Adv. Theor. Appl. Mech., Vol. 7, 2014, no. 1, 1-20 HIKARI Ltd,

Adv. Theor. Appl. Mech., Vol. 7, 2014, no. 1, 1-20 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/atam.2014.31124 Magneto-Hydrodynamic Eect with Temperature Dependent Viscosity on Natural Convection at an Axisymmetric Stagnation Point Saturated in Porous Media M. Modather M. Abdou and S. M. M. El-Kabier Department o Mathematics, Salman Bin Abdul Aziz University, College o Science and Humanity Studies Al-Kharj, Saudi Arabia Copyright 2014 M. Modather M. Abdou and S. M. M. El-Kabier. 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 A three-dimensional boundary layer solution is presented to study the eect o magneto-hydrodynamic with variable viscosity on natural convection in the vicinity o an axisymmetric stagnation point on heated vertical suraces saturated in porous medium. The luid viscosity is assumed to vary as a linear unction o temperature. The governing partial dierential equations or the low and heat transer are transormed into ordinary dierential equations by a suitable similarity transormation then, solved numerically using the Runge-Kutta numerical integration, procedure in conjunction with shooting technique. A parametric study illustrating the inluence o the magnetic parameter M, the viscosity parameter and Darcy number Da are conducted and a representative set o numerical results or the velocity, temperature, concentration, on skin riction, Nusselt number as well as Sherwood number are investigated. The obtained results are shown in graphic and tabulated representation ollowed by a quantitative discussion.

2 M. Modather M. Abdou and S. M. M. El-Kabier Keywords: Magneto-hydrodynamic, Natural convection, Porous media, Variable viscosity, Axisymmetric stagnation low NOMENCLATURE a B C Cp Da D m, F g k K M Nu m w p Pr q w Sc Sh T u,v,w x,y,z α β T β C σ μ θ φ ρ w ' Constant in equation (7) Magnetic ield intensity Concentration Speciic heat o luid Darcy number Mass diusivity Dimensionless velocity unction Acceleration due to gravity Thermal conductivity Permeability o the porous media Magnetic parameter Nusselt number Wall mass lux Pressure Prandtl number Wall heat lux Schmidt number Sherwood number Temperature Velocity component in x, y and z-axis Cartesian coordinates Greek symbols Thermal diusivity Coeicient o thermal expansion Electrical conductivity Coeicient o concentration expansion Absolute dynamic viscosity Dimensionless temperature Dimensionless concentration Viscosity parameter Density Subscripts Reers to condition at wall Reers to condition ar rom the wall Superscript Dierentiation with respect to

Magneto-hydrodynamic eect with temperature dependent viscosity 3 1- INTRODUCTION Magneto-hydrodynamic (MHD) natural convection heat transer low is o considerable interest in the technical ield due to its requent occurrence in industrial technology and geothermal application, high temperature plasmas applicable to nuclear usion energy conversion, liquid metal luid, and (MHD) power generation systems. Sparrow and Cess [1] studied the eect o magnetic ield on the natural convection heat transer. Romig [2] studied the eect o electric and magnetic ield on the heat transer to electrically conducting luid. Riley [3] analyzed the (MHD) ree convection heat transer. Gupta [4] studied laminar ree convection low o an electrically conducting luid rom a vertical plate with uniorm surace heat lux and variable wall temperature in the presence o a magnetic ield. Singh and Cowling [5] investigated thermal convection in magneto-hydrodynamics. Emery [6] has studied the eect o magnetic ield upon the ree convection o a conducting luid. Abo-Eldahab and A. El Aziz [7] investigated the problem o steady, laminar, hydromagnetic heat transer by mixed convection over a continuously stretching surace with power-law variation in the surace temperature or heat lux in the presence o internal heat generation/absorption eect. Chamkha [8] studied hydro-magnetic threedimensional ree convection on a vertical stretching surace with heat generation or absorption, the problem o steady, laminar, ree convection low over a vertical porous surace in the presence o a magnetic ield and heat generation or absorption is considered. Takhar et al. [9] studied the low and heat transer on a stretching surace in a rotating luid, in the presence o a magnetic ield. The combined eects o orced and natural convection heat transer in the presence o transverse magnetic ield orm a vertical suraces with radiation heat transer is studied by Duwairi and Damseh [10]. Duwairi and Damseh [11] investigated the eect o magnetic ield on the ree convection low o a viscous incompressible conductive luid rom a radiative vertical porous plate with uniorm surace temperature, uniorm rate o suction or injection, and uniorm magnetic ield lux. Modather et al. [12] investigated the eect o a transverse magnetic ield and thermal radiation with surace mass transer in ree convection on a vertical stretching surace with suction and blowing. Gorla [13,14] investigated the unsteady low and heat transer characteristics o the boundary layer low o an incompressible luid in the vicinity o an axisymmetric stagnation point or a variation with time o the ree stream such that the inviscid low is steady ater some given inite instant o time. Takhar [15] studied the eects o temperature dependent viscosity on natural convection in axisymmetric stagnation lows around heated vertical surace. The subject o convective low in porous media has attracted considerable attention in the last ew decades, due to its numerous applications in a wide variety o industrial processes as well as in many natural circumstances. Examples o such technological applications are geothermal extraction, storage o nuclear waste material, ground water lows, thermal insulation engineering, ood processing, ibrous insulation, soil pollution and packed-bed reactors to name just a ew. Seddeek [16] studied o the

