MAGNETOHYDRODYNAMIC FLOW AND HEAT TRANSFER OF A JEFFREY FLUID TOWARDS A STRETCHING VERTICAL SURFACE

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1 THERMAL SCIENCE: Year 7, Vol., No. A, pp MAGNETOHYDRODYNAMIC FLOW AND HEAT TRANSFER OF A JEFFREY FLUID TOWARDS A STRETCHING VERTICAL SURFACE by Kartini AHMAD a* and Anuar ISHAK b a Department of Science in Engineering, Kulliyyah of Engineering, International Islamic University, Kuala Lumpur, Malaysia b School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor, Malaysia Original scientific paper DOI:.98/TSCI439A This study investigates the steady-mied convection boundary layer flo near a stagnation point that runs about a linearly stretched vertical surface filled ith a Jeffery fluid in the presence of a transverse magnetic field. It is assumed that the eternal velocity impinges normally to the all and the all temperature varies linearly ith the distance from the stagnation point. The governing partial differential equations that govern the fluid flo are transformed into a set of coupled ordinary differential equations, hich are then solved numerically using a finite difference scheme. The numerical results are presented for some values of parameters, namely Deborah number, Prandtl number, magnetic parameter, and the mied convection parameter, for both assisting and opposing flos. Key ords: stagnation flo, Jeffrey fluid, heat transfer, boundary layer, fluid dynamics Introduction Industrially, the process for stretching sheet has significant relevance to several practical applications, such as the etrusion of metals, plastics and polymers, etc. During the rubber and plastic sheets manufacturing process, a gaseous medium, through the not-yet solidified material, is blon. As such, the study of heat transfer and flo field is necessary to determine the quality of the final products, as eplained by Kare and Jaluria []. It seems that Crane [] as the first to give a similarity solution in closed analytical form for steady -D incompressible boundary layer flo caused by the stretching of a sheet, hich moves in its on plane ith a velocity varying linearly ith distance from a fied point. Flo hich moves toards a vertical surface ill create a flo ith buoyancy force due to the eistence of the temperature difference beteen the all and the free stream. Such flo is knon as the mied convection flo. Hiemenz [3] as the first to investigate the -D stagnation flo toards a stationary semi-infinite all by using similarity transformation in order to reduce the Navier-Stokes equation to a non-linear ordinary differential equation. Since then, lots of research have been carried out to investigate the mied convection flo toards a vertical sheet, [4-9]. It is orth mentioning that the ork of Ishak et al. [4] has been etended by Saleh et al. [] to the case of a shrinking sheet. A study on the mied convection boundary layer flo over a vertical permeable cylinder * Corresponding author, kartini@iium.edu.my

2 68 THERMAL SCIENCE: Year 7, Vol., No. A, pp has been conducted by Ellahi et al. []. On the other hand, MHD flo and heat transfer has been studied in [-4]. In all previously mentioned papers, hoever, investigators confine their studies to Netonian fluid flo problems, hich is inadequate to describe some fluid properties. As such, non-netonian fluids, hich ehibit a non-linear relationship beteen the stresses and the rate of strains, are found to be more interesting than Netonian fluids. There are numerous references on the stagnation point flo toards a vertical surface in various types of non-netonian fluids, hich can be found in the literature, and some of them can be found in [5-9]. The MHD flo of a non-netonian fluid as considered in [-7]. The aim of the present study is to etend the ork done by Ishak et al. [4], by investigating the flo of one kind of Oldroyd model, i. e. Jeffery s version, hich impinges normally on a heated or cooled vertical surface that is being stretched. This fluid model includes elastic and memory effects ehibited by dilute polymer