# Thermal Radiation Effects on Hydromagnetic Mixed Convection Flow along a Magnetized Vertical Porous Plate

Authors: Muhammad Ashraf; S. Asghar; Md. Anwar Hossain
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

Aim of the present work is to investigate the effect of radiation on steady mixed convection boundary layer flow of viscous, incompressible, electrically conducting fluid past a semi-infinite magnetized vertical porous plate with uniform transpiration and variable transverse magnetic field along the surface. The equations governing the flow magnetic and temperature field
are reduced to dimensionless convenient form using the free variable transformations and solved
numerically by using finite difference method. Effects of physical parameters like Prandtl number,
Pr, the conduction-radiation parameterRd, magnetic field parameter S, magnetic Prandtl number
Pm, mixed convection parameter λ, and the surface temperature, θw on the local skin friction coefficient Cfx, local Nusselt number, Nux, and coefficient of magnetic intensity, Mgx against
the local transpiration parameter ξ are shown graphically. Later, the problem is analysed by using
series solution for small and large values of ξ, and the results near and away from the leading
edge are compared with numerical results obtained by finite difference method and found to be in
good agreement.


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## Body

## 1. Introduction

Thermal radiation effects on magnetohydrodynamics of an electrically conducting fluid flows are important in the context of space technology and processes involving high temperature. Physical interests of theses flows encountered in many engineering problems and industrial areas such as propulsion devices for missiles, aircraft, satellites, nuclear power plants, take place at high temperature and radiation effects play a significant role in designing them. One physical interest in this flow lies in the possibility of using such a field to shield a body from excessive heating and radiations. Here the literature survey is being started with the history of the work done by other authors along nonmagnetized, magnetized and then with porous surface and the radiation effects on these surface.Greenspan and Carrier [1] was the first who investigated the flow of viscous, incompressible and electrically conducting fluid in the presence of a symmetrically oriented semi-infinite flat plate in which magnetic field assumed to be coincident with the ambient fluid velocity field. In this investigation fourier transformation together with asymptotic analysis had been incorporated and found that the velocity gradient at the plate approaches zero due to increase in the applied magnetic field intensity. Further contributions to the problem was added by Davies  [2, 3] considering the fact that the flow is opposed by magnetodynamic pressure gradient along an nonmagnetized plate and concluded that for the magnetic field parameter (or Chandrashekhar number) S≃1 the drag coefficient vanishes. Gribben  [4] then considered an axisymmetric magnetohydrodynamic flow of an incompressible, viscous, electrically conducting fluid near a stagnation point considering that the magnetic field lines are circles and parallel to the surface. Later, Gribben [5] who investigated the magnetohydrodynamic boundary layer in steady incompressible flow under the influence of an external magnetic dynamic pressure gradient using the asymptotic analysis and found that the skin friction decreases with the increase of magnetic field. The boundary layer flow and heat transfer of hydromagnetic flow of viscous incompressible fluid flows past an electrically insulated semi-infinite flat plate in the presence of a uniform magnetic field parallel to the plate has been investigated by Ramamoorthy [6] numerically and found that the presence of the magnetic field increases both the momentum and thermal boundary layer thicknesses. On the other hand Tan and Wang [7] studied the effect of applied magnetic field on temperature distribution as well as on the recovery temperature due to the flow of a viscous incompressible electrically conducting fluid past a solid plane surface subject to uniform heat flux. They concluded that the values of recovery factor decreases with the increase of both magnetic field parameter S and magnetic Prandtl number Pm. Hildyard [8] found that the magnetic-field boundary condition used by Gribben was inappropriate and hence making the necessary correction obtained the appropriate asymptotic solutions for large and small values of the magnetic Prandtl number, Pm. Later, Chawla [9] studied the effect of free stream fluctuations on the flow over a semi-infinite plate, with an aligned magnetic field, using von Kármán-Pohlhausen technique and solution for low and high frequency ranges are developed. But, Ingham [10] studied the boundary layer flow on a semi-infinite flat plate placed at zero incidence to a uniform stream of electrically conducting gas with an aligned magnetic field at large distances from the plate. In this analysis the author observed that increasing magnetic field for a given Mach number, or decreasing the Mach number for a given magnetic field thickens the momentum and thermal boundary layer.In all the above investigations, the surface along which the flow of the fluid were considered as nonmagnetized. In recent technological development it is necessary to distorted the attention towards magnetized surface, Glauert [11], first, studied the magnetohydrodynamic boundary layer in uniform flow past a magnetized plate for the small and large values of magnetic Prandtl number, Pm. The observation from this investigation, shows that the velocity and magnetic fields are valid for small value of magnetic field parameter S and for both smaller or larger value of magnetic Prandtl number Pm.Chawla [12] studied the magnetohydrodynamics boundary layer in uniform flow past a semi-infinite magnetize plate, and a magnetic field fluctuating about a nonzero mean in the stream direction, is applied to the plate. He comments that in order to create a fluctuating magnetic field, one needs to join the plate in the manner of Glauert [11] with d.c and a.c generators placed in series. However, Chawla assuming the amplitude of the oscillating transverse magnetic field is much smaller than the uniform magnetic field at the surface. He also considered the basic steady flow using Karman-Pohlhausen technique, and obtained approximate solutions to both steady and oscillating part of the flow.The effects of thermal radiation in different geometries have been discussed by several authors. In this respect, Ali et al. [13] focus on the effect of radiation interaction in boundary layer flow over horizontal surface. Arpaci [14], Cheng and Özişik [15], Sparrow and Cess [16] and highlight the thermal radiation effect with free convection from a heated vertical semi-infinite plate. Soundalgekar et al. [17] have studied radiation effects on free convection flow past a semi-infinite plate using the Colgey-Vincenti equilibrium model. Hossain and Takhar [18] have analyzed the effect of radiation using Rosseland diffusion approximation which leads to nonsimilar solutions for the forced and free convection flow of an optically dense viscous incompressible fluid past a heated vertical plate with uniform free stream velocity and surface temperature. The effect of conduction-radiation on oscillating natural convection boundary layer flow of optically dense viscous incompressible fluids along a vertical plate has been studied by Roy and Hossain [19]. Aboeldahab and Gendy  [20] studied MHD free convection flow of gas past a semi-infinite vertical plate with variable thermophysical properties for high temperature difference by taking into consideration radiation effects and solved the problem numerically using the shooting method. Effects of thermal radiation on unsteady free convection flow past a vertical porous plate with Newtonian heating have recently been demonstrated by Mebine and Adigio [21], who obtained the analytical results by using the Laplace transform technique. Palani and Abbas [22] studied the combined effect of MHD and radiation on free convection flow past an impulsively started isothermal plate with Rosseland diffusion and solved the dimensionless governing equations numerically using the finite element method. Convective boundary layer flows are often controlled by injecting (blowing) or suction (withdrawing) fluid through porous bounding heating surface. This can lead to enhanced heating or cooling of system and can help to delay the transition from laminar to turbulent flow. Eichhorn [23], for example, obtained those power law variations in surface temperature and transpiration velocity which give rise to a similarity solution for the flow from a vertical surface. The case of uniform suction and blowing through an isothermal vertical wall was investigated first by Sparrow and Cess [24], they obtained a series solution which is valid near the leading edge. The problem was considered in more detail by Merkin [25], who obtained asymptotic solutions, valid at large distances from the leading edge, for both the suction and blowing. Using the method of matched asymptotic expansions, the next order correction to the boundary layer solution for this problem was obtained by Clarke [26], who obtained the range of applicability of the analysis by not invoking the the Boussinesq approximation. The effect of strong suction and blowing from general body shapes which admit a similarity solution has been given by Merkin [25]. A transformation of the equations for general blowing and wall temperature variations has been discussed by Vedhanayagam et al. [27]. The case of heated isothermal horizontal surface with transpiration has been discussed in some detail by Clarke and Riley [28] and then by Lin and Yu  [29].The above literature survey shows no existence of any study on the effects of thermal radiation on boundary layer flow of an electrically conducting fluid under both magnetic and buoyancy force along a magnetized porous plate. Hence in the present article the problem investigated is the thermal radiation effects on hydromagnetic mixed convection laminar boundary layer flow of a viscous, incompressible and electrically conducting fluid along a magnetized permeable surface with a variable magnetic field applied in stream direction at the surface. The boundary layer equations for the momentum, energy and magnetic field are reduced to convenient form for integration using appropriate transformations. The solutions of the transformed boundary layer equations are then simulated employing two methods, namely, (i) finite difference method and the (ii) asymptotic series solution for small and large value of local transpiration parameterξ=(V0x/ν)/Rex1/2 that depends on the surface mass-flux, V0, as well as the distance x measured from the leading edge of the plate. The pertinent physical parameters that dominate the flow and other physical quantities, such as the local skin-friction Cfx, rate of heat transfer, Nux and the magnetic intensity Mgx at the surface are the magnetic field parameter, S, and,conduction-radiation parameter Rd, Prandtl number Pr and the magnetic Prandtl number Pm and mixed convection parameter λ also with the surface temperature parameter θw.


## 2. Formulation of the Mathematical Model

We consider the radiation interaction on the laminar two-dimensional magnetohydrodynamic mixed convection flow of an electrically conducting, viscous and incompressible fluid past a uniformly heated vertical porous plate. Thex-axis is taken along the surface and y-axis is normal to it. A schematic diagram illustrating the flow domain and the coordinate system is shown in Figure 1. In Figure 1  δM and δT stands for momentum and thermal boundary layer thicknesses. It is assumed that the surface temperature Tw of the plate is greater than the ambient fluid temperature T∞(2.1)∂u∂x+∂v∂y=0,(2.2)u∂u∂x+v∂v∂y=υ∂2u∂y2+μ¯ρ(Hx∂Hx∂x+Hy∂Hx∂y)+gβ(T-T∞),(2.3)∂Hx∂x+∂Hy∂y=0,(2.4)-γ∂Hx∂y=(uHy-vHx),(2.5)u∂T∂x+V∂T∂y=α∂2T∂y2-∂qr∂y,
where(2.6)qr=-Kr∂T∂y,Kr=-16σsT33αR,∂qr∂y=-Kr∂2T∂y2,
where u and v are fluid velocity components in x- and y-direction, respectively, Hx and Hy are the x- and y-components of magnetic field, qr is the radiative heat flux in the y-direction, α, μ¯ρ, ν and γ are the thermal diffusion, magnetic permeability, density, kinematic coefficient of viscosity and magnetic diffusivity of the medium. The solution of the above equations should satisfy the following boundary conditions:(2.7)u(x,0)=0,v¯(x¯,0)=±V0,Hx(x,0)=Hw(x),T(x,0)=Tw,u(x,∞)=U∞(x),Hx(x,∞)=0,T(x,∞)=T∞.
The nonlinearity of the momentum, hydromagnetic and energy equation makes it difficult to obtain a closed mathematical solution to the problem. However, by introducing the following nondimensional dependent and independent variables we have,(2.8)u=U0u¯,v=vLReL1/2v¯,x¯=xL,y¯=yLReL1/2y,Hx=H0H¯x,Hy=H0ReL1/2H¯y,θ=T-T∞ΔT,
where ΔT is the temperature difference. By using expression (2.8) in (2.1)–(2.15), we have(2.9)∂u¯∂x¯+∂v¯∂y¯=0,(2.10)u¯∂u¯∂x¯+v¯∂v¯∂y¯=∂2u¯∂y¯2+S(H¯x∂H¯x∂x¯+H¯y∂H¯x∂y¯)+λθ¯,(2.11)∂H¯x∂x¯+∂H¯y∂y¯=0,(2.12)-1Pm∂H¯x∂y¯=(u¯H¯y-v¯H¯x),(2.13)u¯∂θ¯∂x¯+V¯∂θ¯∂y¯=1Pr[1+43Rd(1+(θw-1)θ)3]∂2θ¯∂y¯2,
where(2.14)ReL=U0Lν,GrL=gβΔTL3ν2,Pr=να,λ=GrLReL2,Pm=υγ,α=KρCp,S=μ¯H02ρU02,Rd=KαR4σT∞3,
where ReL is the Reynolds number, GrL the Grashof, number, Rd is the Plank number (radiation-conduction parameter), L the reference length, λ is the mixed convection parameter, Pr the Prandtl number and S the magnetic field parameter (also known as Chandrasekhar number), Pm is the magnetic Prandtl number, and α is the thermal diffusion. The corresponding boundary condition take the form:(2.15)u¯(x¯,0)=0,v¯(x¯,0)=SL,H¯x(x¯,0)=H¯w(x),θ(x,0)=T¯w(x¯),u¯(x,∞)=1,H¯x(x,∞)=0,T¯(x,∞)=0.
In the above conditions SL=(V0L/ν)ReL-1/2 is the transpiration parameter.Figure 1
The coordinate system and flow configuration.