4 M. Modather M. Abdou and S. M. M. El-Kabier eect o a magnetic ield and variable viscosity on steady two-dimensional laminar non-darcy orced convection low over a lat plate with variable wall temperature in a porous medium in the presence o blowing (suction). Hassanien et al. [17] investigated variable viscosity and thermal conductivity eects on combined heat and mass transer in mixed convection over a UHF/UMF wedge in porous media in the entire regime. Bagai [18] investigated the eect o temperature dependent viscosity on heat transer rates in the presence o internal heat generation, a similarity solution is proposed or the analysis o the steady ree convection boundary layers over a nonisothermal axisymmetric body embedded in a luid saturated porous medium. Modather and El-Kabeir [19] analyzed the eect o thermal radiation on ree convection low with variable viscosity and uniorm suction velocity along a uniormly heated vertical porous plate embedded in a porous medium in the presence o a uniorm transverse magnetic ield. Modather [20] studied the eect o thermal radiation on unsteady boundary layer low with temperature dependent viscosity and thermal conductivity due to a stretching sheet through porous media. The present study considers the laminar natural convection boundary-layer low in the presence o a transverse magnetic ield with variable viscosity on natural convection in the vicinity o an axisymmetric stagnation point on heated vertical suraces saturated in porous medium. The governing equations or the velocity, temperature and concentration ields are solved numerically using the Runge-Kutta numerical integration, procedure in conjunction with shooting technique. Numerical results are presented in the orm o velocity, temperature and concentration proiles within the boundary layer or dierent parameters entering into the analysis. Also the eects o the pertinent parameters on the local skin riction coeicient the local Nusselt and Sherwood numbers are also discussed. The eects o various parameters entering into the problem have been examined on the velocity and temperature proiles as well as the skin riction coeicient, and Nusselt and Sherwood numbers are presented graphically and in tabular orm. 2- MATHEMATICAL FORMULATIONS Consider a steady MHD incompressible viscous luid past a vertical plane in the y- direction in an axisymmetric stagnation low normal to a heated surace in a porous media. The low is impinging on the plate at the origin. In the ormulation o the problem, the plate is assumed to be at a uniorm temperature and concentration are T w and C w while the ambient luid is maintained at the uniorm temperature and concentration are T and C respectively. The dynamic viscosity μ is taken to be a variable in the equations o motion while the density ρ, the coeicients o thermal and concentration expansion are β T and β C respectively, permeability o the porous media is K, the thermal diusivity and mass diusivity are α and D m respectively. Under the boundary layer assumptions, the governing equations are given by: u v w + + = 0, (1) x y z