solutions and biological fluids, Hayat et al. [8]. Even though the literature on the Jeffery fluid flo is scarce, the fluid is one of great importance. Quite recently, the flo of a Jeffrey fluid in eccentric cylinders as investigated by Ellahi et al. []. Problem formulation Consider a steady -D laminar boundary layer stagnation-point flo of an incompressible electrically-conducting Jeffery s fluid, impinging normally toards a vertical surface. It is assumed that the ambient fluid moves ith velocity u e () = a in the y-direction toards the stagnation point on the plate, ith the temperature varying linearly along it ith T () = T + b, here a (> ) and b are arbitrary constants. The continuous stretching plate is assumed to have a velocity of the form u () = c, here c > is a constant. Assume that a uniform magnetic field of strength, B, is applied in the positive y-direction normal to the plate, and the induced magnetic field due to the magnetic Reynolds number is taken to be small enough and assumed to be negligible in comparison to the applied magnetic field. Under these assumptions, along ith the Boussinesq and boundary layer approimations, the basic equations of the problem can be ritten: u v + = () y du ν u + v = u + + u + + v u v e u u u u u u u e 3 y d y y y y y y σ Bo ( ue u) gβ ( T T ) + ± () ρ subject to the boundary conditions: u + v = α y y T T T u = u ( ), v=, T = T at y = u u ue ( ),, T T as y y (3) (4)

3 THERMAL SCIENCE: Year 7, Vol., No. A, pp here u and v are the velocity components along the - and y-aes, respectively, g the acceleration due to the gravity, and T the fluid temperature in the boundary layer. Further, ν,,, ρ, β, and α are, respectively, the kinematic viscosity, ratio of the relaation and retardation times, relaation time, fluid density, thermal epansion coefficient, and thermal diffusivity. The ± sign in the last term of eq. () represents the influence of the thermal buoyancy force, ith + and signs pertaining to the buoyancy assisting and opposing flo regions, respectively. According to Ramachandran et al. [9], the assisting flo eists if the upper half of the flat plate is heated hile the loer half of the plate is cooled. In this case, the flo near the heated flat surface tends to move upard, and the flo near the cooled flat plate tends to move donard. Therefore, this behaviour acts to assist the flo field. The reverse trend can be observed in the opposing flo. We look at the solutions of eqs. ()-(3) in the folloing forms: T T u ψ = νu f ( ), θ( ) =, = y T T ν here ψ is the stream function, it is defined in the usual ay as u = ψ/ y and ν = ψ/. Thus, e have: u= c f '( ), v= cν f( ) (6) here prime denotes differentiation ith respect to. Substituting variables (5) and (6) into eqs. ()-(3), eq. () is automatically satisfied and eqs. () and (3) are reduced to: iv a '' f + γ ( f ff ) + ( + ) ± θ + M ( f ) f + ff = (7) c θ + Pr fθ f θ = (8) ( ) and the transformed boundary conditions can be ritten: f() =, f '() =, θ() =, at = a f '( ), f ''( ), θ ( ) as c Here, γ = c is the Deborah number, M = aσb /(ρc ) the MHD parameter, Pr = ν/α the Prandtl number, and = Gr /Re the mied convection parameter here Gr = gβ(t T ) 3 /ν is the local Grashof number and Re = u /ν is the local Reynolds number. We note that is a constant, ith > and < corresponding to the opposing and assisting flos, respectively, hile = (i. e. T = T ) for pure forced convection flo. It should be pointed out that for γ = = M =, the problem is reduced to that of Ishak et al. [4]. The physical quantities of principle interest are the skin friction coefficient, C f, and the local Nusselt number, Nu, hich are defined: τ, Nu q C f = = ρue kt ( T ) () here τ and q are the all shear stress and the heat flu from the surface, respectively, hich are given by: (5) (9)