## 3. Method of Solutions

To get the set of equations in convenient form for integration, we will introduce the following one parameter group of transformation for the dependent and independent variables:(3.1)u¯=U(ξ,Y),v¯=x1/2(V+ξ),φ=x1/2ϕ,θ¯=x-1θ(ξ,Y),Y=x-1/2y¯,ξ=SLx¯1/2,H¯x=∂φ∂y,H¯y=-∂φ∂x.Theξ is the local distribution of the surface mass-flux. Here for suction (or withdrawal) ξ is positive and for injection (or blowing) of fluid ξ is negative and for solid surface ξ is zero. We further assume that the surface temperature T¯w(x¯)=x¯-1 and the normal component of the magnetic field at the surface H¯w(x¯)=x¯-1/2. where φ is the potential function that satisfies (2.11). By using this group of transformations, which satisfies equation of continuity and by using in (2.9)–(2.13): we have set of equations:(3.2)12ξ∂U∂ξ-12Y∂U∂Y+∂V∂Y=0,(3.3)12ξU∂U∂ξ+(V+ξ-12YU)∂U∂Y-∂2U∂Y2-S[-12ϕ∂2ϕ∂Y2+12ξ(∂ϕ∂Y∂2ϕ∂ξ∂Y-∂2ϕ∂Y2∂ϕ∂ξ)]-λθ=0,(3.4)1Pm∂2ϕ∂Y2=12Uϕ+12ξU∂ϕ∂ξ+(V+ξ-12YU)∂ϕ∂Y,(3.5)1Pr[1+43Rd(1+(θw-1)θ)3]∂2θ∂Y2=12ξU∂θ∂ξ+(V+ξ-12YU)-∂θ∂Y-Uθ.
The appropriate boundary conditions satisfied by the above system of equations are(3.6)U(ξ,0)=V(ξ,0)=0,ϕ′(ξ,0)=1,θ(ξ,0)=1,V(ξ,∞)=1,ϕ′(ξ,∞)=0,θ(ξ,∞)=0.Once we know the solutions of the above equations, we readily can obtain the values of skin-friction, heat transfer and the normal magnetic intensity at the surface from the following relations in terms of skin-friction, Nusselt number and magnetic intensity from the following relations:(3.7)Rex1/2Cfx=f′′(ξ,0),Rex-1/2Nux=-(1+43Rd)θ′(ξ,0),Rex1/2Mgx=-g(ξ,0).
Now we will discretize the expressions (3.1)–(3.4) with boundary conditions given in (3.5), we have a new system of discretised form of equations as follows:(3.8)A1Ui+1,j+B1Ui,j+C1Ui-1,j=D1,
where(3.9)A1=1+12(Vi,j-ξi-YjUi,j)ΔY,B1=-2+12ξiΔξUi,jΔY2,C1=1-12(Vi,j-ξi-YjUi,j)ΔY,D1=[S2Δξ(H1)i,j((H1)i,j-(H1)i,j-1)-12ξiΔξUi,jUi,j-1]ΔY2-[S4((H1)i+1,j-(H1)i-1,j)+S2((H1)i,j-(H1)i-1,j)(H2)i,j]ΔY,
where (H1)i,j=(∂ϕ/∂y)i,j and (H2)i,j=(∂ϕ/∂ξ)i,j.Similarly for hydromagnetics equation we have(3.10)A2ϕi+1,j+B2ϕi,j+C2ϕi-1,j=D2,A2=1Pm+12(Vi,j-ξi-YjUi,j)ΔY,B2=-2Pm-12(1+ξiΔξ)Ui,jΔY2,C2=1Pm-12(Vi,j-ξi-12YjUi,j)ΔY,D2=-12ξiΔξUi,jϕi,j-1ΔY2,
and the discretised form of energy equation is of the form (3.11)A3θi+1,j+B3θi,j+C3θi-1,j=D3,
where
(3.12)A3=1Pr[1+43Rd(1+(θw-1)θi,j)3]+12(Vi,j-ξi-12YjUi,j)ΔY,B3=-2Pr[1+43Rd(1+(θw-1)θi,j)3]-(1-12ξiΔξ)Ui,jΔY2,C3=1Pr[1+43Rd(1+(θw-1)θi,j)3]-12(Vi,j-ξi-12YjUi,j)ΔY,D3=-12ξiΔξUi,jθi,j-1ΔY2,
velocity can be calculated directly using equation of continuity (3.2) as shown below:
(3.13)Vi,j=Vi-1,j-12(ξΔyΔξ-Yj)Ui,j+12ξΔYΔξUi,j-1-12YjUi-1,j,
where i and j denote the grid points along the X and Y directions, respectively. In order to find the numerical solution we have discretised the expressions (3.2)–(3.5) with boundary conditions (3.6) by using finite difference method, using backward difference for x-direction and central difference for y-direction out of which we get a system of tri-diagonal algebraic equations. These tri-diagonal equations are then solved by Gaussian elimination technique. The computation is started at ξ=0.0, and then marches downstream implicitly. Once we know the solution of these equations, physical quantities of interest such as the coefficient of skin-friction, the coefficient of magnetic intensity, and the coefficient of rate of heat transfer at the surface may be calculated from
(3.14)Rex1/2Cfx=f′′(ξ,0),MgxRex1/2=-ϕ(ξ,0),NuxRex1/2=-(1+43Rd)θ′(ξ,0).


## 4. Results and Discussions

In present investigation we have obtained the solutions of the nonsimilar boundary layer (3.2)–(3.5) with boundary conditions (3.6) that governs the flow of a viscous incompressible and electrically conducting fluid past a magnetized vertical porous plate with surface temperature by using the method discussed in the preceding section for a wide range of physical parameters, S, conduction-radiation parameter Rd, surface temperature θw, Prandtl number Pr, and mixed convection parameter λ, magnetic Prandtl number Pm, against ξ. Below we discuss the effects of the aforementioned physical parameters of the flow fields as well as on the local skin-friction coefficient Rex1/2Cfx, the coefficient of surface magnetic intensity Rex1/2Mgx and rate of heat transfer Rex-1/2Nux on the surface of the plate.