Magneto-hydrodynamic eect with temperature dependent viscosity 5 u u u p u u u ρ u + v + w = + μ + μ + μ x y z x x x y y z z (2) 2 μ + ρg[ βt ( T T ) + βc( C C ) ] σb u u, K v v v p v v v μ ρ u + v + w = + μ + μ + μ v, x y z y x x y y z z K (3) w w w p w w w μ ρ u + v + w = + μ + μ + μ w, x y z z x x y y z z K (4) 2 2 2 T T T T T T u + v + w = α + +, 2 2 2 x y z x y z (5) 2 2 2 C C C C C C u + v + w = Dm + +. 2 2 2 x y z x y z (6) The boundary conditions are given by: u = v = w = 0, T = T, C = C. on z=0 w w u = ax, v = 0, w = az, T = T, C = C, 1 at z = (7) 2 2 2 p = p0 ρa ( x + y ). 2 Here u, v and w are the velocity components along x, y and z-axes; p is the pressure in the low ield; p 0 is the stagnation pressure; g is the acceleration due to gravity, λ is latent heat o the solid phase and the constant a is directly proportional to the ree stream velocity (U ) ar rom the plate and inversely proportional to a characteristic length (L) o the plate. Following Carey and Mollendor [21], we assume that the absolute viscosity can be expressed as 1 dμ μ = μ 1 + ( T T ) (8) μ dt where μ is the value o μ at the ilm temperature o the low. The ollowing transormations would reduce the governing equations (1-6) to ordinary dierential equations: gβ( T Tw ) u = F( ) + ax ( ); v = ay ( ) (9) a aμ w = 2 ( ), T T = ( Tw T ) θ( ), C C = ( Cw C ) φ( ) (10) ρ ρ μ 2 2 2 2 dw p = p0 a ( x + y ) + w 2 (11) 2 ρ dz

6 M. Modather M. Abdou and S. M. M. El-Kabier and = aρ z μ where F,, θ and φ are related to the non-dimensional velocity, temperature and concentration unctions. Using (10), relation (8) can be written as μ = μ [ 1 + γ ( θ )] (13) where γ dμ = Tw 1 ( μ dt T ) (14) It can be veriied that with transormations (9) to (12) the continuity equation (1) is automatically satisied and the momentum, angular momentum and energy equations (2) to (6) reduce to: 2 [1 + γ ( θ )] 1+ γ ( θ ) + (2 + γθ ) + 1 M + = Da (15) [1 + γ ( θ )] 1+ γ ( θ ) F + (2 + γθ ) F + F Nφ θ M + F = 0 (16) Da 1 θ + 2 θ = 0 Pr (17) 1 2 0 Sc φ φ (18) K ρa β C ( Cw C ) where Da = is Darcy number, N = is concentration to thermal μ βt ( Tw T ) C buoyancy parameter, Pr p μ μ = is the ilm Prandtl number, Sc = k Dm is Schmidt number, M 2 σ B = is magnetic parameter. aρ Primes denote dierentiation with respect to only. The transormed boundary conditions may be written as: (0) = (0) = F(0) = 0, θ ( 0) = 1, φ(0) = 1, ( ) = 1, F ( ) = θ ( ) = φ( ) = 0. (19) (12) The shear stress, heat and mass lux at the plate are given by: u τ wx = μ z z= 0 1 aρ gβ ( T = + Tw ) μ 1 F (0) + ax γ (0) 2 μ a (20)