4 7 THERMAL SCIENCE: Year 7, Vol., No. A, pp u T τ = µ, q = k y y y= y= here μ and k being the dynamic viscosity and the thermal conductivity, respectively. Substituting eqs. (5) and (6) into eq. (9), the scaled skin friction coefficient and the local Nusselt number reduce to: Nu Cf Re = f (), = θ () () Re here Re = u /ν is the local Reynolds number. Results and discussions The ordinary differential eqs. (7) and (8), subject to the boundary conditions (9), have been solved numerically using the Keller-bo method for some values of the governing parameters, i. e. the magnetic parameter, M, the material parameter (Deborah number), γ, the mied convection parameter,, the Prandtl number, and the velocity ratio of a/c. The detailed procedure of this method can be found in the books by Cebeci [3] and Cebeci and Bradsha [3]. In order to validate the present code, the present results obtained for the skin friction coefficient C f ) / hen γ = M = = for several values of a/c are compared ith Ishak et al. [4], Mahapatra and Gupta [3], and Nazar et al. [33] and as shon in tab., hereas the present values obtained for the C f ) / and the Nu ) / for different values of Prandtl Table. Comparisons of C f ) / values obtained hen γ = M = = for some values of a/c Mahapatra and Nazar et Ishak et a/c Gupta [3] al. [33] al. [4] Present results Table. Values of C f ) / and Nu ) / obtained for various γ and Prandtl number hen M = and = a/c = ; results in ( ) are those of Ishak et al. [4] Buoyancy assisting flo Buoyancy opposing flo γ Pr C f ) / Nu ) / C f ) / Nu ) / (.3645).84 (.84).75 (.75) (.93) 3.9 (3.9) 5.69 (5.63) (.385).83 (.83).83 (.83) (.93) (3.466) 5.59 (5.593) () number hen M = γ = and = a/c =, for both assisting and opposing flos, are compared ith Ishak et al. [4] as tabulated in tab., along ith the ne results obtained for the C f ) / and the Nu ) / for various values of the Deborah number. For the sake of brevity, the folloing results of this paper are limited to =. It can be seen from tabs. and, the values of C f ) / and Nu ) / obtained are found to be in a good agreement ith previously published results, thus gives great confident to the numerical code used in the present study. Various cases are plotted for the skin friction coefficient and the local Nusselt number for assisting and opposing flos, as shon in figs. -3. The effect of the Deborah number is illus-

5 THERMAL SCIENCE: Year 7, Vol., No. A, pp / C f Re 3 Pr =.7, M =, =, a/c = -/ Nu Re.. Pr =.7, M =, =, a/c = γ =,,, 3 γ =,,, Figure. skin friction coefficient and local Nusselt number vs. for some values of γ hen Pr =.7, M =, and a/c =.9 / C f Re.5.5 γ =, M =, =, a/c = -/ Nu Re 5 4 γ =, M =, =, a/c = Pr = Pr = 7 Pr =, 4.5, 7, 3 Pr = Pr = Figure. skin friction coefficient and local Nusselt number vs. for some values of Prandtl number hen γ =, M =, and a/c = / C f Re γ =.5, M =., =, Pr =.68 a/c =.5,.,.5,.3 -/ Nu Re.6.4 γ =.5, M =., =, Pr = a/c =.3 a/c =.3 a/c =.5 a/c =. a/c = Figure 3. skin friction coefficient and local Nusselt number vs. for some values of a/c hen γ =.5, M =., and Pr = a/c =.5 a/c =. a/c =.5 a/c =.5,.,

6 7 THERMAL SCIENCE: Year 7, Vol., No. A, pp trated in fig. hen Pr =.7, M =, and a/c =. While the influence of the Prandtl number is depicted in fig. for γ =, M =, and a/c =. Figure 3 shos the impact of a/c hen γ =.5, M =., and Pr =.68. The profile of the dimensionless velocity f '() and the dimensionless temperature θ() are depicted in figs. 4-7 for some values of the Deborah number, the Prandtl number, the mied convection parameter,, and the velocity ratio, a/c, for both assisting and opposing flos, respectively. Figures -3 suggest that for all positive values of the mied convection parameter (assisting flo), the values of the skin friction coefficients and the local Nusselt numbers are found to eist, hile there are restricted values of the skin friction coefficient and the local Nusselt number for the opposing flo. The assisting buoyant flo is found to increase the skin friction coefficient, hile the opposing buoyant flo creates decrement of the skin friction coefficient as illustrated in figs.,, and 3. This is due to the assisting buoyant flo, hich enhances the buoyancy force and hence, increases the fluid velocity. This action subsequently increases the all shear stress, hich