Buoyancy Effects on Unsteady MHD Flow of a Reactive Third-Grade Fluid with Asymmetric Convective Cooling

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1 Journal of Applied Fluid Mechanics, Vol. 8, No. 4, pp , 215. Available online at ISSN , EISSN DOI: /acadpub.jafm Buoanc Effects on Unstead MHD Flow of a Reactive Third-Grade Fluid with Asmmetric Convective Cooling T. Chinoka 1,2 and O.D. Makinde 3 1 Center for Research in Computational and Applied Mechanics, Universit of Cape Town, Rondebosch 771, South Africa 2 Department of Chemical Engineering, Universit of California Santa Barbara, CA USA 3 Facult of Militar Science, Stellenbosch Universit, Private Bag X2, Saldanha 7395, South Africa Corresponding Author tchinok@vt.edu (Received April, 1 214; accepted August, 9 214) ABSTRACT This article examines the combined effects of buoanc force and asmmetrical convective cooling on unstead MHD channel flow and heat transfer characteristics of an incompressible, reactive, variable viscosit and electricall conducting third grade fluid. The chemical kinetics in the flow sstem is exothermic and the asmmetric convective heat transfers at the channel walls follow the Newton s law of cooling. The coupled nonlinear partial differential equations governing the problem are derived and solved numericall using a semi-implicit finite difference scheme. Graphical results are presented and phsical aspects of the problem are discussed with respect to various parameters embedded in the sstem. Kewords: Buoanc effect; Unstead reactive flow; Asmmetrical convective cooling; Magnetohdrodnamics (MHD); Third grade fluid; Variable viscosit; Finite difference method. 1. INTRODUCTION Most industrial processes utilize or are affected b reactive fluid flow of Newtonian or non- Newtonian fluid. Examples are abound in the industrial scale polmer and petroleum processing applications (Chinoka and Makinde 21). Unlike with the Navier-Stokes equations, no single constitutive equation can model the rheological properties of the collective of non- Newtonian fluids. A number of models have been developed to capture non-newtonian behavior, see for example (Fosdick and Rajagopal 198; Makinde 27). Among these models is the class of third-grade fluids on which this article will be focused. Magnetohdrodnamic (MHD) processes as well as associated heat transfer processes of non-newtonian fluids are similarl of great interest in industrial scale applications, sa to power engineering, mechanical engineering, nuclear engineering, geophsical fluid dnamics, astrophsical fluid dnamics, etc. (Shercliff 1965). The pioneering work of Hartmann (Hartmann 1937) is considered to be the origin of MHD flow in channels. The main characteristics of MHD flow are based on the observation that the magnetic field enhances the fluid viscosit and thereb offers increased resistance to flow. Several investigations, see for example (Branover 1978) and the references therein, have looked into the effects of applied magnetic fields on channel flow under various conditions. Makinde and Chinoka (Makinde and Chinoka 21; Makinde and Chinoka 211; Chinoka and Makinde 211) analzed the transient heat transfer to hdromagnetic channel flow with, among other effects, radiative heat and convective cooling. The effect of magnetic field on boundar laer flow past a vertical plate embedded in a porous medium under convective boundar conditions has been investigated b Makinde and Aziz (Makinde and Aziz 21). hen the flow is subject to gravitational forces and densit variations, sa, due to temperature changes, buoanc effects become important. The buoanc effects on boundar laer flow

