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The Effect of Induced Magnetic Field and Convective Boundary Condition

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Propulsion and Power Research 2016;5(2):164–175 HOSTED BY http://ppr.buaa.edu.cn/ Propulsion and Power Research www.sciencedirect.com ORIGINAL ARTICLE The effect of induced magnetic field and convective boundary condition on MHD stagnation point flow and heat transfer of upper-convected Maxwell fluid in the presence of nanoparticle past a stretching sheet Wubshet Ibrahimn Department of Mathematics, Ambo University, P.O. Box 19, Ambo, Ethiopia Received 5 February 2015; accepted 17 July 2015 Available online 30 May 2016 KEYWORDS Abstract The present study examines the effect of induced magnetic field and convective fl fl boundary condition on magnetohydrodynamic (MHD) stagnation point ow and heat transfer due Nano uid; fl Stagnation point flow; to upper-convected Maxwell uid over a stretching sheet in the presence of nanoparticles. fi Heat transfer; Boundary layer theory is used to simplify the equation of motion, induced magnetic eld, energy Convective boundary and concentration which results in four coupled non-linear ordinary differential equations. The condition; study takes into account the effect of Brownian motion and thermophoresis parameters. The Induced magnetic governing equations and their associated boundary conditions are initially cast into dimensionless field; form by similarity variables. The resulting system of equations is then solved numerically using Upper-convected Max- fourth order Runge-Kutta-Fehlberg method along with shooting technique. The solution for the well fluid governing equations depends on parameters such as, magnetic, velocity ratio parameter B,Biot number Bi, Prandtl number Pr, Lewis number Le, Brownian motion Nb, reciprocal of magnetic Prandtl number A, the thermophoresis parameter Nt, and Maxwell parameter β. The numerical results are obtained for velocity, temperature, induced magnetic field and concentration profiles as well as skin friction coefficient, the local Nusselt number and Sherwood number. The results indicate that the skin friction coefficient, the local Nusselt number and Sherwood number decrease with an increase in B and M parameters. Moreover, local Sherwood number -ϕ0(0) decreases with an increase in convective parameter Bi, but the nTel.: þ251 911892494. E-mail address: [email protected] Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China. 2212-540X & 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/j.jppr.2016.05.003 The effect of induced magnetic field and convective boundary condition on MHD stagnation point flow 165 local Nusselt number -θ0(0) increases with an increase in Bi. The results are displayed both in graphical and tabular form to illustrate the effect of the governing parameters on the dimensionless velocity, induced magnetic field, temperature and concentration. The numerical results are compared and found to be in good agreement with the previously published results on special cases of the problem. & 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction Recently, the study of heat transfer under convective boundary condition received numerous interests on the The study of MHD stagnation point flow of a viscous researcher's side due to its influence on heat transfer incompressible fluid over a stretching sheet has attracted the characteristics on the surface and consequently, the quality interest of many researchers due to its important applica- of the final product of the manufacturing industries. The tions in industry and manufacturing processing [1–6]. study of a convective heat transfer in magnetic field is Particularly, the study of steady magnetohyderodynamic important in processes, such as gas turbine, nuclear plants, (MHD) stagnation point flow of an electrically fluid has thermal energy storage, etc. Aziz [13] was a pioneer in numerous applications. Some of which are found in the using convective heating process to study boundary layer field of metallurgy and chemical engineering process such flow of the classical Blasius problem over a flat plate. He as, drawing, annealing and thinning of copper wire, etc. studied a similarity solution for laminar thermal boundary Magnetic field is one of the important factors by which layer flow over a flat plate with convective boundary the cooling rate can be controlled and the desired quality of condition. Furthermore, Aziz [14] continued his investiga- the industrial product can be achieved. Many investigations tion on flat plate with convective boundary condition and were made to examine flow characteristics over a stretching analyzed hydrodynamic and thermal slip