### 4.1. Skin Friction, Magnetic Intensity Coefficient, and Rate of Heat Transfer

In first attempt we have obtained the solution of the nonsimilar boundary layer equations governing the mixed convection flow of a viscous incompressible and electrically conducting fluid along a vertical magnetized porous plate againstξ. Tables 1, 2, and 3 exhibiting the effects of radiation parameter or Planks number Rd=1.0,10.0 and for the fixed value of buoyancy force parameter λ=1.0, magnetic Prandtl number Pm=0.1 and Prandtl number Pr=0.1, magnetic force parameter S, and surface temperature θw=1.1 on coefficients of skin friction Rex1/2Cfx, rate of heat transfer Rex-1/2Nux and magnetic intensity Rex1/2Mgx at the surface. From Tables 1, 2, and 3, it can easily be seen that an increase in radiation parameter Rd leads to decrease in coefficient of local skin friction and increases in the rate of heat transfer, magnetic intensity at the surface. This phenomenon can easily be understood from the fact that when radiation parameter Rd increases, the ambient fluid temperature decreases and Roseland mean absorption coefficient increases which reduce the skin friction and enhance the rate of heat transfer and magnetic intensity at the surface. In Figures 2(a), 2(b), and 2(c), where it is observed that with the increase of radiation parameter Rd the skin friction decreases and rate of heat transfer and magnetic intensity at the surface increases. In Figures 3(a), 3(b), and 3(c) it can be seen that the increase in buoyancy force parameter λ=0.0,2.5,5.0,7.5,10 the coefficient of skin friction, heat transfer increases and magnetic intensity at the surface decreases. It is very interesting fact that forced convection is dominant mode of flow and heat transfer when buoyancy parameter λ→0 but with the increase of λ the buoyancy force acts like pressure gradient and increase the the fluid motion, hence the coefficients of skin friction, heat transfer and magnetic intensity increases with the streamwise distance ξ.Table 1
Numerical values ofCfxRex1/2 obtained for Rd=1.0,10.0, and θw=1.1 when Pm=0.1, Pr=0.1, λ=1.0, and S=0.1 against ξ by two methods.
ξRd=1.0Rd=10.0FDMAsymptoticFDMAsymptotic0.051.622251.622809sm1.550371.55809sm0.11.666691.66423sm1.592221.59423sm0.21.757711.75652sm1.677971.67652sm0.41.947331.94108sm1.856811.85108sm0.82.350082.350021sm2.237012.23021sm1.02.55973—2.43497—3.04.70172—4.45669—4.05.71434—5.41609—5.06.68702—6.34363—6.07.635957.63883Lr7.254767.25883Lr7.08.571928.57900Lr8.159098.15900Lr8.09.499469.499912Lr9.060439.06912Lr9.010.9206310.99922Lr9.960129.96174Lr10.011.3362311.39390Lr10.8595810.85930LrTable 2
Numerical values ofMgxRex1/2 obtained for Rd=1.0,10.0, and θw=1.1 when Pm=0.1, Pr=0.1, λ=1.0, and S=0.1 against ξ by two methods.
ξRd=1.0Rd=10.0FDMAsymptoticFDMAsymptotic0.051.350801.35110sm1.376911.37118sm0.11.307721.30005sm1.330241.33005sm0.21.226831.22780sm1.248861.24780sm0.41.083881.08330sm1.101331.10330sm0.80.858530.85429sm0.869240.84291sm1.00.76950—0.77777—3.00.32395—0.32414—4.00.23978—0.23970—5.00.18862—0.18855—6.00.154740.15667Lr0.154700.215667Lr7.00.130680.13286Lr0.130650.13206Lr8.00.112690.12500Lr0.112680.11500Lr9.00.098740.10111Lr0.098730.11111Lr10.00.087580.092741Lr0.087500.081000LrTable 3
Values ofNux/Rex1/2 against ξ for Rd=1.0, 10.0, and θw=1.1 when Pm=0.1, Pr=0.1, λ=1.0, and S=0.1 against ξ by two methods.
ξRd=1.0Rd=10.0FDMAsymptoticFDMAsymptotic0.050.291620.29725sm0.346210.34725sm0.10.293210.29901sm0.349350.34824sm0.20.296360.29954sm0.355580.35654sm0.40.302510.30559sm0.367890.36559sm0.80.314290.31369sm0.391890.39369sm1.00.31992—0.40360—3.00.37116—0.51618—4.00.39612—0.57401—5.00.42193—0.63503—6.00.448870.44860Lr0.699490.69860Lr7.00.476950.47170Lr0.767230.76170Lr8.00.506130.50480Lr0.837910.83480Lr9.00.536360.53790Lr0.911190.91790Lr10.00.567570.563100Lr0.986720.98310Lrsm stand for smallξ, where Lr for large ξ.Numerical solution of (a) skin friction coefficient and (b) coefficient of rate of heat transfer (c) coefficient of magnetic intensity at the surface againstξ for different values of radiation parameter Rd=1.0,5.0,10.0,20.0,50.0, Pm=0.1, Pr=7.0, and S=0.8, θw=1.1, λ=1.0.
(a)(b)(c)Numerical solution of (a) skin friction coefficient and (b) coefficient of rate of heat transfer (c) coefficient of magnetic intensity at the surface againstξ for different values of mixed convection parameter λ=0.0,2.5,5.0,7.5,10.0 when Pm=0.5, Pr=0.1, S=0.1, Rd=10.0, and θw=1.1.
(a)(b)(c)Figures4(a), 4(b), and 4(c) are representing the effects of different values of Prandtl number Pr=0.01,0.1,0.71,7.0, and for fixed value of buoyancy force parameter λ=1.0, magnetic field parameter S=0.4 and magnetic Prandtl number Pm=0.1, radiation parameter Rd=1.0, surface temperature θw=1.1, on the coefficients of skin friction, rate of heat transfer and magnetic intensity at the surface. In these figures, it is observed that with increase of Prandtl number Pr the coefficient of skin friction decreases, coefficient of heat transfer and magnetic intensity at the surface increases. It is very pertinent to mention that the increase in the Prandtl number Pr increases the kinematic viscosity (which ratio of dynamic viscosity to density of the fluid) of the fluid and decreases the thermal diffusion which causes the increase in momentum boundary layer thickness and due to rise in temperature thermal boundary layer becomes thinner. So, these factors are responsible for the aforementioned phenomena. In Figures 5(a), 5(b), and 5(c) the effects of different values of magnetic Prandtl number Pm by keeping other parameters fixed on coefficients of skin friction, heat transfer and magnetic intensity are displayed. From these figures it is shown that the increase in magnetic Prandtl number Pm=1.0,10.0,100.0 increase the coefficients of skin friction, heat transfer and decrease the coefficient of magnetic intensity at the surface. It is also noted that the increase in coefficients of skin friction, heat transfer very remarkable for large values of magnetic Prandtl Pm, that is, for Pm=10.0, 100.0 as compared with magnetic intensity at the surface. The reason is that with the increase of magnetic Prandtl number Pm the magnetic diffusion γ decreases or product of magnetic permeability, electrical conductivity and kinematic viscosity at the surface increases and hence the momentum and thermal boundary layer thicknesses decreases due to which coefficients of skin friction and heat transfer increases and magnetic intensity at the surface decreases.Numerical solution of (a) skin friction coefficient and (b) coefficient of rate of heat transfer (c) coefficient of magnetic intensity at the surface againstξ for different values of Prandtl number Pr=0.01,0.1,0.71,7.0 when Pm=0.1, S=0.4, Rd=1.0, θw=1.1 and λ=1.0.
(a)(b)(c)Numerical solution of (a) skin friction coefficient and (b) coefficient of rate of heat transfer (c) coefficient of magnetic intensity at the surface againstξ for different values of magnetic Prandtl number Pm=1.0,10.0,100.0 when Pr=0.1, S=2.0, Rd=1.0, θw=1.1, and λ=1.0.
(a)(b)(c)


### 4.2. Velocity, Temperature and Magnetic Profiles

Now we will discuss the effects of different physical parameters on the profiles of the velocity, temperature and the transverse component of magnetic field against similarity variableη for transpiration parameter ξ=10.0. The effects of mixed convection parameter λ=0.0,2.5,5.0,7.5,10.0, for two values of magnetic field parameter S=0.0,0.8 and for fixed value of magnetic Prandtl number Pm=1.6, Pr=0.1, ξ=0.5, radiation parameter Rd=10.0, and surface temperature θw=1.1 on velocity, temperature and transverse component of magnetic field profiles are shown in Figures 6(a), 6(b), and 6(c). The dotted and solid lines in Figures 6(a)–6(c) shown the effects of parameter λ for S=0 (absence of magnetic field) and S=0.8 (presence of magnetic field), respectively. It is concluded that the velocity profile is influenced considerably and increase when the value of λ increases and there is no any significant changes shows in the absence of magnetic field as shown by dotted lines in Figure 5(a). In Figure 6(b) it is shown that the temperature decreases with the increase of λ and there is no changes seen for magnetic field parameter S=0 and S=0.8. From Figure 6(c), we note that with the increase of parameter λ the effects of transverse component of magnetic field decreases against η.(a) velocity and (b) temperature (c) transverse component of magnetic field profile againstη at ξ=0.5 for different values of mixed convection parameter λ = 0.0, 2.5, 5.0, 7.5, 10.0 when S=0.0, 0.8 and for Pr=0.1, and Pm=1.6, Rd=10.0, θw=1.1.
(a)(b)(c)Figures7(a), 7(b), and 7(c) are based on the effects of the magnetic field parameter S on the velocity, temperature and component of transverse magnetic field profiles. These figures clearly show that with the increase of magnetic force parameter S the velocity profile decreases and the temperature, transverse component of magnetic field profile increases. In Figures 8(a), 8(b), and 8(c) it is noted that the increase in transpiration parameter ξ increase velocity profile and decrease the temperature and transverse component of magnetic field profiles. From these figures it is also concluded that the momentum boundary layer thickness decreases and thermal boundary layer thickness increases which indicates that transpiration destabilizes the boundary layer. Finally, in Figures 9(a), 9(b), and 9(c) it is shown that with the increase of radiation parameter Rd and keeping other parameters fixed the velocity and temperature distribution decreases and transverse component of magnetic field increases.(a) velocity and (b) temperature (c) transverse component of magnetic field profile againstη at ξ=0.5 for different values magnetic field parameter S=0.0,0.8,1.6 when λ=1.0 and for Pr=0.1, and Pm=1.0, Rd=10.0, θw=1.1.
(a)(b)(c)(a) velocity and (b) temperature (c) transverse component of magnetic field profile againstη for different values transpiration parameter ξ=2.0,1.0,2.0,4.0,8.0,10.0, when S=0.4, λ=1.0, and for Pr=0.1, Pm=2.0, Rd=10.0, θw=1.1
(a)(b)(c)(a) velocity and (b) temperature (c) transverse component of magnetic field profile againstη for different values radiation parameter Rd=0.1,0.5,1.0,5.0,10.0 when S=0.4, λ=1.0 and for ξ=2.0, Pr=0.1, Pm=0.1, θw=1.1.
(a)(b)(c)


### 4.3. Asymptotic Solutions for Small and Largeξ

Now we are heading in finding the solution of the present problem for small and large value of transpiration parameterξ. To do this we first reduce the equations (2.1)–(2.7) to convenient form by introducing the following transformations:(4.1)ψ=x1/2[f(η,ξ)+ξ],φ=x-1/2ϕ(η,ξ),θ¯=x-1θ(η,ξ),η=x-1/2,ξ=sx1/2,
where, η is the similarity variable, ξ be the local transpiration parameter and ψ, ϕ are the function which satisfy the equations of conservation of mass and magnetic field such that:(4.2)u=∂ψ∂y,v=-∂ψ∂x,Hx=∂φ∂y,Hy=-∂φ∂x.
For withdrawal of fluid ξ>0 whereas for blowing of fluid through the surface of the plate ξ<0. Throughout the present computations, value of ξ has been considered positive.By using (4.1) and (4.2) in (2.1)–(2.15), we will obtain the following dimensionless local nonsimilarity equations: (4.3)f′′′+12(ff′′-Sϕϕ′′)+ξf′′+λθ=12ξ[f′∂f′∂ξ-f′′∂f∂ξ-S(ϕ′∂ϕ′∂ξ-ϕ′′∂ϕ∂ξ)],(4.4)1Pmϕ′′+12(fϕ′-f′ϕ)+ξϕ′=12ξ[f′∂ϕ∂ξ-ϕ′∂f∂ξ],(4.5)1Pr[1+43Rd(1+Δθ)3]θ′′+12fθ′+f′θ+ξθ′=12ξ[f′∂θ∂ξ-θ′∂f∂ξ],
where, S=μ¯H02/ρU02, Pm=υ/γ, respectively, are known as magnetic field parameter and magnetic Prandtl number and Δ=θw-1. The corresponding boundary conditions becomes(4.6)f(0,ξ)=0,f′(0,ξ)=0,ϕ′(0,ξ)=1,θ(0,ξ)=1,f′(∞,ξ)=1,ϕ′(∞,ξ)=0,θ(∞,ξ)=0.
It can be seen from equations (4.3)–(4.5) that for ξ=0.0, the set of equations become similar by nature, solutions of which can easily be obtained by standard shooting method, otherwise these equations are locally nonsimilar, solution methodology of which will be discussed in the following sections. Once we know the solution of these equations, physical quantities of interest such as, the skin-friction coefficient, Cfx, and the magnetic intensity Mgx, and the rate of heat transfer Nux at the surface may be calculated from(4.7)Rex1/2Cfx=f′′(ξ,0),Rex1/2Mgx=-ϕ(ξ,0),Rex-1/2Nux=-(1+43Rd)θ′(ξ,0).