Magneto-hydrodynamic eect with temperature dependent viscosity 7 τ wy v = μ z z= 0 1 = μ 1 + γ 2 aρ ay (0) μ T aρ q = ( ) θ w = k k Tw T (0). (22) z μ z= 0 C aρ m = ( ) φ w = Dm Dm Cw C (0). z z= 0 μ One can write Nusselt number and Sherwood number as: a Nu ρ θ (0) (23) μ = a Sh ρ φ (0) (24) μ = qw x mw x Where Nu = and Sh =. (25) Tw T k Cw C Dm Where q w and m w are wall heat and mass lux respectively. 3-NUMERICAL METHOD The set o Equations (15) to (18) under the boundary conditions (19) has been solved numerically using the Runge Kutta integration scheme with shooting method. We let = x 1, =x 2, =x 3, F=x 4, F=x 5, θ =x 6, θ =x 7, φ=x 8, φ =x9 (26) Equations (15) to (18) are transormed into systems o irst-order dierential equations as ollows: x 1 = x 2 x 2 = x 3 1 2 [1 + γ ( x 6 )] x 3 = (2 x1+ γ x 7) x 3+ 1 x 2 M + x 2 ( 1+ γ ( x 6 ) ) Da x 4 = x 5 1 [1 + γ ( x 6 )] x 5 = (2 x1+ γ x7) x5+ x 2x4 Nx8 x6 M + x4 ( 1+ γ ( x 6 ) ) Da x = x 6 7 x = Pr 2 ( xx ) 7 1 7 x = x 8 9 (21)

8 M. Modather M. Abdou and S. M. M. El-Kabier ( 2 ) x = Sc x x (27) 9 1 9 subject to the ollowing initial conditions: x1(0) = x 2(0) = x 4(0) = 0, x 6(0) = 1, x 8(0) = 1, x ( ) = 1, 2 x 4( ) = x 6( ) = x 8( ) = 0. (28) Equation (27) is then integrated numerically as an initial valued problem to a given terminal point. The accuracy o the assumed missing initial condition is then checked by comparing the calculated value o the dependent variable at the terminal point with its given value. I a dierence exists, improved values o the missing initial conditions must be obtained and the process is repeated. A step size o = 01 was selected to be satisactory or a convergence criterion o 10-6 in nearly all cases. The maximum value o to each group o parameters Da, Pr, Sc, M, N and are determined when the values o unknown boundary conditions at = 0 do not change to successul loop with error less than 10_6. From the process o numerical computation, the local skin riction coeicient, the local Nusselt number and the local Sherwood number, which are respectively proportional to ( 0 ), F ( 0 ), θ ( 0 ) and φ (0) are worked out and their numerical values presented in tabular orms. 4- RESULTS AND DISCUSSION The system o equations (15-18) with the boundary conditions (19) respectively was solved numerically by the ourth order Runge-Kutta integration scheme. Calculations were carried out or the value o Prandtl number 0.733, Schmidt number 0.66, Darcy number Da ranging rom to, variable viscosity parameter ranging rom -0.8 to 0.8 and magnetic parameter M ranging rom to are summarized with N=. It should be noted here that, in the case o =0, M=0, Sc=0 and Da= Wang [22] has derived the solution o this problem. On the other hand, or variable viscosity case (M=0, Sc=0 and Da= ) solution o the above equations (15-18) have been obtained by Takhar [23] or Pr=0.7,1,10 and 100 and viscosity variation parameter (-1.6 1.6). We compared these results that we have obtained with the results in re [28] and it is ound that they are in good agreement. Tables (1-3) displays the results or the low rom heated vertical surace which show the surace values o velocity, temperature and concentration gradient components. These are proportional to the riction actor, Nusselt number and Sherwood number respectively. Table (1) show the eect o magnetic parameter M and Darcy number Da on local skin riction coeicient, Nusselt number and Sherwood number, the results indicate that as magnetic parameter M increases, there is an increasing in the local Nusselt number and Sherwood number while the local skin riction coeicient have the opposite behavior. Darcy number parameter also has a noticeable eect on the local skin riction coeicient, increasing Darcy number within the boundary layer