2 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , 215. past a vertical plate was first presented b Ostrach (Ostrach 1953). Thereafter, a number of articles have since appeared on this topic with respect to channel flow, see for example the book b Gebhart et al. (Gebhart and Pera 1971). Makinde and Olanrewaju (Makinde and Olanrewaju 21) studied the buoanc effects on thermal boundar laers over vertical plates under convective surface boundar conditions. A lot of attention has been devoted to the stead channel flows of non-newtonian fluids induced b the action of axial pressure gradient in the presence of magnetic fields. Similar attention has also been devoted to thin film flows over moving or stretching surfaces under various conditions, (Prasad, Vajravelu, and Sujatha 213; Prasad, Vajravelu, Datti, and Raju 213; Redd, Raju, and Varma 213; Loganathan and Stepha 213; Shit and Majee 214). Much less effort has been devoted to examining timedependent flow of similar non-newtonian fluid flow problems and as far as we are aware, no stud appears to have considered the combined effects of buoanc forces, imposed magnetic field and convective heat exchange at the channel surface on the reactive flow, which is the focus of this investigation. The objective of the present work is to stud unstead MHD flow of a reactive, variable viscosit, incompressible and third-grade fluid in a vertical channel under asmmetric convective cooling conditions. The MHD effects result from the assumption that the fluid is electricall conducting and that a uniform transverse magnetic field is applied on the flow field. The mathematical formulation of the problem is established in sections two. In section three the semi-implicit finite difference technique is implemented for the solution process. Both numerical and graphical results are presented and discussed quantitativel with respect to various parameters embedded in the sstem in section four. 2. MATHEMATICAL MODEL Consider the transient flow of an incompressible electricall conducting, variable viscosit, reactive third grade fluid placed inside a vertical channel induced b the action of applied axial pressure gradient. It is assumed that the flow is subjected to the influence of an externall applied homogeneous magnetic field as shown in Fig.1. The fluid has small electrical conductivit and the electromagnetic force produced is also ver small. The channel walls are subjected to unequal convective heat exchange with the ambient. The x-axis is taken along g Convective cooling κ T ȳ = h 1(T a T ) ȳ = ȳ = a B (Magnetic field) B (Magnetic field) u = u = x Reactive fluid ȳ Fig. 1. Geometr of the problem. Convective cooling κ T = h ȳ 2(T T a ) the channel and the ȳ-axis is taken normal to it. The densit variation due to buoanc effects is taken into account in the momentum equation (Boussinesq approximation). Following (Fosdick and Rajagopal 198; Makinde 27), and neglecting the reacting viscous fluid consumption, the governing equations for the momentum and heat balance can be written as; ρ u t = P 6β 3 2 u ȳ 2 x + ȳ ( u ȳ [ µ(t ) u ȳ ] + α 1 3 u ȳ 2 t + ) 2 σb 2 u + gβ (T T ), (1) with the following initial and boundar conditions: u(ȳ,) =, T (ȳ,) = T, (2) u(, t) =, k T ȳ (, t) = h 1[T a T (, t)], u(a, t) =, k T ȳ (a, t) = h 2[T (a, t) T a ], (3) (4) where the additional chemical kinetics term in energ balance equation is due to (Makinde 27). Here T is the absolute temperature, σ is the fluid electrical conductivit, B (= µ e H ) the electromagnetic induction, µ e is the magnetic permeabilit, H is the intensit of the magnetic field, ρ is the densit, β is the volumetric expansion coefficient, g is the gravitational acceleration, c p specific heat at constant pressure, t 932

3 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , t =.4 t = 4 t = 1 t = 4 t = t = 1 t = 2 t = 4 t = 45 t = Fig. 2. Transient and stead state velocit and temperature profiles. 18 where b is a viscosit variation parameter and µ is the initial fluid dnamic viscosit at temperature T. e introduce the following dimensionless variables into Eqs. (1) - (5); 16 max Fig. 3. Blow up of fluid temperature for large λ at time t = 4. λ = ȳ a,x = x a,t = tµ uρa, =,µ = µ, ρa2 µ µ α = brt 2 E, = E(T T ) RT 2, a = E(T a T ) RT 2, γ = β 3µ ρ 2 a 4,δ = α 1 ρa 2,Bi 1 = h 1a k, = h 2a k, Pr = µ ( ) m c p k,λ = KT QEAa 2 C e RT E νl T 2Rk, ε = RT E Pρa 2,P = µ 2,G = P x, is the time, h 1 is the heat transfer coefficient at the left hand side plate, h 2 is the heat transfer coefficient at the right hand side plate, T