#### 4.3.1. Solution for Smallξ

Since near the leading edgeξ is small (ξ≪1), solutions to the equations (4.3)–(4.5) with boundary conditions (4.6) may be obtained by using the perturbation method. We can expand all the depending functions in powers of ξ, we consider that(4.8)f(ξ,η)=∑i∞ξifi(η),ϕ(ξ,η)=∑i=0∞ξiϕi(η),θ(ξ,η)=∑i=0∞ξiθi(η).
Substituting (4.8) into expression (4.3)–(4.5), and taking the terms only up to O(ξ2) we will get the system of equations together with boundary conditions (4.6) which is given as follows:(4.9)f0′′′+12(f0f0′′-Sϕ0ϕ0′′)+λθ0=0,1Pmϕ0′′+12(f0ϕ0′-ϕ0f0′)=0,[1+α(1+Δθ0)3]θ0′′+3αΔ(1+Δθ0)2θ0′2+Pr2f0θ0′+Prf0′θ0=0,f0(0,ξ)=0,f0′(0,ξ)=0,ϕ0′(0,ξ)=1,θ0(0,ξ)=1,f0′(∞,ξ)=1,ϕ0′(∞,ξ)=0,θ0(∞,ξ)=0,f1′′′+12(f0f1′′-f0′f1′-S(ϕ0ϕ1′′-ϕ0′ϕ1′))+(f0′′f1-Sϕ0′′ϕ1)+f0′′+λθ1=0,1Pmϕ1′′+12(f0ϕ1′-f1′ϕ0)+(f0′ϕ1-f1ϕ0′)+ϕ0′=0,[1+α(1+Δθ0)3]θ1′′+3αΔ(1+Δθ0)2(θ1θ0′′+2θ0′θ1′)+6αΔ2θ1(1+Δθ0)θ0′2+Pr2(f0θ1′+f0′θ1)+Pr(f1θ0′+θ0f1′)+Prθ0′=0,f1(0,ξ)=0,f1′(0,ξ)=0,ϕ1′(0,ξ)=0,θ1(0,ξ)=0,f1′(∞,ξ)=0,ϕ1′(∞,ξ)=0,θ1(∞,ξ)=0,f2′′′+12(f0f2′′-f1′2-S(ϕ0ϕ2′′-ϕ1′2))+(f1f1′′-f0′f2′-S(ϕ1ϕ1′′-ϕ0′ϕ2′))+32(f0′′f2-Sϕ0′′ϕ2)+f1′′+λθ2=0,1Pmϕ2′′+12(f0ϕ2′-f0′ϕ2)+(f1ϕ1′-f1′ϕ1)+32(ϕ0′f2-f0′ϕ2)+ϕ1′=0,[1+α(1+Δθ0)3]θ2′′+3αΔθ1′′θ1(1+Δθ0)2+3αΔθ0′′[θ2′′+Δ(2θ2θ0+θ12)+Δ2θ0(θ2θ0+θ12)]+3αΔ[(2θ2′θ0′+θ12)(1+Δθ0)2+4Δθ1′θ1θ0(1+Δθ0)+Δθ02(2θ2+Δ(2θ2θ0+θ12))]+Pr2(f0θ2′+θ1f1′)+(θ1′f1+f2′θ0)+3Pr2f2θ0′+Prθ1′=0,f2(0,ξ)=0,f2′(0,ξ)=0,ϕ2′(0,ξ)=0,θ2(0,ξ)=0,f2′(∞,ξ)=0,ϕ2′(∞,ξ)=0,θ2(∞,ξ)=0.
It is pertinent to mentioned that (4.9) are coupled and nonlinear, so the solutions of these equations can be obtained by the Nachtsheim-Swigert iteration technique together with the sixth-order implicit Runge-Kutta-Butcher initial value solver. After knowing the values of the functions f′′, ϕ and θ′ and their derivatives we can calculate the values of the coefficient of skin friction, surface magnetic intensity and heat transfer in the region near the leading edge against ξ from the following expansion for S=0.1, λ=1.0, Pm=0.1, Pr=0.1, and radiation parameter Rd=1.0, 10.0, and θw=1.1, respectively.(4.10)Rex1/2Cfx=(1.59195+1.12282ξ+1.01047ξ2+⋯),Rex1/2Mgx=-(1.43230+1.02251ξ+1.34209ξ2+⋯),Rex-1/2Nux=-(0.21749-0.00475ξ+0.09878ξ2⋯).The numerical results thus obtained or entered in Tables1, 2, and 3 for coefficients of skin friction, rate of heat transfer and magnetic intensity at the surface. We can see that from these tables the series solution are in excellent agreement with that of finite difference solutions even for ξɛ[0,1].In Table4 the comparison of the solutions obtained by finite difference method and that of Glauert [11] analytically has been given for the coefficient of skin friction and magnetic intensity at the surface. It is observed that for Pm=1.0,10.0, radiation parameter Rd=∞, and the variation in magnetic field parameter S decrease the skin friction and the skin friction approaches to zero as S→1.1 and the local magnetic intensity Mgx increases with the increase of magnetic field parameter S. From this table it can be seen that the present method and the analytical results obtained by Glauert [11] are in good agreement. For magnetic Prandtl number Pm=10.0 the coefficients of skin friction and magnetic intensity at the surface are also noted in excellent agreement. Further, we see that for small value of magnetic Prandtl number Pm=0.1, and for magnetic field parameter S=π as in the case of Glauert [11] the separation occur at the surface. In Table 5 the value of coefficient of skin friction obtained by other authors Glauert [11] for large magnetic Prandtl number Pm by keeping Rd=∞ and Davies [2] for small magnetic Prandtl number by taking magnetic field parameter S=0.1 and sufficiently small S=0.05 are entered and compare with present results and found to be in good agreement. Here we notice that the agreement between present results and results obtained by Glauert [11], Davies [2] are in excellent agreement. The results entered in Table 6 are those obtained from heat transfer in hydromagnetics by Ramamoorthy [6] considering Eckert number equal to zero and radiation parameter Rd=∞are compared with the present results. From this table it can be seen that the numerical results obtained for different values of magnetic field parameter S and for Pm=0.1, Pr=1.0, λ=0.5 are very closed the results obtained by Ramamoorthy.Table 4
Values ofRex1/2Cfx and Rex1/2Mgx obtained by Glauert [2] and present authors while Rd=∞, Pm=1.0 and 10.0 against different values of S.
SPm=1.0Pm=10.0PresentGlauret [2]PresentGlauret [2]f′′(0)g(0)f′′(0)g(0)f′′(0)g(0)f′′(0)g(0)0.00.33212.12900.33212.17970.33210.95250.33211.00950.10.30672.17130.30252.24480.31880.96310.31821.02380.20.28062.21910.27292.30990.30500.97440.30441.03800.40.22572.33720.21382.44020.27700.99950.27671.06650.60.16572.50660.15472.57040.24791.02850.24911.09500.80.09722.80170.09552.70060.21761.06290.22141.12351.00.00144.84880.03642.83080.18571.10510.19381.15201.10.00006.04230.00682.89590.16911.13020.17991.1662Table 5
Values ofRex1/2Cfx when Rd=∞, S=0.1 and 0.05 at ξ=0.0 against Pm obtain by Glauert [2] and Devies [3] and the present authors.
s=0.0s=0.8PmPresentGlauert [2]PmPresentDavies [3]0.10.28880.26690.10.31060.31401.00.30670.30160.20.31070.31572.00.31090.30780.30.31530.31734.00.31450.31280.50.31720.31946.00.31640.31520.70.31830.32048.00.31770.31670.90.31910.320410.00.31860.3178—————————500.32380.3237———750.32480.3247———1000.32540.3254———Table 6
Values ofRex-1/2Nux for different S when Rd=∞, Pm=0.1, Pr=1.0, and λ=0.5 obtained by present authors and Ramamoorthy [6].
SPresent[6]0.10.645510.650480.30.619940.619580.50.591120.57693


#### 4.3.2. Solution for Largeξ

Now, attention is given in finding the solution of equations (4.3)–(4.6) when ξ is large. The order of magnitude analysis of various terms in these equations shows that f′′′ and ξf′′ are largest terms in (4.3) and ϕ′′ and ξϕ′ in (3.14) and θ′′ and ξθ′ in (4.5). In the respective equations, both the terms have to be balanced in magnitude and the only way to do this, is to assume that η is small and hence its derivative is large. It is essential to find appropriate scaling for f, ϕ and θ. On balancing f′′′ and ξf′′ in (4.3) and ϕ′′ and ξϕ′ in (4.4) and θ′′, ξθ′ in (4.5), it is found that η=O(ξ-1), f=O(ξ-1), and ϕ=O(ξ-1). Therefore, following transformations may be introduced(4.11)f(η)=ξ-1f¯(η¯),η¯=ξη,ϕ(η)=ξ-1ϕ¯(η¯),θ(η)=θ¯(η¯).
By using (4.10), the transformed equation will take the form:(4.12)f¯′′′+f¯′′+λξ-2θ=12f[f¯′∂f¯∂ξ-f¯′′∂f¯∂ξ-S(ϕ¯′∂ϕ¯′∂ξ-ϕ¯′′∂ϕ¯∂ξ)],(4.13)1Pmϕ¯′′+ϕ¯′=12ξ-1[f¯′∂ϕ¯∂ξ-f¯′∂f¯∂ξ],(4.14)1Pr[1+43Rd(1+Δθ)3]+θ¯′=12ξ-1[f¯′∂θ¯∂ξ-θ¯′∂f¯∂ξ].
The regular perturbation of the functions f¯, ϕ¯ and θ¯ in power of ξ-2 is given as follows:(4.15)f̅(ξ,η̅)=∑m=01ξ-2mf̅m(η̅),ϕ̅(ξ,η̅)=∑m=01ξ-2mϕ̅m(η̅),θ̅(ξ,η̅)=∑m=01ξ-2mθ̅m(η̅).
By substituting (4.14) into (4.11)–(4.13), and equating like power of ξ and by dropping bars we have the following set of equations:O(ξ0):
(4.16)f0′′′+f0′′=0,1Pmϕ0′′+ϕ0′=0,[1+α(1+Δθ0)3]θ0′′+3αΔ(1+Δθ0)2θ0′2+Prθ0′=0.
The boundary conditions regarding to the perturbation series expansion are of the following form:(4.17)f0(0)=0,f0′(0)=0,ϕ0′(0)=1,θ0(0)=1,f0′(∞)=1,ϕ0′(∞)=0,θ0(∞)=0.O(ξ-2):(4.18)f1′′′+f1′′+λθ0=0,1Pmϕ1′′+ϕ1′=0,[1+α(1+Δθ0)3]θ1′′+3αΔ(1+Δθ0)2(θ1θ0′′+2θ0′θ1′)+6αΔ2θ1(1+Δθ0)θ0′2+Prθ1′=0.
And the boundary conditions regarding to the perturbation series expansion are of the following form:(4.19)f1(0)=0,f1′(0)=0,ϕ1′(0)=0,θ1(0)=0,f1′(∞)=0,ϕ1′(∞)=0,θ1(∞)=0.
From the solutions of (4.15)–(4.16) and (4.17)–(4.18), we obtain(4.20)f0(η)=η+e-η-1,f0′′(0)=1,ϕ0(η)=-1Pme-ηPm,ϕ0(0)=-1Pm,θ0′(η)=[1+43Rd(1+Δθ0)3]Pre-ηPr,θ0′(0)=[1+43Rd(1+Δθ0)3]Pr,f1′′(η)=λe-ηPr(Pr-1)+Prλe-η-λe-η2(Pr-1),f1′′(0)=λ(Pr-1)+Prλ-λ2(Pr-1),ϕ1(η)=0,ϕ1(0)=0,θ1′(η)=0,θ1′(0)=0.
Since, now, we know the values of f0′′(0), ϕ0(0), and θ0′(0), and f1′′(0), ϕ1(0), and θ1′(0) from the above solutions we calculate the friction coefficient, Rex1/2Cfx, local rate of heat transfer, Rex-1/2Nux, and the local magnetic intensity, Rex1/2Mgx at the surface from the following expressions:(4.21)Rex1/2Cfx≃ξ+λ(Pr-1)+Prλ-λ2(Pr-1)ξ-1,Rex1/2Mgx≃ξPm,Rex-1/2Nux≃[1+43Rd(1+Δθ0)3]Prξ.Numerical value of the local skin friction coefficient, surface magnetic intensity and the local rate of heat transfer are obtained from the relations (4.21) for different values of magnetic field parameter S and magnetic Prandtl number Pm and radiation parameter Rd, and surface temperature θw, Prandtl number Pr, respectively, in the down stream region are entered in Tables 1, 2, and 3, respectively. From these tables it can be seen that for large value of transpiration parameter ξ the skin friction Rex1/2Cfx approaches to ξ and the values of coefficient magnetic intensity Rex1/2Mgx approaches to ξ/Pm and Nusselt number Rex-1/2Nux approaches to [1+(4/3Rd)(1+Δθ0)3]Prξ. The comparison of the present results with the numerical results obtained by FDM shows excellent agreement in the down stream region.