Magneto-hydrodynamic eect with temperature dependent viscosity 9 leads to an increase in the velocity within the layer and thus increases the local skin riction coeicient, on the other hand as it increases the thermal boundary layer thickness decreases and thus the rate o the heat and mass transer increase. The eect o temperature dependent viscosity with Darcy number on local skin riction coeicient, Nusselt number and Sherwood number is shown in Table (3), It can be seen rom this table that the viscosity variation parameter has a signiicant eect on the local Nusselt number. As the viscosity variation parameter decreases, the thermal boundary layer thickness decreases and, thus, the rate o the heat transer increases. The viscosity variation parameter also has a noticeable eect on the local skin riction coeicient and slight change in the local mass transer coeicient, decreasing the viscosity within the boundary layer leads to an increase in the velocity within the layer and, thus, increases the local skin riction coeicient while the local mass transer coeicient has the opposite behavior. Figures (2-5) display the dimensionless o velocity, temperature and concentration proiles or various values o magnetic parameter M, it can be seen that the velocity o the luid decreases as magnetic parameter M increases while temperature and concentration increase with it. Figures (6-9) describes the behavior o the velocity, temperature and concentration ields or various values o Darcy number parameter Da, it can be seen that, the concentration and temperature ields o the luid decrease as the Darcy number Da increases, while the velocity increases as the Darcy number Da increases. Figures (10-13) display the dimensionless o velocity, temperature and concentration proiles or various values o variable viscosity. It is observed that the temperature and concentration across the boundary layer, increase with increasing viscosity variation parameter while the velocity decreases. From equation (20) it ollows that the stagnation point is at: gβ ( Tw T ) F (0) xs = at y=z=0 (26) 2 a (0) Since F'(0) is negative, the stagnation point is below the centre line o the low when Tw>T (heated wall) and above the centre line o the low or Tw<T (cooled wall). Finally, the variation o F'(0)/"(0) with magnetic parameter M or some values o Darcy number Da and variable viscosity parameter is shown in igures (14-15), we note that this ratio increases with the increase o the magnetic parameter M and variable viscosity parameter and decreases with the increase o Darcy number Da. 5- CONCLUDING REMARKS In the present work, we have studied the problem o steady laminar natural convection boundary-layer low in the presence o a transverse magnetic ield with variable viscosity on natural convection in the vicinity o an axisymmetric stagnation point on heated vertical suraces saturated in porous medium. The viscosity o the luid is taken as a unction o temperature. The eect o Darcy number parameter Da,

10 M. Modather M. Abdou and S. M. M. El-Kabier magnetic parameter M and viscosity parameter on velocity, temperature, concentration, skin riction, Nusselt number as well as Sherwood number are investigated. As magnetic parameter M increases, there is an increasing in the local Nusselt number and Sherwood number while the local skin riction coeicient have the opposite behavior. Darcy number parameter also has a noticeable eect on the local skin riction coeicient, local heat and mass transer. Acknowledgements. This project was supported by the deanship o scientiic research at Salman bin Abdulaziz University under the research project No.5/T/33 REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] Sparrow, E. M. and Cess, R. D. "Eect o magnetic ield on ree convection heat transer", Int. J. Heat Mass Transer, Vol. 3, pp. 267-274, (1961). Romig, M. "The inluence o electric and magnetic ield on heat transer to electrically conducting luids", Adv. Heat Transer, Vol. 1, pp. 268-352, (1964). Riley, N. "Magnetohydrodynamic ree convection", J. Fluid Mech., Vol. 18, pp. 577-586, (1964). Gupta, A.S. "Laminar ree convection low o an electrically conducting luid rom a vertical plate with uniorm surace heat lux and variable wall temperature in the presence o a magnetic ield", J. Appl. Math. Phys. (ZAMP) 13, 324 332, (1963). Singh, K.R. and Cowling, T.G. "Thermal convection in magnetohydrodynamics", Q. J. Mech. Appl. Math. 16, 1 15, (1963). Emery, A.F. "The eect o magnetic ield upon the ree convection o a conducting luid", J. Heat Transer 85 (2), 119 124, Series C, (1963). Emad M. Abo-Eldahab and Mohamed A. El Aziz "Blowing/suction eect on hydromagnetic heat transer by mixed convection rom an inclined continuously stretching surace with internal heat generation/absorption" Int. J. o Therm. Sciences, Vol. 43, pp. 709-719, (2004). Chamkha, Ali J. "Hydromagnetic three-dimensional ree convection on a vertical stretching surace with heat generation or absorption", International Journal o Heat and Fluid Flow, Vol.20, pp. 84-92, (1999). Takhar, H. S. Chamkha, A. J. and Nath, G. " Flow and heat transer on a stretching surace in a rotating luid with a magnetic ield", Int. J. Therm. Sci., vol. 42, pp. 23-31, (2003) Duwairi, M and. Damseh, Rebhi A "MHD-buoyancy aiding and opposing lows with viscous dissipation eect rom radiate vertical suraces", Can. J. Chem. Vol. 82, pp. 613-618, (2004). Duwairi, H. M. and Damseh, Rebhi A. "Magnetohydrodynamic natural convection heat transer rom radiate vertical porous suraces", Heat and Mass Transer, Vol. 40, pp. 787-792, (2004).

