of the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul.. 14, 119–132 (2009). Publisher Full Text

5. Grimm, RG: Squeezing flows of Newtonian liquid films an analysis include the fluid inertia. Appl. Sci. Res.. 32, 149–166 (1976). Publisher Full Text

6. Kamiyama, S: Inertia effects in MHD hydrostatic thrust bearing. J. Lubr. Technol.. 91, 589–596 (1969). Publisher Full Text

7. Hamza, EA: The magnetohydrodynamic squeeze film. J. Tribol.. 110, 375–377 (1988). Publisher Full Text

8. Bhattacharya, S, Pal, A: Unsteady MHD squeezing flow between two parallel rotating discs. Mech. Res. Commun.. 24, 615–623 (1997). Publisher Full Text

9. Geng, F, Cui, M: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl.. 327, 1167–1181 (2007). Publisher Full Text

10. Zhou, Y, Lin, Y, Cui, M: An efficient computational method for second order boundary value problems of nonlinear differential equations. Appl. Math. Comput.. 194, 357–365 (2007)

11. Yao, H, Cui, M: A new algorithm for a class of singular boundary value problems. Appl. Math. Comput.. 186, 1183–1191 (2007). Publisher Full Text

12. Wang, W, Cui, M, Han, B: A new method for solving a class of singular two-point boundary value problems. Appl. Math. Comput.. 206, 721–727 (2008). Publisher Full Text

13. Wang, YL, Chao, L: Using reproducing kernel for solving a class of partial differential equation with variable-coefficients. Appl. Math. Mech.. 29, 129–137 (2008). Publisher Full Text

14. Li, F, Cui, M: A best approximation for the solution of one-dimensional variable-coefficient Burgers’ equation. Numer. Methods Partial Differ. Equ.. 25, 1353–1365 (2009). Publisher Full Text

15. Zhou, S, Cui, M: Approximate solution for a variable-coefficient semilinear heat equation with nonlocal boundary conditions. Int. J. Comput. Math.. 86, 2248–2258 (2009). Publisher Full Text

16. Du, J, Cui, M: Solving the forced Duffing equations with integral boundary conditions in the reproducing kernel space. Int. J. Comput. Math.. 87, 2088–2100 (2010). Publisher Full Text

17. Lv, X, Cui, M: An efficient computational method for linear fifth-order two-point boundary value problems. J. Comput. Appl. Math.. 234, 1551–1558 (2010). Publisher Full Text

18. Du, J, Cui, M: Constructive proof of existence for a class of fourth-order nonlinear BVPs. Comput. Math. Appl.. 59, 903–911 (2010). Publisher Full Text

19. Wu, BY, Li, XY: Iterative reproducing kernel method for nonlinear oscillator with discontinuity. Appl. Math. Lett.. 23, 1301–1304 (2010). Publisher Full Text

20. Cui, M, Lin, Y: Nonlinear Numerical Analysis in the Reproducing Kernel Spaces, Nova Science Publishers, New York (2009)

21. Lü, X, Cui, M: Analytic solutions to a class of nonlinear infinite-delay-differential equations. J. Math. Anal. Appl.. 343, 724–732 (2008). Publisher Full Text

22. Jiang, W, Cui, M: Constructive proof for existence of nonlinear two-point boundary value problems. Appl. Math. Comput.. 215, 1937–1948 (2009). Publisher Full Text

23. Cui, M, Du, H: Representation of exact solution for the nonlinear Volterra-Fredholm integral equations. Appl. Math. Comput.. 182, 1795–1802 (2006). Publisher Full Text

24. Jiang, W, Lin, Y: Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space. Commun. Nonlinear Sci. Numer. Simul.. 16, 3639–3645 (2011). Publisher Full Text

25. Lin, Y, Cui, M: A numerical solution to nonlinear multi-point boundary-value problems in the reproducing kernel space. Math. Methods Appl. Sci.. 34, 44–47 (2011). Publisher Full Text

26. Mohammadi, M, Mokhtari, R: Solving the generalized regularized long wave equation on the basis of a reproducing kernel space. J. Comput. Appl. Math.. 235, 4003–4014 (2011). Publisher Full Text

27. Wu, BY, Li, XY: A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method. Appl. Math. Lett.. 24, 156–159 (2011). Publisher Full Text

28. Yao, H, Lin, Y: New algorithm for solving a nonlinear hyperbolic telegraph equation with an integral condition. Int. J. Numer. Methods Biomed. Eng.. 27, 1558–1568 (2011). Publisher Full Text

29. Inc, M, Akgül, A: The reproducing kernel Hilbert space method for solving Troesch’s problem. J. Assoc. Arab Univ. Basic. Appl. Sci.. 14, 19–27 (2013)

30. Inc, M, Akgül, A, Geng, F: Reproducing kernel Hilbert space method for solving Bratu’s problem. Bul. Malays. Math. Sci. Soc. (in press)

31. Inc, M, Akgül, A, Kilicman, A: Explicit solution of telegraph equation based on reproducing kernel method. J. Funct. Spaces Appl.. 2012, (2012) Article ID 984682

32. Inc, M, Akgül, A, Kilicman, A: A new application of the reproducing kernel Hilbert space method to solve MHD Jeffery-Hamel flows problem in non-parallel walls. Abstr. Appl. Anal.. 2013, (2013) Article ID 239454

33. Inc, M, Akgül, A, Kilicman, A: On solving KdV equation using reproducing kernel Hilbert space method. Abstr. Appl. Anal.. 2013, (2013) Article ID 578942

34. Inc, M, Akgül, A, Kilicman, A: Numerical solutions of the second-order one-dimensional telegraph equation based on reproducing kernel Hilbert space method. Abstr. Appl. Anal.. 2013, (2013) Article ID 768963

35. Akram, G, Rehman, HU: Numerical solution of eighth order boundary value problems in reproducing Kernel space. Numer. Algorithms. 62(3), 527–540 (2013). Publisher Full Text

36. Wenyan, W, Bo, H, Masahiro, Y: Inverse heat problem of determining time-dependent source parameter in reproducing kernel space. Nonlinear Anal., Real World Appl.. 14(1), 875–887 (2013). Publisher Full Text

37. Mokhtari, R, İsfahani, FT, Mohammadi, M: Reproducing kernel method for solving nonlinear differential-difference equations. Abstr. Appl. Anal.. 2012, (2012) Article ID 514103

38. Islam, S, Ullah, M, Zaman, G, Idrees, M: Approximate solutions to MHD squeezing fluid flow. J. Appl. Math. Inform.. 29(5-6), 1081–1096 (2011)

39. Idrees, M, Islam, S, Haq, S, Islam, S: Application of the optimal homotopy asymptotic method to squeezing flow. Comput. Math. Appl.. 59, 3858–3866 (2010). Publisher Full Text

40. Mohyuddin, MR, Gotz, T: Resonance behavior of viscoelastic fluid in Poiseuille flow in the presence of a transversal magnetic field. Int. J. Numer. Methods Fluids. 49, 837–847 (2005). Publisher Full Text