- Research
- Open Access
- Published:

# Dual solutions in MHD flow on a nonlinear porous shrinking sheet in a viscous fluid

*Boundary Value Problems*
**volume 2013**, Article number: 32 (2013)

## Abstract

In this paper, the problem of magnetohydrodynamic (MHD) flow of a viscous fluid on a nonlinear porous shrinking sheet is studied. The boundary layer partial differential equations are first transformed into an ordinary differential equation, which is then solved numerically by the shooting method. The features of the flow for various governing parameters are presented and discussed in detail. It is found that dual solutions only exist for positive values of the controlling parameter.

**MSC:** 34B15, 76D10.

## 1 Introduction

Boundary layer flow over a stretching sheet has been studied in various aspects since the pioneering work done by Sakiadis [1]. Lately, many researchers have studied the shrinking sheet boundary layer flow problem due to its important applications in industries which involve packaging process, for example, shrink wrapping. The study on shrinking sheet was first initiated by Wang [2] by considering the stretching deceleration surface. Miklavcic and Wang [3] proved the existence and uniqueness for viscous flow due to a shrinking sheet, and dual solutions were reported for certain range of the suction parameter. Later, Wang [4] also studied the stagnation flow towards a shrinking sheet by considering the two-dimensional and axisymmetric stagnation flows, and dual solutions were also reported only for the two-dimensional case. Further, Sajid *et al.*[5] and Hayat *et al.*[6] considered the rotating flow over a shrinking surface. The steady boundary layer flow problems induced by a shrinking sheet can be found in [7–11] in different aspects. On the other hand, the unsteady case is described in papers by Fang *et al.*[12] and Ali *et al.*[13, 14]. It is worth mentioning that dual solutions are also found in the papers by Fang [11], Fang *et al.*[12] and Ali *et al.*[14].

Recently, Nadeem and Hussain [15] solved analytically the problem of magnetohydrodynamic (MHD) flow of a viscous fluid on a nonlinear porous shrinking sheet using the homotopy analysis method and dual solutions were not reported. Hence, the present paper aims to obtain the dual solutions numerically for the problem considered in Nadeem and Hussain [15] for various controlling parameters and magnetic parameters.

## 2 Basic equations

Consider the steady two-dimensional flow of an incompressible electrically conducting fluid towards a nonlinear porous shrinking sheet. The magnetic field $B(x)$ is applied normal to the shrinking and porous sheet. The governing equations of the present problem are

where *u* and *v* are the velocity components along the *x* and *y* directions, respectively, *ν* is the kinematic viscosity, *ρ* is the fluid density and *σ* is the fluid electrical conductivity. The boundary conditions of Eqs. (1) and (2) are

where ${V}_{0}$ is the porosity of the sheet. Following Chiam [16], we assume the external electric field and polarization effects in Eq. (2) are negligible, therefore the magnetic field $B(x)$ is in the form of

We assume the magnetic Reynolds number is small, so that the induced magnetic field is negligible. Applying the following similarity transformations:

to Eqs. (1) and (2), we obtain the following ordinary differential equation:

subject to the boundary conditions

where primes denote differentiation with respect to *η*, while

are the wall mass transfer (suction) parameter, the magnetic parameter and the non-dimensional (controlling) parameter, respectively.

The physical quantities of interest is the skin friction coefficient ${C}_{f}$ which is defined as

where the shear stress ${\tau}_{w}$ is defined as

with *μ* being the dynamic viscosity. Using (5) and (10), we get

where $R{e}_{x}={u}_{w}x/\nu $ is the local Reynolds number.

## 3 Results and discussion

Equation (6) subject to the boundary conditions (7) has been solved numerically using the shooting method as described in the paper by Meade *et al.*[17]. This technique is an iterative algorithm which attempts to identify appropriate initial conditions for a related initial value problem (IVP) that provides the solution to the original boundary value problem (BVP). The results of the skin friction coefficient ${f}^{\u2033}(0)$ obtained in this study are compared with those of Nadeem and Hussain [15], as displayed in Table 1. The agreement between these results is very good. Therefore, we are confident that the present method is accurate.

Figure 1 shows the variations of the skin friction coefficient ${f}^{\u2033}(0)$ with the wall mass transfer (suction) parameter *s* for different values of the controlling parameter *β*. Dual solutions are found to exist only for positive values of *β*. It is also found that ${f}^{\u2033}(0)$ decreases with *β*. On the other hand, the skin friction coefficient ${f}^{\u2033}(0)$ increases with *s*. This is because, physically, suction produces more resistance to the transport phenomena. The critical value of the suction parameter ${s}_{c}$ decreases as *β* increases; therefore, the controlling parameter accelerates the boundary layer separation.

The variations of the skin friction coefficients with the suction parameter for $\beta =0.5,1.0\text{and}1.5$ are displayed in Figures 2, 3 and 4, respectively. The skin friction coefficients increase with both the magnetic parameter and the suction parameter. Figures 2 to 4 also show the existence of the dual solutions. The boundary layer separates from the surface at $s={s}_{c}$, and beyond this value, the boundary layer approximation is no longer valid. Therefore, the full Navier-Stokes equation need to be used. Normally, the first solutions are physically stable and this can be verified by performing a stability analysis; see Merkin [18] and Weidman *et al.*[19]. In Figures 2 to 4, it is found that ${s}_{c}$ increases with *M*. Therefore, we can conclude that the magnetic parameter delays the boundary layer separation from the surface.