Magneto-hydrodynamic eect with temperature dependent viscosity 11 [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] S. M. M. EL-Kabeir, M. Modather M. Abdou and R.S.R. Gorla, "Hydromagnetic combined convection rom radiating vertical stretching sheet with surace mass transer", Int. J. Fluid Mechanics Research, 35(6), 544-556, 2008. Gorla, R. S. R. "The inal approach to steady state in an axisymmetric stagnation low ollowing a change in ree stream velocity", Appl. Sci. Res. Vol. 40, pp. 247-251, (1983). Gorla, R. S. R. "The inal approach to steady state in a nonsteady axisymmetric stagnation point heat transer", Warme-Stoubertrag, Vol. 22, pp. 33-44, (1988). Takhar, H. S. (1993): "Variable viscosity eects on natural convection in axisymmetric stagnation lows adjacent to vertical suraces", Int. Comm. Heat Mass Transer, v. 20, pp. 255-264. Seddeek, M. A., Eects o magnetic ield and variable viscosity on orced non-darcy low about a lat plate with variable wall temperature in porous media in the presence o suction and blowing, Journal o Applied Mechanics and Technical Physics, vol. 43, no. 1, pp. 13-17, 2002. Hassanien, I. A. Essawy, A. H. and Moursy, N. M. "Variable viscosity and thermal conductivity eects on combined heat and mass transer in mixed convection over a UHF/UMF wedge in porous media: the entire regime", Applied Mathematics and Computation, Vol.145, 2-3, pp. 667-682, (2003). Bagai, S., "Eect o variable viscosity on ree convection over a non-isothermal axisymmetric body in a porous medium with internal heat generation", Acta Mechanica, vol. 169, no. 1-4, pp.187-194, 2004. M. Modather M. Abdou and EL-Kabeir, S. M. M. "Magnetohydrodynamics and radiative eect on ree convection low o luid with variable viscosity rom a vertical plate through a porous medium", J. o Porous Media, vol.10, No. 5, pp. 503-514, (2007). M. Modather M. Abdou "Eect o Radiation with Temperature Dependent Viscosity and Thermal Conductivity on Unsteady a Stretching Sheet through Porous Media" Nonlinear Analysis: Modelling and Control, 2010, Vol. 15, No. 3, 257 270 (2010) Carey, P. V., and Mollendor, C. J. "Natural convection in liquids with temperaturedependent viscosity", in Proceedings o the Sixth International Heat Transer Conerence, Toronto, Vol. 2, pp. 211-217, Hemisphere, Washington, D. C. (1978). Wang, C. Y. J. Appl. Mech. Vol. 2, pp. 724-725, (1987). Takhar, H.S. (1993): "Variable viscosity eects on natural convection in axisymmetric stagnation lows adjacent to vertical suraces", Int. Comm. Heat Mass Transer, v. 20, pp. 255-264.

