Similarity Solutions of the MHD Boundary Layer Flow Past a Constant Wedge within Porous Media

Hindawi Mathematical Problems in Engineering Volume 2017, Article ID 1428137, 11 pages https://doi.org/10.1155/2017/1428137 Research Article Similarity Solutions of the MHD Boundary Layer Flow Past a Constant Wedge within Porous Media Ramesh B. Kudenatti,1 Shreenivas R. Kirsur,2 Achala L. Nargund,3 and N. M. Bujurke4 1 Department of Mathematics, Bangalore University, Bangalore 560 001, India Department of Mathematics, Gogte Institute of Technology, Belagavi 590 008, India 3 P. G. Department of Mathematics and Research Centre in Applied Mathematics, MES College, Malleswaram, Bangalore 560 003, India 4 Department of Mathematics, Karnatak University, Dharwad 580 003, India 2 Correspondence should be addressed to Ramesh B. Kudenatti; Received 25 September 2016; Revised 25 November 2016; Accepted 13 December 2016; Published 19 January 2017 Academic Editor: Hang Xu Copyright © 2017 Ramesh B. Kudenatti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The two-dimensional magnetohydrodynamic flow of a viscous fluid over a constant wedge immersed in a porous medium is studied. The flow is induced by suction/injection and also by the mainstream flow that is assumed to vary in a power-law manner with coordinate distance along the boundary. The governing nonlinear boundary layer equations have been transformed into a thirdorder nonlinear Falkner-Skan equation through similarity transformations. This equation has been solved analytically for a wide range of parameters involved in the study. Various results for the dimensionless velocity profiles and skin frictions are discussed for the pressure gradient parameter, Hartmann number, permeability parameter, and suction/injection. A far-field asymptotic solution is also obtained which has revealed oscillatory velocity profiles when the flow has an adverse pressure gradient. The results show that, for the positive pressure gradient and mass transfer parameters, the thickness of the boundary layer becomes thin and the flow is directed entirely towards the wedge surface whereas for negative values the solutions have very different characters. Also it is found that MHD effects on the boundary layer are exactly the same as the porous medium in which both reduce the boundary layer thickness. 1. Introduction The research in MHD boundary layer flow has many important engineering applications such as power generators, the cooling of reactors, polymer industry, and spinning of filaments. In industrial applications, when sheets or filaments are made to cool, these get stretched. This cooling can be controlled by applying the magnetic field; we would then expect the final products with desired shapes. Because of these significant applications, Pavlov [1] considered the boundary layer flow of a conducting incompressible viscous fluid due to deformation of an elastic surface in uniformly applied magnetic field. Andersson [2] studied the MHD boundary layer flow of a viscous fluid past a stretching surface and showed that external magnetic field has the same effect on the flow as the viscoelasticity. There are numerous papers available in the literature on classical MHD boundary layer flows (Watanabe and Pop [3]; Chaturvedi [4]) and along with heat and mass transfer flows (Yih [5], Sobha and Ramkrishna [6], Aly et al. [7], etc.). Joneidi et al. [8] undertook the study of heat and mass transfer of viscous fluid in an electrically conducting fluid and showed that MHD decreases the boundary layer thickness. Hayata et al. [9] have modeled the two-dimensional magnetohydrodynamic boundary layer flow in a channel with porous walls, and they considered Maxwell’s fluid in the porous space between the channel walls and solved the governing ordinary differential equations by homotopy analysis method. The effects of all embedded flow parameters on the dimensionless velocity components and temperature along with Nusselt number are analyzed. Pantokratoras [10] has obtained the exact solutions to the boundary layer flow along a vertical plate in the 2 Mathematical Problems in Engineering presence of the applied magnetic field and obtained analytical expressions in terms of series form for Blasius-Sakiadis and Sakiadis flows. Xu et al. [11] have investigated the boundary layer flow and heat transfer in an incompressible viscous electrically conducting fluid that is caused by impulsive stretching of the surface and used a well-developed homotopy analysis method. They showed that the magnetic parameter reduces the boundary layer thickness but enhances thermal boundary layer thickness. The study of fluid flows and mass transfer problems has significant applications in a wide variety of geophysical and engineering application such as flow of ground water energy storage and chemical reactors (Nield and Bejan [12]; Ingham et al. [13]). Some materials such as sintered bronze or metal sheet perforated with numerous small holes are porous. If these materials are used as the boundary of a region of fluid flow and free stream velocity is maintained on the other side away from the flow, the fluid will be sucked/injected through the boundary. Then the appropriate boundary conditions for the flow region is the normal component of the relative velocity of fluid and surface at the boundary should be equal to some value determined by the porosity that is represented by the normal relative velocity 𝛼 (𝛼 > 0 for suction and 𝛼 < 0 is injection) (Batchelor [14]). Yao [15] studied the FalknerSkan flow over a wedge in a porous medium using the approximate analytical solution given by homotopy analysis. Guedda et al. [16] have explored numerically for the case of MHD mixed boundary layer flow over a vertical flat plate in a porous medium and shown that there exist multiple solutions for specific parameters. It is noticed that the exact solutions in boundary layer flow problems through similarity transformations are very few. In fact the classical Falkner-Skan equation, 𝑓󸀠󸀠󸀠 (𝜂) + 𝑓 (𝜂) 𝑓󸀠󸀠 (𝜂) + 𝛽 (1 − 𝑓󸀠2 (𝜂)) = 0, (1) with the boundary conditions 𝑓 (0) = 0, 𝑓󸀠 (0) = 0, (2) 𝑓󸀠 (∞) = 1, admits no analytical solution, because the nonlinearities in the equation and appearance of the pressure gradient parameter make the above system susceptible to obtain an analytical solution. However, when one or both of the initial conditions are nonzero and for 𝛽 = −1, the above system yields an analytical solution in the form of error and exponential functions (Sachdev et al. [17]). Exploring this known analytical solution, they modified and rewrote it to obtain an exact solution of the Falkner-Skan equation for all possible values of 𝛽. Pioneered by this work, Kudenatti et al. [18] obtained an analytical solution of the MHD Falkner-Skan equation for various values of 𝛽 and the Hartmann number 𝑀 and explored that their so (...truncated)