Abstract
The present analysis considers the steady magnetohydrodynamic (MHD) laminar boundary
layer flow of an incompressible electrically conducting fluid caused by a continuous
moving wedge in a parallel free stream with a variable induced magnetic field parallel
to the wedge walls outside the boundary layer. Using a similarity transformation,
the governing system of partial differential equations is first transformed into a
system of ordinary differential equations in the form of a twopoint boundary value
problem (BVP) and then solved numerically using a finite difference scheme known as
the Keller box method. Numerical results are obtained for the velocity profiles and
the skin friction coefficient for various values of the moving parameter λ, the wedge parameter β, the reciprocal magnetic Prandtl number α and the magnetic parameter S. Results indicate that when the wedge and the fluid move in the opposite directions,
multiple solutions exist up to a critical value
MSC: 34B15, 76D10.
Keywords:
boundary layer; magnetohydrodynamic; induced magnetic field; moving wedge1 Introduction
Magnetohydrodynamics (MHD) is a subject that studies the behavior of an electrically conducting fluid in the presence of an electromagnetic field with applications in many different fields of engineering as well as geophysics, astrophysics, manufacturing, etc. The subject of MHD has been applied, for example, in problems associated with the confinement of plasma by magnetic fields and in projects involving thermonuclear generation of energy. In recent years it has been widely used in metallurgy industries involving sheetlike materials such as production of paper, polymer sheets and wire drawing and in horizontal continuous casting of hollow billets. For examples of these applications, see Li et al.[1] and Yan et al.[2]. Historically, the study of the hydrodynamic behavior of the boundary layer on a semiinfinite flat plate in the presence of a uniform transverse magnetic field has been first considered by Rossow [3]. Since then, the study of MHD flow and heat transfer fields past moving surfaces has drawn considerable attention with variations in types of geometrical surfaces and types of fluids.
The steady laminar flow of a viscous and incompressible fluid passing a fixed wedge
was first analyzed in the early 1930s by Falkner and Skan [4] to illustrate the application of Prandtl’s boundary layer theory, in which a similarity
transformation was used to reduce the boundary layer equations to an ordinary differential
equation known as the FalknerSkan equation. The FalknerSkan equation also represents
the boundary layer flow with streamwise pressure gradient. The general cases with
The present work aims to study the boundary layer flow over a moving wedge in a parallel
free stream of an electrically conducting fluid with the induced magnetic field. It
considers an extension of the results reported by Riley and Weidman [14] and Ishak et al.[16] on the flow characteristics of a moving wedge in a parallel free stream. Both studies
reported the existence of multiple solutions when the fluid and the wedge move in
the opposite directions within a specific range of moving parameter λ and a critical value
2 Basic equations
Consider the steady laminar flow of an incompressible electrically conducting fluid caused by a continuous moving wedge in a parallel free stream with a variable induced magnetic field applied parallel to the wedge walls outside the boundary layer (inviscid flow). Following Apelblat [24] or Cowling [28], the basic equations for the flow of a viscous, electrically conducting, incompressible fluid can be written in a vectorial form as follows:
where V is the fluid velocity vector, H is the induced magnetic field vector,
where
We will take the boundary conditions of equations (4)(6) to be
where
equations (4)(6) can be reduced to the following system of nonlinear ordinary differential equations:
subject to the boundary conditions (7) which are now transformed to
where primes denote differentiation with respect to η. Further, λ is the moving parameter, α is the reciprocal magnetic Prandtl number, β is the wedge parameter and S, the ratio of the magnetic to dynamic pressure, is the magnetic parameter. These parameters are defined as
We notice that different values of β characterize a number of mainstream flows. For
The physical quantity of interest is the skin friction coefficient which is defined as
where the wall shear stress is given by
where
We also notice that for
3 Results and discussion
Nonlinear ordinary differential equations (9) and (10) subject to the boundary conditions (11) form a twopoint boundary value problem (BVP) and are solved numerically using the Keller box method as described in the book by Cebeci and Bradshaw [29]. In this method, the solution is obtained using the following four steps:
(i) Reduce equations (9) and (10) to a firstorder system.
(ii) Write the difference equations using centered differences.
(iii) Linearize the resulting algebraic equations by Newton’s method and write them in the matrixvector form.
(iv) Solve the linear systems by the blocktridiagonalelimination technique.
The numerical method is then programmed using MATLAB R2010a software. To obtain a
numerical solution, it is required to make an appropriate guess for the step size
of η, Δη and the thickness of the boundary layer
Table 1
. Values of
Table 2
. Values of
Variations of the velocity profiles
Figure 1
. Variation of velocity profiles
Figure 2
. Variation of velocity profiles
Figure 3
. Variation of velocity profiles
Figure 4
. Variation of velocity profiles
Figure 5 presents the variation of the skin friction coefficient
Figure 5
. Skin friction coefficient
Figure 6 presents the variation of the skin friction coefficient
Figure 6
. Skin friction coefficient
Table 3
. Values of
Figure 7 shows the velocity profiles
Figure 7
. Velocity profiles at critical values
Figure 8
. Velocity profile
Figure 9 illustrates the variation of the skin friction coefficient
Figure 9
. Skin friction coefficient
Figure 10 presents the variation of the skin friction coefficient
Figure 10
. Skin friction coefficient
Figure 11
. (a) Velocity profiles
Figure 12
. (a) Velocity profiles
Figure 13
. (a) Velocity profiles
Figure 14 shows the variation of the induced magnetic gradient
Figure 14
. Induced magnetic gradient
Following the convention adopted by earlier researchers, we define the first two upper
branches of solutions as those for which
A reduction in the skin friction
According to the Lorenz law, the induced magnetic field will oppose the change in
the original magnetic field rather than the field itself. If, for example, the original
field is decreasing, then the induced magnetic field must be in the same direction
as the original field to oppose the decrease. From Figures 6 and 9, we see that the induced magnetic gradient
4 Conclusions
In this paper, we have considered similarity solutions for the steady MHD boundary
layer flow due to a continuous moving wedge in a parallel free stream with the induced
magnetic field. We investigated the effects of the moving parameter λ, the ratio of magnetic to dynamic pressure S, the wedge parameter β and the reciprocal magnetic Prandtl number α on the flow field and the induced magnetic field characteristics. It has been found
that increasing the values of the moving parameter λ and the wedge parameter β speeds up the fluid flow. In contrast, increasing the ratio of magnetic to dynamic
pressure S and the reciprocal magnetic Prandtl number α slows down the fluid flow. Furthermore, the skin friction or the surface shear stress
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The paper is the result of joint work of all authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Acknowledgements
The authors gratefully acknowledge the financial support received in the form of a FRGS research grant from the Ministry of Higher Education, Malaysia, and DIP201231 from the Universiti Kebangsaan, Malaysia. They also wish to express their sincere thanks to the reviewers for the valuable comments and suggestions.
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