## 4.1. Skin Friction, Magnetic Intensity Coefficient, and Rate of Heat Transfer

In first attempt we have obtained the solution of the nonsimilar boundary layer equations governing the mixed convection flow of a viscous incompressible and electrically conducting fluid along a vertical magnetized porous plate againstξ. Tables 1, 2, and 3 exhibiting the effects of radiation parameter or Planks number Rd=1.0,10.0 and for the fixed value of buoyancy force parameter λ=1.0, magnetic Prandtl number Pm=0.1 and Prandtl number Pr=0.1, magnetic force parameter S, and surface temperature θw=1.1 on coefficients of skin friction Rex1/2Cfx, rate of heat transfer Rex-1/2Nux and magnetic intensity Rex1/2Mgx at the surface. From Tables 1, 2, and 3, it can easily be seen that an increase in radiation parameter Rd leads to decrease in coefficient of local skin friction and increases in the rate of heat transfer, magnetic intensity at the surface. This phenomenon can easily be understood from the fact that when radiation parameter Rd increases, the ambient fluid temperature decreases and Roseland mean absorption coefficient increases which reduce the skin friction and enhance the rate of heat transfer and magnetic intensity at the surface. In Figures 2(a), 2(b), and 2(c), where it is observed that with the increase of radiation parameter Rd the skin friction decreases and rate of heat transfer and magnetic intensity at the surface increases. In Figures 3(a), 3(b), and 3(c) it can be seen that the increase in buoyancy force parameter λ=0.0,2.5,5.0,7.5,10 the coefficient of skin friction, heat transfer increases and magnetic intensity at the surface decreases. It is very interesting fact that forced convection is dominant mode of flow and heat transfer when buoyancy parameter λ→0 but with the increase of λ the buoyancy force acts like pressure gradient and increase the the fluid motion, hence the coefficients of skin friction, heat transfer and magnetic intensity increases with the streamwise distance ξ.Table 1
Numerical values ofCfxRex1/2 obtained for Rd=1.0,10.0, and θw=1.1 when Pm=0.1, Pr=0.1, λ=1.0, and S=0.1 against ξ by two methods.
ξRd=1.0Rd=10.0FDMAsymptoticFDMAsymptotic0.051.622251.622809sm1.550371.55809sm0.11.666691.66423sm1.592221.59423sm0.21.757711.75652sm1.677971.67652sm0.41.947331.94108sm1.856811.85108sm0.82.350082.350021sm2.237012.23021sm1.02.55973—2.43497—3.04.70172—4.45669—4.05.71434—5.41609—5.06.68702—6.34363—6.07.635957.63883Lr7.254767.25883Lr7.08.571928.57900Lr8.159098.15900Lr8.09.499469.499912Lr9.060439.06912Lr9.010.9206310.99922Lr9.960129.96174Lr10.011.3362311.39390Lr10.8595810.85930LrTable 2
Numerical values ofMgxRex1/2 obtained for Rd=1.0,10.0, and θw=1.1 when Pm=0.1, Pr=0.1, λ=1.0, and S=0.1 against ξ by two methods.
ξRd=1.0Rd=10.0FDMAsymptoticFDMAsymptotic0.051.350801.35110sm1.376911.37118sm0.11.307721.30005sm1.330241.33005sm0.21.226831.22780sm1.248861.24780sm0.41.083881.08330sm1.101331.10330sm0.80.858530.85429sm0.869240.84291sm1.00.76950—0.77777—3.00.32395—0.32414—4.00.23978—0.23970—5.00.18862—0.18855—6.00.154740.15667Lr0.154700.215667Lr7.00.130680.13286Lr0.130650.13206Lr8.00.112690.12500Lr0.112680.11500Lr9.00.098740.10111Lr0.098730.11111Lr10.00.087580.092741Lr0.087500.081000LrTable 3
Values ofNux/Rex1/2 against ξ for Rd=1.0, 10.0, and θw=1.1 when Pm=0.1, Pr=0.1, λ=1.0, and S=0.1 against ξ by two methods.
ξRd=1.0Rd=10.0FDMAsymptoticFDMAsymptotic0.050.291620.29725sm0.346210.34725sm0.10.293210.29901sm0.349350.34824sm0.20.296360.29954sm0.355580.35654sm0.40.302510.30559sm0.367890.36559sm0.80.314290.31369sm0.391890.39369sm1.00.31992—0.40360—3.00.37116—0.51618—4.00.39612—0.57401—5.00.42193—0.63503—6.00.448870.44860Lr0.699490.69860Lr7.00.476950.47170Lr0.767230.76170Lr8.00.506130.50480Lr0.837910.83480Lr9.00.536360.53790Lr0.911190.91790Lr10.00.567570.563100Lr0.986720.98310Lrsm stand for smallξ, where Lr for large ξ.Numerical solution of (a) skin friction coefficient and (b) coefficient of rate of heat transfer (c) coefficient of magnetic intensity at the surface againstξ for different values of radiation parameter Rd=1.0,5.0,10.0,20.0,50.0, Pm=0.1, Pr=7.0, and S=0.8, θw=1.1, λ=1.0.
(a)(b)(c)Numerical solution of (a) skin friction coefficient and (b) coefficient of rate of heat transfer (c) coefficient of magnetic intensity at the surface againstξ for different values of mixed convection parameter λ=0.0,2.5,5.0,7.5,10.0 when Pm=0.5, Pr=0.1, S=0.1, Rd=10.0, and θw=1.1.
(a)(b)(c)Figures4(a), 4(b), and 4(c) are representing the effects of different values of Prandtl number Pr=0.01,0.1,0.71,7.0, and for fixed value of buoyancy force parameter λ=1.0, magnetic field parameter S=0.4 and magnetic Prandtl number Pm=0.1, radiation parameter Rd=1.0, surface temperature θw=1.1, on the coefficients of skin friction, rate of heat transfer and magnetic intensity at the surface. In these figures, it is observed that with increase of Prandtl number Pr the coefficient of skin friction decreases, coefficient of heat transfer and magnetic intensity at the surface increases. It is very pertinent to mention that the increase in the Prandtl number Pr increases the kinematic viscosity (which ratio of dynamic viscosity to density of the fluid) of the fluid and decreases the thermal diffusion which causes the increase in momentum boundary layer thickness and due to rise in temperature thermal boundary layer becomes thinner. So, these factors are responsible for the aforementioned phenomena. In Figures 5(a), 5(b), and 5(c) the effects of different values of magnetic Prandtl number Pm by keeping other parameters fixed on coefficients of skin friction, heat transfer and magnetic intensity are displayed. From these figures it is shown that the increase in magnetic Prandtl number Pm=1.0,10.0,100.0 increase the coefficients of skin friction, heat transfer and decrease the coefficient of magnetic intensity at the surface. It is also noted that the increase in coefficients of skin friction, heat transfer very remarkable for large values of magnetic Prandtl Pm, that is, for Pm=10.0, 100.0 as compared with magnetic intensity at the surface. The reason is that with the increase of magnetic Prandtl number Pm the magnetic diffusion γ decreases or product of magnetic permeability, electrical conductivity and kinematic viscosity at the surface increases and hence the momentum and thermal boundary layer thicknesses decreases due to which coefficients of skin friction and heat transfer increases and magnetic intensity at the surface decreases.Numerical solution of (a) skin friction coefficient and (b) coefficient of rate of heat transfer (c) coefficient of magnetic intensity at the surface againstξ for different values of Prandtl number Pr=0.01,0.1,0.71,7.0 when Pm=0.1, S=0.4, Rd=1.0, θw=1.1 and λ=1.0.
(a)(b)(c)Numerical solution of (a) skin friction coefficient and (b) coefficient of rate of heat transfer (c) coefficient of magnetic intensity at the surface againstξ for different values of magnetic Prandtl number Pm=1.0,10.0,100.0 when Pr=0.1, S=2.0, Rd=1.0, θw=1.1, and λ=1.0.
(a)(b)(c)