increases the skin friction coefficient, Ishak et al. [4]. Referring to figs. and, it should be noted that all curves intersect at a point hen the buoyancy force is zero, =. In this case, a/c = gives C f ) / =. Hoever, if a/c, it should be epected that the intersection point is no longer fied at, as C f ) / is very much influenced by the stretching velocity and the velocity of the eternal stream given by the constants a and c, respectively, as depicted in fig. 3. The figure also provides an idea of the point of intersection taking various values of a/c. The effects of the material parameter γ on the skin friction coefficient and the local Nusselt number can be seen in figs. and, hen Pr =.7, M =, and a/c = for both assisting and opposing flos. The values of Nu ) / are found to coincide at.458. The eistence of the material parameter/deborah number decreases the skin friction coefficient and surface heat transfer for the assisting flo. This is in line ith the results obtained in tab.. Flo ith a high Deborah number indicates that the fluid is dominated by elasticity, demonstrating solid-like behaviour, Reiner [34]. As such, the result is epected. The effects of the Prandtl number on the skin friction coefficient and the local Nusselt number are depicted in figs. and hen γ =, M =, and a/c =. It is seen that for a fied value of, as Prandtl number increases, the value of the skin friction coefficient decreases and the local Nusselt number is found to increase for the assisting flo. Flo ith a high value Prandtl number shos that the fluid is more viscous and, in return, retards the movement of the flo, hence reducing both the shear stress and the thermal conductivity on the surface. The implication can be seen in figs. and, here the skin friction C f ) / decreases but the local Nusselt number Nu ) / increases. The opposite trend occurs for the opposing flo. For a fied value of Prandtl number, it is noted that the local Nusselt number slightly increases as the buoyant parameter is increased for assisting flo, as shon in fig.. Figures 3 and 3 sho the effect of a/c on the skin friction coefficient and the local Nusselt number hen γ =.5, M =., and Pr =.68. It is noted that for the assisting flo, as increases, the skin friction coefficient and the local Nusselt number ill increase, too. This is due to a large, hich produces a large buoyancy force and in turn yields high kinetic energy to accelerate the fluid flo and increase the skin friction coefficient and the local Nusselt number. An increment of a/c ill result in the increment of the skin friction coefficient and the local Nusselt number. As prescribed in eq. (9), the velocity profiles f ʹ() in figs. 4, 5, and 6 begin at at the beginning of the motion for both the assisting and opposing flos. For the assisting flo ( > ), the velocity increases sloly (f ʹ > in the boundary layer) until it achieved a cer-

7 THERMAL SCIENCE: Year 7, Vol., No. A, pp f ().5 Pr =.7, =, M =, =, a/c = Pr =.7, =, M =, =, a/c = θ(). γ = 3,,,,, γ =,,, 3.95 γ = 3, 3,,,,, Figure 4. velocity and temperature profiles for some values of γ hen =, Pr =.7, M =., and a/c =. f ().4. Pr =, 7, 4.5, γ =, =, M =, =, a/c = γ =, =, M =, =, a/c = θ() Pr =, 7, 4.5,.98 Pr =, 7, 4.5, Figure 5. velocity and temperature profiles for some values of Prandtl number hen γ =, =, M =, and a/c = f ().5..5 γ =, M =, Pr =.7, =, a/c = γ =, M =, Pr =.7, =, a/c =.9 θ().8 =,.5,, =.5,,.9 =,.5,, Figure 6. velocity and temperature profiles for some values of hen γ =, M =, Pr =.7 and a/c =.. =.5,,