flow boundary sheet under different conditions of MHD. Concerning this, layer over a flat plate with constant heat flux boundary Pavlov [7] and Hayat et al. [8] have investigated the effect condition. Since then, different researchers extended the of a magnetic field on the viscous flow of an electrically idea to various configurations of the flat plate and stretching conducting fluid past a stretching sheet. sheet. Accordingly, researchers in the references [15–19] The study of MHD boundary layer flow over a stretching studied the heat transfer under convective boundary condi- sheet has extended and included other parameters, such as tion over a surface. induced magnetic field. Accordingly, Ali et al. [9] examined Furthermore, Makinde and Olanrewaju [20] studied the MHD boundary layer flow and heat transfer over a stretching combined effects of buoyancy force and convective heat sheet with induced magnetic field. The result shows that the transfer on thermal boundary layer over a vertical plate with velocity and the induced magnetic field profiles increase with an convective boundary condition. Still further, investigators in increase in the applied magnetic field. Furthermore, they refs. [21–22] studied the convective heating condition by discussed some important application areas of MHD flow, such taking into account the condition, such as slip, permeability, as in designating cooling system with liquid metals, MHD porosity for Newtonian and non-Newtonian fluids over a generator and other devices in many manufacturing industries. surface. Ali et al. [10] also studied the effect of induced magnetic Nowadays, the studies on boundary layer flow and heat field on the stagnation point flow of Newtonian fluids. The transfer of a nanofluid is active research area in fluid science numerical results of their study indicated that the skin and engineering, and many research papers have been friction coefficient decreases as the magnetic parameter published on this area. Moreover, investigators in the increases. Furthermore, Ali et al. [11] studied numerically references [23–27] have studied the convective boundary- the MHD mixed convection boundary layer flow towards a layer flow of a nanofluid past a stretching surface. They stagnation point flow on a vertical surface with the effect of used the model in which Brownian motion and thermo- induced magnetic field. It is indicated that the induced field phoresis effects were taken into account. Their result is most affected by the reciprocal of the magnetic Prandtl indicates that the local Nusselt number is a decreasing number when compared with the skin friction and heat function of the parameters, such as Brownian motion and transfer coefficients. Still further, Kumari and Nath [12] thermophoresis parameter. analyzed numerically steady mixed convection stagnation Furthermore, Olanrewaju and Makinde [28] have ana- point flow upper convected Maxwell fluids with induced lyzed a boundary layer stagnation point flow of a nanofluid magnetic field. Their study shows that the surface velocity over a permeable flat surface with Newtonian heating. Still gradient and heat transfer are increased by increasing further, Wubshet and Shanker [29] have numerically magnetic and buoyancy parameters. examined the boundary-layer flow and heat transfer of 166 Wubshet Ibrahim Nomenclature Tw uniform temperature over the surface of a sheet Tf temperature of hot fluid A the reciprocal of the magnetic Prandtl number T1 ambient temperature B velocity ratio u, v velocity component along x and y direction Bi Biot number B0 magnetic field parameter Greek letters Cf skin friction coefficient C1 ambient concentration α thermal diffusivity DB Brownian diffusion coefficient β Deborah number DT thermophoresis diffusion coefficient η dimensionless similarity variable f dimensionless velocity stream function m dynamic viscosity of the fluid h dimensionless magnetic stream function υ kinematic viscosity of the fluid hf heat transfer coefficient ψ stream function k thermal conductivity θ dimensionless temperature Le Lewis number τ parameter defined by (ρc)p/(ρc)f M magnetic parameter σ electrical conductivity Nb Brownian motion parameter Nt theromophoresis parameter Subscripts Nux local Nusselt number Pr Prandtl number 1 condition at the free stream Rex local Reynolds number w condition at the surface Shx local Sherwood number T temperature of the fluid inside the boundary layer Nanofluid over a vertical plate with convective surface [36] studied the effect of double stratification on boundary boundary condition. Their study indicated that the local layer flow and heat transfer of nanofluid over a vertical Nusselt number
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