Mathematical Problems in Engineering

Volume 2018, Article ID 9402836, 11 pages

https://doi.org/10.1155/2018/9402836

## Influences of Slip Velocity and Induced Magnetic Field on MHD Stagnation-Point Flow and Heat Transfer of Casson Fluid over a Stretching Sheet

^{1}Department of Mathematics, Faculty of Science, King Khalid University, Abha 9004, Saudi Arabia^{2}Department of Mathematics, Deanship of Educational Services, Qassim University, P.O. Box 6595, Buraidah 51452, Saudi Arabia

Correspondence should be addressed to Mohamed Abd El-Aziz; moc.oohay@999zizaledba_m

Received 3 March 2018; Revised 3 May 2018; Accepted 22 May 2018; Published 11 July 2018

Academic Editor: Efstratios Tzirtzilakis

Copyright © 2018 Mohamed Abd El-Aziz and Ahmed A. Afify. 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.

#### Abstract

The steady MHD boundary layer flow near the stagnation point over a stretching surface in the presence of the induced magnetic field, viscous dissipation, magnetic dissipation, slip velocity phenomenon, and heat generation/absorption effects has been investigated numerically. The Casson fluid model is used to characterize the non-Newtonian fluid behavior. The governing partial differential equations using appropriate similarity transformations are reduced into a set of nonlinear ordinary differential equations, which are solved numerically using a shooting method with fourth-order Runge-Kutta integration scheme. Comparisons with the earlier results have been made and good agreements were found. Numerical results for the velocity, induced magnetic field, temperature profiles, skin friction coefficient, and Nusselt number are presented through graphs and tables for various values of physical parameters. Results predicted that the magnetic parameter with has the tendency to enhance the heat transfer rate, whereas the reverse trend is seen with . It is also noticed that the rate of heat transfer is a decreasing function of the reciprocal of a magnetic Prandtl number, whereas the opposite phenomenon occurs with the magnitude of the friction factor.

#### 1. Introduction

The steady MHD boundary layer flow of an incompressible viscous fluid near the stagnation point has received great attention owing to their wide applications in the various fields of industry and engineering applications such as design of thrust bearings, transpiration cooling, and aerodynamics extrusion of plastic sheets. The classical two-dimensional stagnation point flow on a flat plate was first studied by Hiemenz [1]. Hiemenz problem was extended to the axisymmetric case by Homann [2]. The impact of an external magnetic field on Hiemenz flow of an electrically conducting fluid was investigated by some researchers [3–7]. Recently, the MHD stagnation point flow past a stretching sheet with the influences of radiation, velocity, and thermal slip phenomena was analyzed by Khan et al. [8].

In recent years, it has been observed that a number of industrial fluids such as molten plastics, polymeric liquids, foodstuff, and slurries exhibit non-Newtonian fluid behavior. For non-Newtonian fluids, various models have been proposed. The vast majorities of non-Newtonian fluid models are concerned with simple models like the power law and grade two or three etc. Reviews of non-Newtonian fluid problems have been presented in [9–15]. There is another non-Newtonian fluid model, known as the Casson fluid model. The Casson fluid can be defined as a shear thinning the liquid which is assumed to have an infinite viscosity at zero the rate of shear, a yield stress below where no flow occurs, and a zero viscosity at an infinite rate of shear. This fluid has significant applications in polymer processing industries and biomechanics. Boundary layer flow of Casson fluid over different geometries is considered by many authors [16–21]. Recently, Khan et al. [22] numerically discussed the influence of chemical reaction on an unsteady Casson fluid over the stretching surface.

The induced magnetic field has received considerable interest owing to its use in many scientific and technological phenomena, for example, in MHD energy generator systems and magnetohydrodynamic boundary layer control technologies. The influence of induced magnetic field on unsteady MHD free convective flow over a semi-infinite vertical surface was investigated by Kumar and Singh [23]. Beg et al. [24] investigated the hydromagnetic convection flow of a Newtonian, electrically conducting fluid over a translating, nonconducting plate with the aligned magnetic field. The impacts of a transverse magnetic field and magnetic induction on MHD natural convection boundary layer flow over an infinite vertical flat plate were analytically studied by Ghosh et al. [25]. Ali et al. [26] discussed the effect of an induced magnetic field on boundary layer stagnation-point flow over a stretching surface. Iqbal et al. [27] studied the combined effects entropy generation and induced the magnetic field on stagnation point flow and heat transfer due to nanofluid towards a stretching sheet.

Slip boundary condition is a very developed phenomenon which includes the nonadherence of fluids to surfaces. Fluids exhibiting slip are important in the areas of technology and industry such as in the polishing of artificial heart valves and internal cavities. In this context, Kundsen number is a deciding coefficient, which is a measure of the molecular mean free path to characteristic length. When Knudsen number is very small, no slip is noticed between the surface and the fluid and is in tune with the essence of continuum mechanics. Beavers and Joseph [28] proposed a slip flow boundary condition. The influences of thermal radiation, Newtonian heating, and slip velocity phenomenon on MHD flow and heat transfer past a permeable stretching sheet were numerically studied by Afify et al. [29]. The impacts of slip flow, convective boundary condition, and thermal radiation on mixed convection heat and mass transfer flow over a vertical surface were numerically discussed by Uddin et al. [30]. The impact of viscous dissipation and velocity slip phenomenon on ferrofluid flows past a slender stretching sheet was investigated by Ramana Reddy et al. [31]. The influence of nonuniform heat source and slip flow on MHD nanofluid flow past a slandering stretching sheet was analyzed by Ramana Reddy et al. [32]. Recently, Hosseini et al. [33] analyzed the flow and heat transfer characteristics of an unsteady flow past a permeable stretching sheet in the presence of the velocity slip factor and temperature jump influences.

To the best of the author's knowledge, this work has not been previously studied in the scientific research. The main aim of this paper is to analyze the MHD stagnation-point flow and heat transfer of a non-Newtonian fluid known as Casson fluid over a stretching surface in the presence of the induced magnetic field, viscous dissipation, velocity slip boundary condition, and heat generation/absorption effects. Diagrams and tables are presented and discussed for various physical parameters entering into the problem.

#### 2. Mathematical Formulation

Consider the steady magnetohydrodynamic (MHD) flow of a non-Newtonian Casson fluid near the stagnation point over a stretching surface coinciding with the plane , the flow being confined to . Two equal and opposite forces are applied along the x-axis so that the surface is stretched keeping the origin fixed. The effect of the induced magnetic field is taken into account. The flow configuration is shown in Figure 1. The viscous dissipation, magnetic dissipation, and heat generation/absorption terms are included in the energy equation. The rheological equation of state for an isotropic and incompressible flow of Casson fluid can be expressed as follows (Eldabe and Salwa [34]): where is the plastic dynamic viscosity of the non-Newtonian fluid, is the yield stress of fluid, is the product of the component of deformation rate by itself, namely, , is the, (*i, j*)-the component of the deformation rate, and is a critical value of based on non-Newtonian model. Under the above-mentioned assumptions and the boundary layer approximations, the governing equations of Casson fluid can be written as (Cowling [35])