8 74 THERMAL SCIENCE: Year 7, Vol., No. A, pp tain value, then decreased asymptotically to a/c =, (in this case) at the outside of the boundary layer. This is epected as the thermal epansion caused by the high all temperature assists the flo. On the contrary, for the opposing flo, f ʹ < in the boundary layer due to the resistance of thermal epansion to the flo, Abbas et al. [6]. Hoever, no boundary layer is formed hen the buoyancy or mied convection parameter is absent and the velocity f ʹ = throughout the flo domain, as illustrated in fig. 6. Figure 4 shos that the magnitude of the velocity f ʹ decreases ith the increase of the Deborah number. As such, it may be epected that as γ, f ʹ = in the hole flo domain. This is clear from the fact that γ measures the viscosity/elasticity of a fluid, as previously eplained. The same phenomenon can be observed hen Pr, as shon in fig. 5. Increasing the Prandtl number causes the decrease of the velocity flo and the temperature at the surface, as depicted in figs. 5 and 5. This is due to the fact that an increase in Prandtl number indicates the increase of the fluid heat capacity or the decrease of the thermal diffusivity, causing a diminution of the influence of the thermal epansion to the flo, Abbas et al. [6]. This results in faster formation of the thermal boundary layer and in turn, decreases the thermal boundary layer thickness as Prandtl number increases for both assisting and opposing flos. The effect of the Deborah number and the mied convection parameter are less pronounced ith a variation in temperature, as can be seen in figs. 4 and 6. These figures clearly sho that for the assisting flo, no temperature difference occurs for various values of γ and, hile the temperature difference is conspicuous in the opposing flo. Hoever, there is a slight difference in the temperature distribution for the opposing flo, i. e. at any point in the boundary layer, the temperature increases ith the increase of γ, fig, 4 and, fig. 6, respectively. The thermal boundary layer thickness for both assisting and opposing flos are found to be more or less equal, as depicted in these to figures. Figure 7 demonstrates the effect of a/c hen =, Pr =.68, M =., and γ =.5. It is noted that the flo, hen the stretching velocity is less than the free stream velocity (a/c > ), has a boundary layer structure and the thickness of the boundary layer decreases ith an increase in a/c for both assisting and opposing flos. According to Mahapatra and Gupta [3], an increase in a in relation to c (a/c > ) implies an increase in straining motion near the stagnation region, resulting in increased acceleration of the eternal stream, leading to increased velocity, hich is then thinned by the boundary layer ith an increase of a/c. This fact as eperimentally confirmed by Flachsbart, Schlichting [35]. An inverted boundary layer is f () 3.5 =, Pr =.68, M =,, γ =,5, = =, Pr =.68, M =,, γ =,5, = a/c =.3 θ().8.5 a/c = a/c = a/c =.5 a/c = Figure 7. velocity and temperature profiles for some values of a/c hen =, Pr =.68, M =., and γ =.5.4. a/c =.3,.5,,.5,.

9 THERMAL SCIENCE: Year 7, Vol., No. A, pp seen to form for a/c < hen the stretching velocity c of the surface eceeds the velocity a of the eternal stream. Figure 7 shos the temperature distribution in the thermal boundary layer. It is observed that the temperature of the fluid decreases as the distance from the surface is increased. The temperature at a point is found to decrease ith the increase in a/c for both assisting and opposing flos. As such, flo ith high a/c produces a thinner thermal boundary layer. The temperature for the opposing flo is slightly higher compared ith the assisting flo at all points in the boundary layer, and the buoyancy effect is more pronounced for small a/c. Despite various profiles depicted in figs. 4-7, all the velocity and temperature profiles presented in those figures satisfy the far field boundary conditions (9) asymptotically, thus supporting the validity of the numerical results obtained. Figure 8 shos the streamlines obtained for Pr = 7, γ =, =, M =, and a/c = toards a stagnation point on a stretched Figure 8. Streamlines for -D stretching sheet hen Pr = 7, γ =, =, M =, and a/c = vertical surface. The oncoming flo generated by the code is found to be satisfactory and hence e are confident of the results obtained in the present paper. Conclusion In this paper, the steady -D stagnation-point flo of a Jeffery fluid toards a vertical stretched surface in the presence of buoyancy force and magnetic field is investigated