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This article is part of the series Recent Trends on Boundary Value Problems and Related Topics.

Open Access Research

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

Fadzilah Md Ali1*, Roslinda Nazar2, Norihan Md Arifin1 and Ioan Pop3

Author affiliations

1 Department of Mathematics & Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Selangor, 43400 UPM, Malaysia

2 School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, 43600 UKM, Malaysia

3 Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, 400084, Romania

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Citation and License

Boundary Value Problems 2013, 2013:32  doi:10.1186/1687-2770-2013-32


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/32


Received:21 September 2012
Accepted:28 January 2013
Published:18 February 2013

© 2013 Ali et al.; licensee Springer

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, 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.

Keywords:
boundary layer; dual solutions; magnetohydrodynamic; shrinking sheet; numerical solution

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M1">View MathML</a> is applied normal to the shrinking and porous sheet. The governing equations of the present problem are

(1)

(2)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M4">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M5">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M1">View MathML</a> is in the form of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M7">View MathML</a>

(4)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M8">View MathML</a>

(5)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M9">View MathML</a>

(6)

subject to the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M10">View MathML</a>

(7)

where primes denote differentiation with respect to η, while

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M11">View MathML</a>

(8)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M12">View MathML</a> which is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M13">View MathML</a>

(9)

where the shear stress <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M14">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M15">View MathML</a>

(10)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M16">View MathML</a>

(11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M17">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18">View MathML</a> 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.

Table 1 . Various values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18">View MathML</a>withβwhen<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M20">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M21">View MathML</a>

Figure 1 shows the variations of the skin friction coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18">View MathML</a> decreases with β. On the other hand, the skin friction coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M18">View MathML</a> increases with s. This is because, physically, suction produces more resistance to the transport phenomena. The critical value of the suction parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M25">View MathML</a> decreases as β increases; therefore, the controlling parameter accelerates the boundary layer separation.

thumbnail Figure 1 . Variation of the skin friction coefficient for different values ofβwhen<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M26">View MathML</a>.

The variations of the skin friction coefficients with the suction parameter for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M27">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M28">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M25">View MathML</a> increases with M. Therefore, we can conclude that the magnetic parameter delays the boundary layer separation from the surface.

thumbnail Figure 2 . Variation of the skin friction coefficient for different values ofMwhen<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M30">View MathML</a>.

thumbnail Figure 3 . Variation of the skin friction coefficient for different values ofMwhen<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M31">View MathML</a>.

thumbnail Figure 4 . Variation of the skin friction coefficient for different values ofMwhen<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M32">View MathML</a>.

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.

thumbnail Figure 5 . Velocity profiles for different values ofβwhen<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M26">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M34">View MathML</a>.

thumbnail Figure 6 . Velocity profiles for different values ofMwhen<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/32/mathml/M34">View MathML</a>.

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.

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.

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.

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