## 4.2. Velocity, Temperature and Magnetic Profiles

Now we will discuss the effects of different physical parameters on the profiles of the velocity, temperature and the transverse component of magnetic field against similarity variableη for transpiration parameter ξ=10.0. The effects of mixed convection parameter λ=0.0,2.5,5.0,7.5,10.0, for two values of magnetic field parameter S=0.0,0.8 and for fixed value of magnetic Prandtl number Pm=1.6, Pr=0.1, ξ=0.5, radiation parameter Rd=10.0, and surface temperature θw=1.1 on velocity, temperature and transverse component of magnetic field profiles are shown in Figures 6(a), 6(b), and 6(c). The dotted and solid lines in Figures 6(a)–6(c) shown the effects of parameter λ for S=0 (absence of magnetic field) and S=0.8 (presence of magnetic field), respectively. It is concluded that the velocity profile is influenced considerably and increase when the value of λ increases and there is no any significant changes shows in the absence of magnetic field as shown by dotted lines in Figure 5(a). In Figure 6(b) it is shown that the temperature decreases with the increase of λ and there is no changes seen for magnetic field parameter S=0 and S=0.8. From Figure 6(c), we note that with the increase of parameter λ the effects of transverse component of magnetic field decreases against η.(a) velocity and (b) temperature (c) transverse component of magnetic field profile againstη at ξ=0.5 for different values of mixed convection parameter λ = 0.0, 2.5, 5.0, 7.5, 10.0 when S=0.0, 0.8 and for Pr=0.1, and Pm=1.6, Rd=10.0, θw=1.1.
(a)(b)(c)Figures7(a), 7(b), and 7(c) are based on the effects of the magnetic field parameter S on the velocity, temperature and component of transverse magnetic field profiles. These figures clearly show that with the increase of magnetic force parameter S the velocity profile decreases and the temperature, transverse component of magnetic field profile increases. In Figures 8(a), 8(b), and 8(c) it is noted that the increase in transpiration parameter ξ increase velocity profile and decrease the temperature and transverse component of magnetic field profiles. From these figures it is also concluded that the momentum boundary layer thickness decreases and thermal boundary layer thickness increases which indicates that transpiration destabilizes the boundary layer. Finally, in Figures 9(a), 9(b), and 9(c) it is shown that with the increase of radiation parameter Rd and keeping other parameters fixed the velocity and temperature distribution decreases and transverse component of magnetic field increases.(a) velocity and (b) temperature (c) transverse component of magnetic field profile againstη at ξ=0.5 for different values magnetic field parameter S=0.0,0.8,1.6 when λ=1.0 and for Pr=0.1, and Pm=1.0, Rd=10.0, θw=1.1.
(a)(b)(c)(a) velocity and (b) temperature (c) transverse component of magnetic field profile againstη for different values transpiration parameter ξ=2.0,1.0,2.0,4.0,8.0,10.0, when S=0.4, λ=1.0, and for Pr=0.1, Pm=2.0, Rd=10.0, θw=1.1
(a)(b)(c)(a) velocity and (b) temperature (c) transverse component of magnetic field profile againstη for different values radiation parameter Rd=0.1,0.5,1.0,5.0,10.0 when S=0.4, λ=1.0 and for ξ=2.0, Pr=0.1, Pm=0.1, θw=1.1.
(a)(b)(c)


## 4.3. Asymptotic Solutions for Small and Largeξ

Now we are heading in finding the solution of the present problem for small and large value of transpiration parameterξ. To do this we first reduce the equations (2.1)–(2.7) to convenient form by introducing the following transformations:(4.1)ψ=x1/2[f(η,ξ)+ξ],φ=x-1/2ϕ(η,ξ),θ¯=x-1θ(η,ξ),η=x-1/2,ξ=sx1/2,
where, η is the similarity variable, ξ be the local transpiration parameter and ψ, ϕ are the function which satisfy the equations of conservation of mass and magnetic field such that:(4.2)u=∂ψ∂y,v=-∂ψ∂x,Hx=∂φ∂y,Hy=-∂φ∂x.
For withdrawal of fluid ξ>0 whereas for blowing of fluid through the surface of the plate ξ<0. Throughout the present computations, value of ξ has been considered positive.By using (4.1) and (4.2) in (2.1)–(2.15), we will obtain the following dimensionless local nonsimilarity equations: (4.3)f′′′+12(ff′′-Sϕϕ′′)+ξf′′+λθ=12ξ[f′∂f′∂ξ-f′′∂f∂ξ-S(ϕ′∂ϕ′∂ξ-ϕ′′∂ϕ∂ξ)],(4.4)1Pmϕ′′+12(fϕ′-f′ϕ)+ξϕ′=12ξ[f′∂ϕ∂ξ-ϕ′∂f∂ξ],(4.5)1Pr[1+43Rd(1+Δθ)3]θ′′+12fθ′+f′θ+ξθ′=12ξ[f′∂θ∂ξ-θ′∂f∂ξ],
where, S=μ¯H02/ρU02, Pm=υ/γ, respectively, are known as magnetic field parameter and magnetic Prandtl number and Δ=θw-1. The corresponding boundary conditions becomes(4.6)f(0,ξ)=0,f′(0,ξ)=0,ϕ′(0,ξ)=1,θ(0,ξ)=1,f′(∞,ξ)=1,ϕ′(∞,ξ)=0,θ(∞,ξ)=0.
It can be seen from equations (4.3)–(4.5) that for ξ=0.0, the set of equations become similar by nature, solutions of which can easily be obtained by standard shooting method, otherwise these equations are locally nonsimilar, solution methodology of which will be discussed in the following sections. Once we know the solution of these equations, physical quantities of interest such as, the skin-friction coefficient, Cfx, and the magnetic intensity Mgx, and the rate of heat transfer Nux at the surface may be calculated from(4.7)Rex1/2Cfx=f′′(ξ,0),Rex1/2Mgx=-ϕ(ξ,0),Rex-1/2Nux=-(1+43Rd)θ′(ξ,0).


### 4.3.1. Solution for Smallξ

Since near the leading edgeξ is small (ξ≪1), solutions to the equations (4.3)–(4.5) with boundary conditions (4.6) may be obtained by using the perturbation method. We can expand all the depending functions in powers of ξ, we consider that(4.8)f(ξ,η)=∑i∞ξifi(η),ϕ(ξ,η)=∑i=0∞ξiϕi(η),θ(ξ,η)=∑i=0∞ξiθi(η).
Substituting (4.8) into expression (4.3)–(4.5), and taking the terms only up to O(ξ2) we will get the system of equations together with boundary conditions (4.6) which is given as follows:(4.9)f0′′′+12(f0f0′′-Sϕ0ϕ0′′)+λθ0=0,1Pmϕ0′′+12(f0ϕ0′-ϕ0f0′)=0,[1+α(1+Δθ0)3]θ0′′+3αΔ(1+Δθ0)2θ0′2+Pr2f0θ0′+Prf0′θ0=0,f0(0,ξ)=0,f0′(0,ξ)=0,ϕ0′(0,ξ)=1,θ0(0,ξ)=1,f0′(∞,ξ)=1,ϕ0′(∞,ξ)=0,θ0(∞,ξ)=0,f1′′′+12(f0f1′′-f0′f1′-S(ϕ0ϕ1′′-ϕ0′ϕ1′))+(f0′′f1-Sϕ0′′ϕ1)+f0′′+λθ1=0,1Pmϕ1′′+12(f0ϕ1′-f1′ϕ0)+(f0′ϕ1-f1ϕ0′)+ϕ0′=0,[1+α(1+Δθ0)3]θ1′′+3αΔ(1+Δθ0)2(θ1θ0′′+2θ0′θ1′)+6αΔ2θ1(1+Δθ0)θ0′2+Pr2(f0θ1′+f0′θ1)+Pr(f1θ0′+θ0f1′)+Prθ0′=0,f1(0,ξ)=0,f1′(0,ξ)=0,ϕ1′(0,ξ)=0,θ1(0,ξ)=0,f1′(∞,ξ)=0,ϕ1′(∞,ξ)=0,θ1(∞,ξ)=0,f2′′′+12(f0f2′′-f1′2-S(ϕ0ϕ2′′-ϕ1′2))+(f1f1′′-f0′f2′-S(ϕ1ϕ1′′-ϕ0′ϕ2′))+32(f0′′f2-Sϕ0′′ϕ2)+f1′′+λθ2=0,1Pmϕ2′′+12(f0ϕ2′-f0′ϕ2)+(f1ϕ1′-f1′ϕ1)+32(ϕ0′f2-f0′ϕ2)+ϕ1′=0,[1+α(1+Δθ0)3]θ2′′+3αΔθ1′′θ1(1+Δθ0)2+3αΔθ0′′[θ2′′+Δ(2θ2θ0+θ12)+Δ2θ0(θ2θ0+θ12)]+3αΔ[(2θ2′θ0′+θ12)(1+Δθ0)2+4Δθ1′θ1θ0(1+Δθ0)+Δθ02(2θ2+Δ(2θ2θ0+θ12))]+Pr2(f0θ2′+θ1f1′)+(θ1′f1+f2′θ0)+3Pr2f2θ0′+Prθ1′=0,f2(0,ξ)=0,f2′(0,ξ)=0,ϕ2′(0,ξ)=0,θ2(0,ξ)=0,f2′(∞,ξ)=0,ϕ2′(∞,ξ)=0,θ2(∞,ξ)=0.
It is pertinent to mentioned that (4.9) are coupled and nonlinear, so the solutions of these equations can be obtained by the Nachtsheim-Swigert iteration technique together with the sixth-order implicit Runge-Kutta-Butcher initial value solver. After knowing the values of the functions f′′, ϕ and θ′ and their derivatives we can calculate the values of the coefficient of skin friction, surface magnetic intensity and heat transfer in the region near the leading edge against ξ from the following expansion for S=0.1, λ=1.0, Pm=0.1, Pr=0.1, and radiation parameter Rd=1.0, 10.0, and θw=1.1, respectively.(4.10)Rex1/2Cfx=(1.59195+1.12282ξ+1.01047ξ2+⋯),Rex1/2Mgx=-(1.43230+1.02251ξ+1.34209ξ2+⋯),Rex-1/2Nux=-(0.21749-0.00475ξ+0.09878ξ2⋯).The numerical results thus obtained or entered in Tables1, 2, and 3 for coefficients of skin friction, rate of heat transfer and magnetic intensity at the surface. We can see that from these tables the series solution are in excellent agreement with that of finite difference solutions even for ξɛ[0,1].In Table4 the comparison of the solutions obtained by finite difference method and that of Glauert [11] analytically has been given for the coefficient of skin friction and magnetic intensity at the surface. It is observed that for Pm=1.0,10.0, radiation parameter Rd=∞, and the variation in magnetic field parameter S decrease the skin friction and the skin friction approaches to zero as S→1.1 and the local magnetic intensity Mgx increases with the increase of magnetic field parameter S. From this table it can be seen that the present method and the analytical results obtained by Glauert [11] are in good agreement. For magnetic Prandtl number Pm=10.0 the coefficients of skin friction and magnetic intensity at the surface are also noted in excellent agreement. Further, we see that for small value of magnetic Prandtl number Pm=0.1, and for magnetic field parameter S=π as in the case of Glauert [11] the separation occur at the surface. In Table 5 the value of coefficient of skin friction obtained by other authors Glauert [11] for large magnetic Prandtl number Pm by keeping Rd=∞ and Davies [2] for small magnetic Prandtl number by taking magnetic field parameter S=0.1 and sufficiently small S=0.05 are entered and compare with present results and found to be in good agreement. Here we notice that the agreement between present results and results obtained by Glauert [11], Davies [2] are in excellent agreement. The results entered in Table 6 are those obtained from heat transfer in hydromagnetics by Ramamoorthy [6] considering Eckert number equal to zero and radiation parameter Rd=∞are compared with the present results. From this table it can be seen that the numerical results obtained for different values of magnetic field parameter S and for Pm=0.1, Pr=1.0, λ=0.5 are very closed the results obtained by Ramamoorthy.Table 4
Values ofRex1/2Cfx and Rex1/2Mgx obtained by Glauert [2] and present authors while Rd=∞, Pm=1.0 and 10.0 against different values of S.
SPm=1.0Pm=10.0PresentGlauret [2]PresentGlauret [2]f′′(0)g(0)f′′(0)g(0)f′′(0)g(0)f′′(0)g(0)0.00.33212.12900.33212.17970.33210.95250.33211.00950.10.30672.17130.30252.24480.31880.96310.31821.02380.20.28062.21910.27292.30990.30500.97440.30441.03800.40.22572.33720.21382.44020.27700.99950.27671.06650.60.16572.50660.15472.57040.24791.02850.24911.09500.80.09722.80170.09552.70060.21761.06290.22141.12351.00.00144.84880.03642.83080.18571.10510.19381.15201.10.00006.04230.00682.89590.16911.13020.17991.1662Table 5
Values ofRex1/2Cfx when Rd=∞, S=0.1 and 0.05 at ξ=0.0 against Pm obtain by Glauert [2] and Devies [3] and the present authors.
s=0.0s=0.8PmPresentGlauert [2]PmPresentDavies [3]0.10.28880.26690.10.31060.31401.00.30670.30160.20.31070.31572.00.31090.30780.30.31530.31734.00.31450.31280.50.31720.31946.00.31640.31520.70.31830.32048.00.31770.31670.90.31910.320410.00.31860.3178—————————500.32380.3237———750.32480.3247———1000.32540.3254———Table 6
Values ofRex-1/2Nux for different S when Rd=∞, Pm=0.1, Pr=1.0, and λ=0.5 obtained by present authors and Ramamoorthy [6].
SPresent[6]0.10.645510.650480.30.619940.619580.50.591120.57693