numerically using a finite difference scheme ith an iterative technique. The problem is formulated in such a ay that the stretching velocity and the surface temperature vary linearly ith the distance from the stagnation point. The numerical solution obtained for the quantities of interest, hich are the skin friction coefficient and the local Nusselt number for some values of Pr = 7, γ =, =, M =, =, a/c = ψ(aν) / =,.,.5,, ψ(aν) / =.,.5,, the parameters, are displayed and discussed for both assisting and opposing flos. The flo ith a high Deborah number is found to decrease the magnitude of the skin friction and the local Nusselt number for the assisting flo, hile the opposite trend occurs for the opposing flo. Nomenclature a, b, c constants, [ ] B uniform magnetic field, [T] C f skin friction coefficient, [ ] f dimensionless stream function, [ ] g acceleration to gravity, [ms ] Gr local Grashof number, [ ] k thermal conductivity, [Wm K ] M magnetic parameter, [ ] Nu local Nusselt number, [ ] Pr Prandtl number, [ ] q all heat flu, [Wm ] Re local Reynolds number, [ ] T fluid temperature, [K] T () temperature of the stretching sheet, [K] T ambient temperature, [K] u, v velocity components along the - and y-directions, respectively, [ms ] u e velocity of the ambient fluid, [ms ], y Cartesian co-ordinates along the surface and normal to it, respectively, Greek symbols α thermal diffusit, [m s ] β thermal epansion coefficient, [K ] γ Deborah number, [ ] similarity variable, [ ]

10 76 THERMAL SCIENCE: Year 7, Vol., No. A, pp θ dimensionless parameter, [ ] buoyancy or mied convection parameter, [ ] ratio of the relaation and retardation times, [ ] relaation time, [s] μ dynamic viscosity, [kgm s ] ν kinematic viscosity, [m s ] ρ fluid density, [kgm 3 ] σ electrical conductivity, [Sm ] τ shear stress, [kgm s ] ψ stream function, [ ] Subscripts condition at the stretching sheet condition at infinity Superscript ʹ differentiation ith respect to Acknoledgement The first author gratefully acknoledge the financial support received in the form of a research grant (RAGS project code: RAGS3-3-66) from the Ministry of Higher Education, Malaysia. References [] Kare, M. V., Jaluria, Y., Numerical Simulation of Thermal Transport Associated ith a Continuously Moving Flat Sheet in Material Processing, ASME Journal of Heat Transfer, 3 (99), 3, pp [] Crane, L. J., Flo Past a Stretching Plate. Zeitschrift für angeandte Mathematik und Physik ZAMP, (97), 4, pp [3] Hiemenz, K., The Boundary Layer Analysis of a Uniformly Floing Liquid Circulating in Straight Circular Cylinder (in German), Dinglers Polytechn. J, 36 (9), pp [4] Ishak, A., et al., Mied Convection Boundary Layers in the Stagnation-Point Flo toard a Stretching Vertical Sheet, Meccanica, 4 (6), 5, pp [5] Ishak, A., et al., Unsteady Mied Convection Boundary Layer Flo due to a Stretching Vertical Surface, Arabian Journal for Science and Engineering, 3 (6), B, pp [6] Ishak, A., et al., MHD Mied Convection Boundary Layer Flo toards a Stretching Vertical Surface ith Constant Wall Temperature, International Journal of Heat and Mass Transfer, 53 (), 3-4, pp [7] Pal, D., Heat and Mass Transfer in Stagnation-Point Flo toards a Stretching Surface in the Presence of Buoyant Force and Thermal Radiation, Meccanica, 44 (9),, pp [8] Ali, F. M., et al., MHD Mied Convection Boundary Layer Flo toard a Stagnation Point on a Vertical Surface ith Induced Magnetic Field, ASME Journal of Heat Transfer, 33 (),, pp [9] Chen, C. H., Mied Convection Unsteady Stagnation-Point Flo toards a Stretching Sheet ith Slip Effects, Mathematical Problems in Engineering, 4 (4), ID [] Saleh, S. H. M., et al., Mied Convection Stagnation-Flo toards a Vertical Shrinking Sheet, International Journal of Heat and Mass Transfer, 73 (4), June, pp [] Ellahi, R., et al., A Study on the Mied Convection Boundary Layer Flo and Heat Transfer over a Vertical Slender Cylinder, Thermal Science, 8 (4), 4, pp [] Rashidi, M. M., Mehr, N. F., Series Solutions for the Flo in the Vicinity of the Equator of an