12 M. Modather M. Abdou and S. M. M. El-Kabier

Magneto-hydrodynamic eect with temperature dependent viscosity 13 Table 1: Values o ( 0), F (0), θ (0) and φ (0) with Pr =0.733,, N= and. Da M ( 0 ) F ( 0 ) θ ( 0 ) φ (0) -F'(0)/"(0) 0.72780 0.62576 5187 0.45666 0.75261 0.73853 0.71873 0.67236 7083 2936 0.49446 0.44230 4926 0941 0.47597 0.42638 3408 1.18021 1.30237 1.47234 0.85242 0.72024 0.62331 0.49993 0.77276 0.76471 0.74870 0.70345 0.60926 6337 2364 0.46285 8606 4191 0377 0.44575 0.90654 6174 1.20117 1.40709 5.0 0.94130 0.78869 0.67528 3080 0.78236 0.77863 0.76589 0.72286 0.63358 8539 4291 0.47669 0.60935 6299 2217 0.45883 0.83115 0.98726 1.13419 1.36183 2 0.98994 0.82656 0.70420 4786 0.78641 0.78496 0.77411 0.73270 0.64602 9682 5303 0.48405 0.62127 7393 3184 0.46581 0.79440 0.94967 9929 1.33738 5 0001 0.83444 0.71023 5142 0.78715 0.78617 0.77572 0.73468 0.64853 9913 5509 0.48556 0.62367 7614 3381 0.46724 0.78714 0.94215 9221 1.33235 10 0340 0.83709 0.71226 5261 0.78740 0.78657 0.77626 0.73534 0.64936 9991 5578 0.48607 0.62447 7689 3447 0.46772 0.78473 0.93964 8985 1.33066 0679 0.83975 0.71430 5381 0.78764 0.78696 0.77679 0.73600 0.65020 0.60068 5647 0.48658 0.62528 7763 3513 0.46820 0.78233 0.93714 8748 1.32896

14 M. Modather M. Abdou and S. M. M. El-Kabier Table 2: Values o ( 0), F (0), θ (0) and φ (0) with Pr =0.733,, N= and M=. Da ( 0 ) F ( 0 ) θ ( 0 ) φ (0) -F'(0)/"(0) -0.80 5.0 2 5 10 0.98784 1.12069 1.22831 1.29245 1.30622 1.31088 1.31558 1.32880 1.36155 1.37539 1.38002 1.38073 1.38095 1.38117 2063 5946 8741 0.60281 0.60601 0.60708 0.60816 0.49948 3658 6338 7816 8123 8226 8330 1.34515 1.21492 1.11974 6775 5704 5345 4985-0.60 5.0 2 5 10 0.88871 0816 1.10347 1.15971 1.17173 1.17580 1.17990 1.18933 1.22123 1.23559 40871.2 1.24176 1.24204 1.24231 1564 5322 7999 9464 9767 9869 9971 0.49495 3086 5651 7057 7348 7446 7544 1.33827 1.21134 1.11973 6998 5976 5633 5290 5.0 2 5 10 0.69564 0.78833 0.85920 0.89989 0.90849 0.91139 0.91431 0.91891 0.94945 0.96514 0.97193 0.97318 0.97360 0.97400 0423 3825 6169 7429 7687 7774 7861 0.48467 1714 3957 5164 5412 5495 5579 1.32095 1.20437 1.12330 8005 7121 6825 6529 0.60 5.0 2 5 10 8076 0.65664 0.71247 0.74377 0.75031 0.75252 0.75474 0.75888 0.78891 0.80574 0.81364 0.81517 0.81568 0.81618 6460.49 2688 4718 5790 6009 6082 6155 0.47774 0673 2613 3638 3847 3917 3987 1.30669 1.20143 1.13091 9395 8644 8393 8141 0.80 5.0 2 5 10 5187 0.62331 0.67528 0.70420 0.71023 0.71226 0.71430 0.71873 0.74870 0.76589 0.77411 0.77572 0.77626 0.77679 0.49446 2364 4291 5303 5509 5578 5647 0.47597 0377 2217 3184 3381 3447 3513 1.30237 1.20117 1.13419 9929 9221 8985 8748