### 4.3.2. Solution for Largeξ

Now, attention is given in finding the solution of equations (4.3)–(4.6) when ξ is large. The order of magnitude analysis of various terms in these equations shows that f′′′ and ξf′′ are largest terms in (4.3) and ϕ′′ and ξϕ′ in (3.14) and θ′′ and ξθ′ in (4.5). In the respective equations, both the terms have to be balanced in magnitude and the only way to do this, is to assume that η is small and hence its derivative is large. It is essential to find appropriate scaling for f, ϕ and θ. On balancing f′′′ and ξf′′ in (4.3) and ϕ′′ and ξϕ′ in (4.4) and θ′′, ξθ′ in (4.5), it is found that η=O(ξ-1), f=O(ξ-1), and ϕ=O(ξ-1). Therefore, following transformations may be introduced(4.11)f(η)=ξ-1f¯(η¯),η¯=ξη,ϕ(η)=ξ-1ϕ¯(η¯),θ(η)=θ¯(η¯).
By using (4.10), the transformed equation will take the form:(4.12)f¯′′′+f¯′′+λξ-2θ=12f[f¯′∂f¯∂ξ-f¯′′∂f¯∂ξ-S(ϕ¯′∂ϕ¯′∂ξ-ϕ¯′′∂ϕ¯∂ξ)],(4.13)1Pmϕ¯′′+ϕ¯′=12ξ-1[f¯′∂ϕ¯∂ξ-f¯′∂f¯∂ξ],(4.14)1Pr[1+43Rd(1+Δθ)3]+θ¯′=12ξ-1[f¯′∂θ¯∂ξ-θ¯′∂f¯∂ξ].
The regular perturbation of the functions f¯, ϕ¯ and θ¯ in power of ξ-2 is given as follows:(4.15)f̅(ξ,η̅)=∑m=01ξ-2mf̅m(η̅),ϕ̅(ξ,η̅)=∑m=01ξ-2mϕ̅m(η̅),θ̅(ξ,η̅)=∑m=01ξ-2mθ̅m(η̅).
By substituting (4.14) into (4.11)–(4.13), and equating like power of ξ and by dropping bars we have the following set of equations:O(ξ0):
(4.16)f0′′′+f0′′=0,1Pmϕ0′′+ϕ0′=0,[1+α(1+Δθ0)3]θ0′′+3αΔ(1+Δθ0)2θ0′2+Prθ0′=0.
The boundary conditions regarding to the perturbation series expansion are of the following form:(4.17)f0(0)=0,f0′(0)=0,ϕ0′(0)=1,θ0(0)=1,f0′(∞)=1,ϕ0′(∞)=0,θ0(∞)=0.O(ξ-2):(4.18)f1′′′+f1′′+λθ0=0,1Pmϕ1′′+ϕ1′=0,[1+α(1+Δθ0)3]θ1′′+3αΔ(1+Δθ0)2(θ1θ0′′+2θ0′θ1′)+6αΔ2θ1(1+Δθ0)θ0′2+Prθ1′=0.
And the boundary conditions regarding to the perturbation series expansion are of the following form:(4.19)f1(0)=0,f1′(0)=0,ϕ1′(0)=0,θ1(0)=0,f1′(∞)=0,ϕ1′(∞)=0,θ1(∞)=0.
From the solutions of (4.15)–(4.16) and (4.17)–(4.18), we obtain(4.20)f0(η)=η+e-η-1,f0′′(0)=1,ϕ0(η)=-1Pme-ηPm,ϕ0(0)=-1Pm,θ0′(η)=[1+43Rd(1+Δθ0)3]Pre-ηPr,θ0′(0)=[1+43Rd(1+Δθ0)3]Pr,f1′′(η)=λe-ηPr(Pr-1)+Prλe-η-λe-η2(Pr-1),f1′′(0)=λ(Pr-1)+Prλ-λ2(Pr-1),ϕ1(η)=0,ϕ1(0)=0,θ1′(η)=0,θ1′(0)=0.
Since, now, we know the values of f0′′(0), ϕ0(0), and θ0′(0), and f1′′(0), ϕ1(0), and θ1′(0) from the above solutions we calculate the friction coefficient, Rex1/2Cfx, local rate of heat transfer, Rex-1/2Nux, and the local magnetic intensity, Rex1/2Mgx at the surface from the following expressions:(4.21)Rex1/2Cfx≃ξ+λ(Pr-1)+Prλ-λ2(Pr-1)ξ-1,Rex1/2Mgx≃ξPm,Rex-1/2Nux≃[1+43Rd(1+Δθ0)3]Prξ.Numerical value of the local skin friction coefficient, surface magnetic intensity and the local rate of heat transfer are obtained from the relations (4.21) for different values of magnetic field parameter S and magnetic Prandtl number Pm and radiation parameter Rd, and surface temperature θw, Prandtl number Pr, respectively, in the down stream region are entered in Tables 1, 2, and 3, respectively. From these tables it can be seen that for large value of transpiration parameter ξ the skin friction Rex1/2Cfx approaches to ξ and the values of coefficient magnetic intensity Rex1/2Mgx approaches to ξ/Pm and Nusselt number Rex-1/2Nux approaches to [1+(4/3Rd)(1+Δθ0)3]Prξ. The comparison of the present results with the numerical results obtained by FDM shows excellent agreement in the down stream region.