Magnetohydrodynamic Boundary-Layer over a Porous Rotating Sphere ith Heat Transfer, Thermal Science, 8 (4), Suppl., pp. S57-S537 [3] Rashidi, S., et al. Study of Stream Wise Transverse Magnetic Fluid Flo ith Heat Transfer around a Porous Obstacle, Journal of Magnetism and Magnetic Materials, 378 (5), Mar., pp [4] Boričić, A. Z., et al., Magnetohydrodynamic Effects on Unsteady Dynamic, Thermal and Diffusion Boundary Layer Flo over a Horizontal Circular Cylinder, Thermal Science, 6 (), Suppl., pp. S3-S3 [5] Lok, Y., et al., Unsteady Mied Convection Flo of a Micropolar Fluid Near the Stagnation- Point on a Vertical Surface, International Journal of Thermal Sciences, 45 (6),, pp

11 THERMAL SCIENCE: Year 7, Vol., No. A, pp [6] Abbas, Z., et al., Mied Convection in the Stagnation-Point Flo of a Maell Fluid toards a Vertical Stretching Surface, Nonlinear Analysis: Real World Applications, (), 4, pp [7] Ahmad, K., Nazar, R., Unsteady Magnetohydrodynamic Mied Convection Stagnation-Point Flo of a Viscoelastic Fluid on a Vertical Surface, Journal of Quality Measurement and Analysis, 6 (),, pp. 5-7 [8] Das, K., Slip Effects on MHD Mied Convection Stagnation-Point Flo of a Micropolar Fluid toards a Shrinking Vertical Sheet, Computers & Mathematics ith Applications, 63 (),, pp [9] Makinde, O. D., et al., Buoyancy Effects on MHD Stagnation-Point Flo and Heat Transfer of a Nanofluid Past a Convectively Heated Stretching/Shrinking Sheet, International Journal of Heat and Mass Transfer, 6 (3), July, pp [] Ellahi, R., The Effects of MHD and Temperature Dependent Viscosity on the Flo of Non-Netonian Nanofluid in a Pipe: Analytical Solutions, Applied Mathematical Modelling, 37 (3), 3, pp [] Ellahi, R., et al., M., Series Solutions of Magnetohydrodynamic Peristaltic Flo of a Jeffrey Fluid in Eccentric Cylinders, Applied Mathematics & Information Sciences, 7 (3), 4, pp [] Singh, V., Agaral, S., MHD Flo and Heat Transfer for Maell Fluid over an Eponentially Stretching Sheet ith Variable Thermal Conductivity in Porous Medium, Thermal Science, 8 (4), Suppl., pp. S599-S65 [3] Abdel-Rahman, G. M., Effects of Variable Viscosity and Thermal Conductivity on Unsteady MHD Flo of Non-Netonian Fluid over a Stretching Porous Sheet, Thermal Science, 7 (3), 4, pp [4] Yacob, N. A. et al., Hydromagnetic Flo and Heat Transfer Adjacent to a Stretching Vertical Sheet in a Micropolar Fluid, Thermal Science, 7 (3),, pp [5] Sheikholeslami, M., et al., Effects of MHD on Cu-Water Nanofluid Flo and Heat Transfer by Means of CVFEM, Journal of Magnetism and Magnetic Materials, 349 (4), Jan., pp. 88- [6] Zeeshan, A., et al., Magnetohydrodynamic Flo of Water/Ethylene Glycol Based Nanofluids ith Natural Convection Through Porous Medium, The European Physical Journal Plus, 9 (4), Dec., pp. 6-7 [7] Lin, Y., et al. MHD Thin Film and Heat Transfer of Poer La Fluids over an Unsteady Stretching Sheet ith Variable Thermal Conductivity, Thermal Science, (5), 6, pp [8] Hayat, T., et al., An Analysis of Peristaltic Transport for Flo of a Jeffrey Fluid, Acta Mechanica, 93 (7), -, pp. - [9] Ramachandran, N., et al., Mied Convection in Stagnation Flos Adjacent to Vertical Surfaces, ASME Journal of Heat Transfer, (988),, pp [3] Cebeci, T., Convective Heat Transfer, Horizon Publishing, Bishop, Cal., USA, [3] Cebeci, T., Bradsha, P., Physical and Computational Aspects of Convective Heat Transfer, Springer, Ne York, USA, 988 [3] Mahapatra, T. R., Gupta, A. S., Heat Transfer in Stagnation-Point Toards a Stretching Sheet, Heat Mass Transfer, 38 (), 6, pp [33] Nazar, R., et al., Unsteady Boundary Layer Flo in the Region of the Stagnation Point on a Stretching Sheet, International Journal of Engineering Science, 4 (4), -, pp [34] Reiner, M., The Deborah Number, Physics Today, 7 (964),, p. 6 [35] Schlichting, H., Boundary Layer Theory, McGra-Hill, Ne York, USA, 968 Paper submitted: November 3, 4 Paper revised: February, 5 Paper accepted: February 4, 5 7 Society of Thermal Engineers of Serbia Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-ND 4. terms and conditions