MHD boundary layer flow due to a moving wedge in a parallel stream with the induced magnetic field
© Jafar et al.; licensee Springer. 2013
Received: 30 September 2012
Accepted: 14 January 2013
Published: 11 February 2013
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 two-point 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 of the moving parameter λ, whose value depends on the values of S and β.
MSC: 34B15, 76D10.
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 sheet-like 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.  and Yan et al. . Historically, the study of the hydrodynamic behavior of the boundary layer on a semi-infinite flat plate in the presence of a uniform transverse magnetic field has been first considered by Rossow . 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  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 Falkner-Skan equation. The Falkner-Skan equation also represents the boundary layer flow with stream-wise pressure gradient. The general cases with were numerically studied by Fang  and Weidman et al.  independently. There are many references on the solutions of Falkner-Skan equations; for example, see Hartree , Hastings , Brodie and Banks , Pantokratoras , Alizadeh et al. , Yao , and Abbasbandy and Hayat . Similarity solutions for pressure gradient driven flow over a stretching boundary were analyzed by Riley and Weidman  for the case of external velocity and boundary velocity being proportional to the same powers of the downstream coordinate. Very interesting and extensive results were reported demonstrating a rich variety of solutions available, including the existence of multiple solutions, and an exact solution was also presented for . Fang and Zhang  studied a special case of the Falkner-Skan equation with in the presence of wall suction and injection. An exact solution was presented for the boundary conditions with both wall mass transfer and wall movement, with different solution behavior identified in different solution regions. On the other hand, Ishak et al.  considered the steady MHD boundary layer flow in a conducting fluid flowing transverse to a variable magnetic field along a moving wedge in a free stream. The results reported were consistent with those found by Riley and Weidman  and with earlier studies by the same authors Ishak et al. [17, 18]. More recent studies on similar problems were done by Van Gorder and Vajravelu , Postelnicu and Pop  and Parand et al. .
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  and Ishak et al.  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 beyond which the solution is non-existent. The present study considers the corresponding MHD flow of the paper by Ishak et al. , but with the induced magnetic field, and investigates how this magnetic field affects the flow and the critical value . The induced magnetic field is assumed to be applied parallel to the wedge walls at the outer edge of the boundary layer. Such an induced magnetic field has been also considered by Davies , Apelblat [23, 24], Kumari et al. , Takhar et al.  and more recently by Kumari and Nath . To obtain the solutions, the governing partial differential equations are first transformed into ordinary differential equations using a similarity transformation. The ordinary differential equations obtained are then solved numerically by a very efficient finite difference scheme known as the Keller box method for some values of the selected parameters. The effect of the induced magnetic field on the flow field for different values of the wedge parameter β is included in the analysis. Particular cases of the present results are compared with those reported by Riley and Weidman  and Ishak et al. [16, 17].
2 Basic equations
where and are the x-velocity and magnetic field at the edge of the boundary layer, respectively. We assume here that and , where is the constant velocity at the outer edge of the boundary layer and is the value of at . Further, m is also a constant, which varies in the range .
We notice that different values of β characterize a number of main-stream flows. For , equations (9) and (10) are reduced to the MHD Blasius problem. The values and are equivalent to the flow past a wedge placed symmetrically in a stream. For MHD boundary layers, we take the values of the parameters S and α to be in the range and ; see Davies  and Kumari et al. . This is the same range of magnetic parameter adopted by Takhar et al.  and several earlier researchers investigating similar problems. It is also consistent with the existence of the steady-state solution of the ‘super Alfven’ flow.
where is the local Reynolds number.
We also notice that for the present problem corresponds to the MHD boundary layer flow over a static wedge, which has been considered by Apelblat , in which the MHD wedge problem was solved using the Laplace transform method to give an infinite series approximation solution for and . On the other hand, it may be noted that for (without a magnetic field), equation (9) reduces to that of Ishak et al. . Therefore, as implies the absence of a magnetic field, equation (10) governing the induced magnetic field is no longer necessary.
3 Results and discussion
Nonlinear ordinary differential equations (9) and (10) subject to the boundary conditions (11) form a two-point boundary value problem (BVP) and are solved numerically using the Keller box method as described in the book by Cebeci and Bradshaw . In this method, the solution is obtained using the following four steps:
Write the difference equations using centered differences.
Linearize the resulting algebraic equations by Newton’s method and write them in the matrix-vector form.
Solve the linear systems by the block-tridiagonal-elimination technique.
Values of for , and various β
Values of for ,
Riley and Weidman 
Values of for different values of m when
Following the convention adopted by earlier researchers, we define the first two upper branches of solutions as those for which is greater for a given value of β, while the third branch is that with the smallest value of . We notice that the velocity profiles for the first two upper branches of solutions exhibit the same monotonic behavior. The boundary layer for the first branch is usually very thin and the velocity profile rapidly attains the value . In general, the third branch of solutions usually involves a much larger boundary layer thickness compared to the other two branches. It is usually characterized by starting off with a rather small value of , with a non-monotonic behavior in the development of the velocity profiles , before assuming its final asymptotic value . Similar non-monotonic behavior was reported by Riley and Weidman  when they considered the velocity profiles of the upper branch solution for . Following Ishak et al. , we postulate that the upper branch of solutions with the highest value of (first solutions) are physically stable and occur in practice since it is the only solution for , i.e., when the fluid and the solid surface move in the same direction.
A reduction in the skin friction implies a reduction in the drag force. Thus, the magnetic field reduces the drag force and speeds up the separation. On the other hand, increasing the included angle of the wedge will increase the drag force, hence delaying the separation. This result is consistent with that reported by Ishak et al. .
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 increases monotonically with the increasing value of λ. This increase is supposedly opposing a decrease in the original magnetic field. Furthermore, the induced magnetic gradient also decreases with the increase in S, which is consistent with the Lorenz law. We also notice that the effect of both S and β is more pronounced on the skin friction compared to the induced magnetic gradient .
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 and the induced magnetic gradient decrease with the increase of the ratio of magnetic to dynamic pressure S, but increase with the wedge parameter β. We have also demonstrated the existence of a rich variety of solutions by varying the value of the wedge parameter β. We have also found that when the wedge and the fluid move in the same direction, the solution is unique for all values of the parameters β and S. However, when the wedge and the free stream move in the opposite directions, multiple solutions exist for some range of values of the moving parameter λ as soon as the value of the moving parameter is greater than a critical value . This critical value of λ is dependent on both parameters β and S. It has been found that increasing the wedge parameter β will increase the value of , while increasing the ratio of magnetic to dynamic pressure S will reduce it. Thus, increasing the ratio of magnetic to dynamic pressure speeds up the boundary layer separation, while increasing the wedge parameter β delays it.
The authors gratefully acknowledge the financial support received in the form of a FRGS research grant from the Ministry of Higher Education, Malaysia, and DIP-2012-31 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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