Figures 5 and 6 illustrate the dual velocity profiles for various *β* and *M*, respectively, with both figures satisfying the boundary conditions (7). This is to prove the dual nature of the present problem. From Figure 5, the boundary layer thickness increases with *β*. However, the boundary layer thickness decreases as *M* increases, as displayed in Figure 6. This phenomenon is due to the fact that *M* creates the Lorentz force which slows down the fluid flow, hence reducing the boundary layer thickness.

## 4 Conclusions

A study is performed for the problem of MHD flow of a viscous fluid on a nonlinear porous shrinking sheet. It is observed that the dual solutions existed only for positive values of the controlling parameter. In this study, we can conclude that the controlling parameter accelerated the boundary layer separation, however, the magnetic parameter delayed the boundary layer separation.

## References

- 1.
Sakiadis BC:

**Boundary layers on continuous solid surfaces.***AIChE J.*1961, 7: 26-28. 10.1002/aic.690070108 - 2.
Wang CY:

**Liquid film on an unsteady stretching surface.***Q. Appl. Math.*1990, 48(4):601-610. - 3.
Miklavcic M, Wang CY:

**Viscous flow due to a shrinking sheet.***Q. Appl. Math.*2006, 64: 283-290. - 4.
Wang CY:

**Stagnation flow towards a shrinking sheet.***Int. J. Non-Linear Mech.*2008, 43: 377-382. 10.1016/j.ijnonlinmec.2007.12.021 - 5.
Sajid M, Javed T, Hayat T:

**MHD rotating flow of a viscous fluid over a shrinking surface.***Nonlinear Dyn.*2008, 51: 259-265. - 6.
Hayat T, Abbas Z, Javed T, Sajid M:

**Three-dimensional rotating flow induced by a shrinking sheet for suction.***Chaos Solitons Fractals*2009, 39: 1615-1626. 10.1016/j.chaos.2007.06.045 - 7.
Fang T, Liang W, Lee CF:

**A new solution branch for the Blasius equation - a shrinking sheet problem.***Comput. Math. Appl.*2008, 56: 3088-3095. 10.1016/j.camwa.2008.07.027 - 8.
Fang T:

**Boundary layer flow over a shrinking sheet with power-law velocity.***Int. J. Heat Mass Transf.*2008, 51: 5838-5843. 10.1016/j.ijheatmasstransfer.2008.04.067 - 9.
Fang T, Zhang J:

**Thermal boundary layers over a shrinking sheet: an analytical solution.***Acta Mech.*2010, 209: 325-343. 10.1007/s00707-009-0183-2 - 10.
Sajid M, Hayat T:

**The application of homotopy analysis method for MHD viscous flow due to a shrinking sheet.***Chaos Solitons Fractals*2009, 39: 1317-1323. 10.1016/j.chaos.2007.06.019 - 11.
Fang T:

**Boundary layer flow over a shrinking sheet with power-law velocity.***Int. J. Heat Mass Transf.*2008, 51: 5838-5843. 10.1016/j.ijheatmasstransfer.2008.04.067 - 12.
Fang T-G, Zhang J, Yao S-S:

**Viscous flow over an unsteady shrinking sheet with mass transfer.***Chin. Phys. Lett.*2009., 26: Article ID 014703 - 13.
Ali FM, Nazar R, Arifin NM, Pop I:

**Unsteady shrinking sheet with mass transfer in a rotating fluid.***Int. J. Numer. Methods Fluids*2011, 66: 1465-1474. 10.1002/fld.2325 - 14.
Ali FM, Nazar R, Arifin NM, Pop I:

**Unsteady flow and heat transfer past an axisymmetric permeable shrinking sheet with radiation effect.***Int. J. Numer. Methods Fluids*2011, 67: 1310-1320. 10.1002/fld.2435 - 15.
Nadeem S, Hussain A:

**MHD flow of a viscous fluid on a nonlinear porous shrinking sheet with homotopy analysis method.***Appl. Math. Mech.*2009, 30(12):1569-1578. 10.1007/s10483-009-1208-6 - 16.
Chiam TC:

**Hydromagnetic flow over a surface stretching with a power-law velocity.***Int. J. Eng. Sci.*1995, 33(3):429-435. 10.1016/0020-7225(94)00066-S - 17.
Meade DB, Haran BS, White RE:

**The shooting technique for the solution of two-point boundary value problems.***Maple Tech. Newsl.*1996, 3: 85-93. - 18.
Merkin JH:

**On dual solutions occurring in mixed convection in a porous medium.***J. Eng. Math.*1985, 20: 171-179. - 19.
Weidman PD, Kubitschek DG, Davis AMJ:

**The effect of transpiration on self-similar boundary layer flow over moving surfaces.***Int. J. Eng. Sci.*2006, 44: 730-737. 10.1016/j.ijengsci.2006.04.005

## Acknowledgements

The authors gratefully acknowledged the financial support received in the form of a FRGS research grant and a LRGS research grant (LRGS/TD/2011/UKM/ICT/03/02) from the Ministry of Higher Education, Malaysia, and DIP-2012-31 from the Universiti Kebangsaan Malaysia.

## Author information

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors have contributed their parts equally and have also read and approved the final manuscript.

## Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

## Rights and permissions

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

## About this article

#### Received

#### Accepted

#### Published

#### DOI

### Keywords

- boundary layer
- dual solutions
- magnetohydrodynamic
- shrinking sheet
- numerical solution