## 4.3.1. Solution for Smallξ

Since near the leading edgeξ is small (ξ≪1), solutions to the equations (4.3)–(4.5) with boundary conditions (4.6) may be obtained by using the perturbation method. We can expand all the depending functions in powers of ξ, we consider that(4.8)f(ξ,η)=∑i∞ξifi(η),ϕ(ξ,η)=∑i=0∞ξiϕi(η),θ(ξ,η)=∑i=0∞ξiθi(η).
Substituting (4.8) into expression (4.3)–(4.5), and taking the terms only up to O(ξ2) we will get the system of equations together with boundary conditions (4.6) which is given as follows:(4.9)f0′′′+12(f0f0′′-Sϕ0ϕ0′′)+λθ0=0,1Pmϕ0′′+12(f0ϕ0′-ϕ0f0′)=0,[1+α(1+Δθ0)3]θ0′′+3αΔ(1+Δθ0)2θ0′2+Pr2f0θ0′+Prf0′θ0=0,f0(0,ξ)=0,f0′(0,ξ)=0,ϕ0′(0,ξ)=1,θ0(0,ξ)=1,f0′(∞,ξ)=1,ϕ0′(∞,ξ)=0,θ0(∞,ξ)=0,f1′′′+12(f0f1′′-f0′f1′-S(ϕ0ϕ1′′-ϕ0′ϕ1′))+(f0′′f1-Sϕ0′′ϕ1)+f0′′+λθ1=0,1Pmϕ1′′+12(f0ϕ1′-f1′ϕ0)+(f0′ϕ1-f1ϕ0′)+ϕ0′=0,[1+α(1+Δθ0)3]θ1′′+3αΔ(1+Δθ0)2(θ1θ0′′+2θ0′θ1′)+6αΔ2θ1(1+Δθ0)θ0′2+Pr2(f0θ1′+f0′θ1)+Pr(f1θ0′+θ0f1′)+Prθ0′=0,f1(0,ξ)=0,f1′(0,ξ)=0,ϕ1′(0,ξ)=0,θ1(0,ξ)=0,f1′(∞,ξ)=0,ϕ1′(∞,ξ)=0,θ1(∞,ξ)=0,f2′′′+12(f0f2′′-f1′2-S(ϕ0ϕ2′′-ϕ1′2))+(f1f1′′-f0′f2′-S(ϕ1ϕ1′′-ϕ0′ϕ2′))+32(f0′′f2-Sϕ0′′ϕ2)+f1′′+λθ2=0,1Pmϕ2′′+12(f0ϕ2′-f0′ϕ2)+(f1ϕ1′-f1′ϕ1)+32(ϕ0′f2-f0′ϕ2)+ϕ1′=0,[1+α(1+Δθ0)3]θ2′′+3αΔθ1′′θ1(1+Δθ0)2+3αΔθ0′′[θ2′′+Δ(2θ2θ0+θ12)+Δ2θ0(θ2θ0+θ12)]+3αΔ[(2θ2′θ0′+θ12)(1+Δθ0)2+4Δθ1′θ1θ0(1+Δθ0)+Δθ02(2θ2+Δ(2θ2θ0+θ12))]+Pr2(f0θ2′+θ1f1′)+(θ1′f1+f2′θ0)+3Pr2f2θ0′+Prθ1′=0,f2(0,ξ)=0,f2′(0,ξ)=0,ϕ2′(0,ξ)=0,θ2(0,ξ)=0,f2′(∞,ξ)=0,ϕ2′(∞,ξ)=0,θ2(∞,ξ)=0.
It is pertinent to mentioned that (4.9) are coupled and nonlinear, so the solutions of these equations can be obtained by the Nachtsheim-Swigert iteration technique together with the sixth-order implicit Runge-Kutta-Butcher initial value solver. After knowing the values of the functions f′′, ϕ and θ′ and their derivatives we can calculate the values of the coefficient of skin friction, surface magnetic intensity and heat transfer in the region near the leading edge against ξ from the following expansion for S=0.1, λ=1.0, Pm=0.1, Pr=0.1, and radiation parameter Rd=1.0, 10.0, and θw=1.1, respectively.(4.10)Rex1/2Cfx=(1.59195+1.12282ξ+1.01047ξ2+⋯),Rex1/2Mgx=-(1.43230+1.02251ξ+1.34209ξ2+⋯),Rex-1/2Nux=-(0.21749-0.00475ξ+0.09878ξ2⋯).The numerical results thus obtained or entered in Tables1, 2, and 3 for coefficients of skin friction, rate of heat transfer and magnetic intensity at the surface. We can see that from these tables the series solution are in excellent agreement with that of finite difference solutions even for ξɛ[0,1].In Table4 the comparison of the solutions obtained by finite difference method and that of Glauert [11] analytically has been given for the coefficient of skin friction and magnetic intensity at the surface. It is observed that for Pm=1.0,10.0, radiation parameter Rd=∞, and the variation in magnetic field parameter S decrease the skin friction and the skin friction approaches to zero as S→1.1 and the local magnetic intensity Mgx increases with the increase of magnetic field parameter S. From this table it can be seen that the present method and the analytical results obtained by Glauert [11] are in good agreement. For magnetic Prandtl number Pm=10.0 the coefficients of skin friction and magnetic intensity at the surface are also noted in excellent agreement. Further, we see that for small value of magnetic Prandtl number Pm=0.1, and for magnetic field parameter S=π as in the case of Glauert [11] the separation occur at the surface. In Table 5 the value of coefficient of skin friction obtained by other authors Glauert [11] for large magnetic Prandtl number Pm by keeping Rd=∞ and Davies [2] for small magnetic Prandtl number by taking magnetic field parameter S=0.1 and sufficiently small S=0.05 are entered and compare with present results and found to be in good agreement. Here we notice that the agreement between present results and results obtained by Glauert [11], Davies [2] are in excellent agreement. The results entered in Table 6 are those obtained from heat transfer in hydromagnetics by Ramamoorthy [6] considering Eckert number equal to zero and radiation parameter Rd=∞are compared with the present results. From this table it can be seen that the numerical results obtained for different values of magnetic field parameter S and for Pm=0.1, Pr=1.0, λ=0.5 are very closed the results obtained by Ramamoorthy.Table 4
Values ofRex1/2Cfx and Rex1/2Mgx obtained by Glauert [2] and present authors while Rd=∞, Pm=1.0 and 10.0 against different values of S.
SPm=1.0Pm=10.0PresentGlauret [2]PresentGlauret [2]f′′(0)g(0)f′′(0)g(0)f′′(0)g(0)f′′(0)g(0)0.00.33212.12900.33212.17970.33210.95250.33211.00950.10.30672.17130.30252.24480.31880.96310.31821.02380.20.28062.21910.27292.30990.30500.97440.30441.03800.40.22572.33720.21382.44020.27700.99950.27671.06650.60.16572.50660.15472.57040.24791.02850.24911.09500.80.09722.80170.09552.70060.21761.06290.22141.12351.00.00144.84880.03642.83080.18571.10510.19381.15201.10.00006.04230.00682.89590.16911.13020.17991.1662Table 5
Values ofRex1/2Cfx when Rd=∞, S=0.1 and 0.05 at ξ=0.0 against Pm obtain by Glauert [2] and Devies [3] and the present authors.
s=0.0s=0.8PmPresentGlauert [2]PmPresentDavies [3]0.10.28880.26690.10.31060.31401.00.30670.30160.20.31070.31572.00.31090.30780.30.31530.31734.00.31450.31280.50.31720.31946.00.31640.31520.70.31830.32048.00.31770.31670.90.31910.320410.00.31860.3178—————————500.32380.3237———750.32480.3247———1000.32540.3254———Table 6
Values ofRex-1/2Nux for different S when Rd=∞, Pm=0.1, Pr=1.0, and λ=0.5 obtained by present authors and Ramamoorthy [6].
SPresent[6]0.10.645510.650480.30.619940.619580.50.591120.57693


## 4.3.2. Solution for Largeξ

Now, attention is given in finding the solution of equations (4.3)–(4.6) when ξ is large. The order of magnitude analysis of various terms in these equations shows that f′′′ and ξf′′ are largest terms in (4.3) and ϕ′′ and ξϕ′ in (3.14) and θ′′ and ξθ′ in (4.5). In the respective equations, both the terms have to be balanced in magnitude and the only way to do this, is to assume that η is small and hence its derivative is large. It is essential to find appropriate scaling for f, ϕ and θ. On balancing f′′′ and ξf′′ in (4.3) and ϕ′′ and ξϕ′ in (4.4) and θ′′, ξθ′ in (4.5), it is found that η=O(ξ-1), f=O(ξ-1), and ϕ=O(ξ-1). Therefore, following transformations may be introduced(4.11)f(η)=ξ-1f¯(η¯),η¯=ξη,ϕ(η)=ξ-1ϕ¯(η¯),θ(η)=θ¯(η¯).
By using (4.10), the transformed equation will take the form:(4.12)f¯′′′+f¯′′+λξ-2θ=12f[f¯′∂f¯∂ξ-f¯′′∂f¯∂ξ-S(ϕ¯′∂ϕ¯′∂ξ-ϕ¯′′∂ϕ¯∂ξ)],(4.13)1Pmϕ¯′′+ϕ¯′=12ξ-1[f¯′∂ϕ¯∂ξ-f¯′∂f¯∂ξ],(4.14)1Pr[1+43Rd(1+Δθ)3]+θ¯′=12ξ-1[f¯′∂θ¯∂ξ-θ¯′∂f¯∂ξ].
The regular perturbation of the functions f¯, ϕ¯ and θ¯ in power of ξ-2 is given as follows:(4.15)f̅(ξ,η̅)=∑m=01ξ-2mf̅m(η̅),ϕ̅(ξ,η̅)=∑m=01ξ-2mϕ̅m(η̅),θ̅(ξ,η̅)=∑m=01ξ-2mθ̅m(η̅).
By substituting (4.14) into (4.11)–(4.13), and equating like power of ξ and by dropping bars we have the following set of equations:O(ξ0):
(4.16)f0′′′+f0′′=0,1Pmϕ0′′+ϕ0′=0,[1+α(1+Δθ0)3]θ0′′+3αΔ(1+Δθ0)2θ0′2+Prθ0′=0.
The boundary conditions regarding to the perturbation series expansion are of the following form:(4.17)f0(0)=0,f0′(0)=0,ϕ0′(0)=1,θ0(0)=1,f0′(∞)=1,ϕ0′(∞)=0,θ0(∞)=0.O(ξ-2):(4.18)f1′′′+f1′′+λθ0=0,1Pmϕ1′′+ϕ1′=0,[1+α(1+Δθ0)3]θ1′′+3αΔ(1+Δθ0)2(θ1θ0′′+2θ0′θ1′)+6αΔ2θ1(1+Δθ0)θ0′2+Prθ1′=0.
And the boundary conditions regarding to the perturbation series expansion are of the following form:(4.19)f1(0)=0,f1′(0)=0,ϕ1′(0)=0,θ1(0)=0,f1′(∞)=0,ϕ1′(∞)=0,θ1(∞)=0.
From the solutions of (4.15)–(4.16) and (4.17)–(4.18), we obtain(4.20)f0(η)=η+e-η-1,f0′′(0)=1,ϕ0(η)=-1Pme-ηPm,ϕ0(0)=-1Pm,θ0′(η)=[1+43Rd(1+Δθ0)3]Pre-ηPr,θ0′(0)=[1+43Rd(1+Δθ0)3]Pr,f1′′(η)=λe-ηPr(Pr-1)+Prλe-η-λe-η2(Pr-1),f1′′(0)=λ(Pr-1)+Prλ-λ2(Pr-1),ϕ1(η)=0,ϕ1(0)=0,θ1′(η)=0,θ1′(0)=0.
Since, now, we know the values of f0′′(0), ϕ0(0), and θ0′(0), and f1′′(0), ϕ1(0), and θ1′(0) from the above solutions we calculate the friction coefficient, Rex1/2Cfx, local rate of heat transfer, Rex-1/2Nux, and the local magnetic intensity, Rex1/2Mgx at the surface from the following expressions:(4.21)Rex1/2Cfx≃ξ+λ(Pr-1)+Prλ-λ2(Pr-1)ξ-1,Rex1/2Mgx≃ξPm,Rex-1/2Nux≃[1+43Rd(1+Δθ0)3]Prξ.Numerical value of the local skin friction coefficient, surface magnetic intensity and the local rate of heat transfer are obtained from the relations (4.21) for different values of magnetic field parameter S and magnetic Prandtl number Pm and radiation parameter Rd, and surface temperature θw, Prandtl number Pr, respectively, in the down stream region are entered in Tables 1, 2, and 3, respectively. From these tables it can be seen that for large value of transpiration parameter ξ the skin friction Rex1/2Cfx approaches to ξ and the values of coefficient magnetic intensity Rex1/2Mgx approaches to ξ/Pm and Nusselt number Rex-1/2Nux approaches to [1+(4/3Rd)(1+Δθ0)3]Prξ. The comparison of the present results with the numerical results obtained by FDM shows excellent agreement in the down stream region.


## 5. Conclusion

The physical parameters such as mixed convection parameterλ, transpiration parameter ξ, magnetic field parameter S, magnetic Prandtl number Pm and radiation parameter Rd, Prandtl number Pr and surface temperature θw exerts significant influence on coefficients of skin friction Rex1/2Cfx, heat transfer Rex-1/2Nux and magnetic intensity Rex1/2Mgx at the surface.(i)
The coefficient of skin friction decreases, and the coefficient of rate of heat transfer and magnetic intensity at the surface increases with the increase of radiation parameterRd.(ii)
The coefficients of skin friction, heat transfer increases and magnetic intensity at the surface decreases with the increase of mixed convection parameterλ by keeping radiation parameter Rd, surface temperature θw, magnetic Prandtl Pm, magnetic force parameter S fixed. The momentum and thermal boundary layer thicknesses decreases and velocity and temperature profiles increases with the increase of the mixed convection parameter λ. It is also noted that the increase in mixed convection parameter λ reduce the transverse component of magnetic field profile.(iii)
The increase in Prandtl numberPr reduce the coefficient of skin friction and enhance the coefficient of heat transfer and magnetic intensity at the surface.(iv)
The coefficient of skin friction, heat transfer increases and the coefficient of magnetic intensity decreases with the increase of magnetic Prandtl numberPm.(v)
The transpiration parameterξ play a significant role in boundary layer, due to increase in transpiration parameter ξ the momentum and thermal boundary layer thicknesses decreases and the transverse component of magnetic field profile also reduced.(vi)
It is also concluded that an increase of the conduction radiation parameterRd decrease the local velocity as well as temperature distribution and enhance the transverse component of magnetic field at the surface.



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*Source: 686594-2011-02-09.xml*