Magneto-hydrodynamic eect with temperature dependent viscosity 15 Table 3: Values o ( 0), F (0), θ (0) and φ (0) with,, N= and Da=2. M ( 0 ) F ( 0 ) θ ( 0 ) φ (0) -F'(0)/"(0) -0.80 1.89051 1.54914 1.29245 0.96504 1.42109 1.40899 1.38002 1.28846 0.70764 0.65237 0.60281 2349 0.67855 0.62560 7816 0247 0.75170 0.90953 6775 1.33513-0.60 1.68502 1.38518 1.15971 0.87179 1.27447 1.26519 1.24087 1.16192 0.69727 0.64313 9464 1717 0.66892 0.61702 7057 0.49661 0.75636 0.91338 6998 1.33280 1.28648 6573 0.89989 0.68780 0.99243 0.98802.971930 0.91566 0.67186 0.62032 7429 0116 0.64531 9581 5164 0.48174 0.77143 0.92708 8005 1.33129 0.60 4964 0.87482 0.74377 7625 0.82746 0.82546 0.81364 0.76942 0.65187 0.60217 5790 0.48801 0.62672 7892 3638 0.46950 0.78832 0.94358 9395 1.33522 0.80 0.98994 0.82656 0.70420 4786 0.78641 0.78496 0.77411 0.73270 0.64602 9682 5303 0.48405 0.62127 7393 3184 0.46581 0.79440 0.94967 9929 1.33738

16 M. Modather M. Abdou and S. M. M. El-Kabier M= 0.8 ' 0.6 M= M= =-0.8 = 0.4 0.2 Da=2 Fig. 2 Velocity distribution or varous values o viscosity and magnetic M parameters 0.8 =-0.8 = Da=2 θ 0.6 M=,, 0.4 0.2 Fig. 3 Temperature distribution or varous values o viscosity and magnetic M parameters 0.8 =-0.8 = Da=2 φ 0.6 M=,, 0.4 0.2 Fig. 4 Concentration distribution or varous values o viscosity and magnetic M parameters

Magneto-hydrodynamic eect with temperature dependent viscosity 17 0-5 -0.10 = =-0.8 F -0.15-0.20 M=,, -0.25-0.30 Da=2-0.35 Fig. 5 Velocity distribution (F) or varous values o viscosity and magnetic M parameters 0.8 0.6 Da=1, 2, 3, 5, 20, 50, 100, ' 0.4 0.2 M= Fig. 6 Velocity distribution or varous values o Darcy number Da 0.8 0.6 Da=1, 2, 3, 5, 20, 50, 100, θ 0.4 0.2 M= Fig. 7 Temperature distribution or varous values o Darcy number Da

18 M. Modather M. Abdou and S. M. M. El-Kabier 0.8 0.6 Da=1, 2, 3, 5, 20, 50, 100, φ 0.4 0.2 M= Fig. 8 Concentration distribution or varous values o Darcy number Da -0.1 F Da=1, 2, 3, 5, 20, 50, 100, -0.2 M= -0.3 Fig. 9 Velocity distribution (F) or varous values o Darcy number Da ' 0.8 0.6 =-0.8 =-0.6 = =0.6 0.4 0.2 Da=5 M= Fig. 10 Velocity distribution or varous values o viscosity parameter

Magneto-hydrodynamic eect with temperature dependent viscosity 19 0.8 0.6 =0.6 = =-0.6 =-0.8 θ 0.4 0.2 Da=5 M= Fig. 11 Temperature distribution or varous values o viscosity parameter 0.8 0.6 =0.6 = =-0.6 =-0.8 φ 0.4 0.2 Da=5 M= Fig. 12 Concentration distribution or varous values o viscosity parameter 0-5 -0.10-0.15 =0.6 = =-0.6 =-0.8 F -0.20-0.25-0.30 Da=5 M= -0.35 Fig. 13 Velocity distribution (F) or varous values o viscosity parameter

20 M. Modather M. Abdou and S. M. M. El-Kabier 1.50 1.35 Da=1, 2, 3, 5, 20, 50, 100, -F'(0)/"(0) 1.20 5 0.90 M= 0.75 1.5 M Fig. 14 Eect o magnetic parameter M and Darcy number Da -F'(0)/"(0) 1.35 1.20 5 =-0.8 =-0.6 = =0.6 0.90 Da=2 0.75 1.5 M Fig. 15 Eect o magnetic parameter M and viscosity parameter Received: November 1, 2013