the fluid initial temperature, T a is the ambient temperature, k the thermal conductivit of the material, Q the heat of reaction, A is the rate constant, E is the activation energ, R is the universal gas constant, C is the initial concentration of the reactant species, a is the channel width, l is the Planck s number, K is the Boltzmann s constant, ν is the vibration frequenc, α 1 and β 3 are the material coefficients, P is the modified pressure, m is the numerical exponent such that m { 2,,.5}, where the three values represent numerical exponents for Sensitised, Arrhenius and Bimolecular kinetics respectivel (see (Chinoka and Makinde 21; Makinde 27)). The temperature dependant viscosit ( µ) can be expressed as µ(t ) = µ e b(t T ), (5) a3 ρ 2 Eµ 2 Ha 2 = σb2 a2,gr = gβ RT 2 µ ( νl Ω = KT ) m µ 3 e E RT ρ 2 QAa 4 C, (6) and obtain the following dimensionless governing equations; t δ 3 2 t = G Ha 2 + e α Gr + ( αe α + 6γ 2 2, ) 2, (7) Pr t = { ( ) [ λ (1 + ε) m exp + Ω Ha ε ( ) ( 2 ( ) )]} 2 e α + 2γ, (8) 933

4 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , 215. (,) =, (,) =, (9) (,t) =, (,t) = Bi 1 [ a (,t)], (1) (1,t) =, (1,t) = [(1,t) a ], (11) where λ,pr,bi,ε,δ,γ,g,gr,α,ω, a,ha represent the Frank-Kamenetskii parameter, Prandtl number, Biot number, activation energ parameter, material parameter, non-newtonian parameter, pressure gradient parameter, Buoanc paramter, variable viscosit parameter, viscous heating parameter, the ambient temperature parameter and Hartmann number, respectivel. The other dimensionless quantities of interest are the skin friction (C f ) and the wall heat transfer rate (Nu) given as C f = d d (1,t), Nu = d(1,t). (12) d In the following section, the Eqs. (7) - (12) are solved numericall using a semi-implicit finite difference scheme. 3. NUMERICAL SOLUTION Our numerical algorithm is based on the semi-implicit finite difference scheme given in Chinoka et al. (Chinoka, Renard, Renard, and Khismatullin 25; Chinoka 28; Chinoka 211; Ireka and Chinoka 213; Chinoka, Goqo, and Olajuwon 213) for the isothermal viscoelastic case. As in Chinoka (Chinoka 28), we extend the algorithm to the temperature equation and take the implicit terms at the intermediate time level (N + ξ) where ξ 1. The algorithm emploed in Chinoka (Chinoka 28) uses ξ = 1/2, we will however follow the formulation in Chinoka et al. (Chinoka, Renard, Renard, and Khismatullin 25) and thus take ξ = 1 in this article so that we can use larger time steps. 4. RESULTS AND DISCUSSION Unless otherwise stated, we emplo the parameter values: G = 1, Gr =.1, Pr = 1, a =.1, δ =.1,λ =.1, Bi 1 = 1, = 1 m =.5, ε =.1, α =.1,Ω =.1,γ =.1,Ha = 1, =.2, t =.1 and t = 2. These will be the default values in this work and hence in an graph where an of these parameters is not explicitl mentioned, it will be understood that such parameters take on the default values. 4.1 Transient and Stead Flow Profiles e displa the transient solutions in Fig. 2. The figures show a transient increase in both fluid velocit and temperature until a stead state is reached Blow up of solutions e need to point out earl on that depending on certain parameter values in the problem, the stead temperature and velocit profiles, such as those shown in Fig. 2, ma not be attainable. In particular, the reaction parameter λ will need to be carefull controlled as large values can easil lead to blow up of solutions as illustrated in Fig. 3. As shown in Fig. 3, larger values of λ would lead to finite time temperature blow up since the terms associated with λ are strong heat sources Parameter dependance of solutions The response of the velocit and temperature to varing values of the Hartmann number (Ha) is illustrated in Fig. 4. An increase in the parameter Ha leads to corresponding increases in damping magnetic properties of the fluid. These forces result in increased resistance to flow and thus explain the reduction in fluid velocit with increasing Hartmann number, see Fig. 4. The reduced velocit in turn decreases the viscous heating source terms in the temperature equation and hence correspondingl decreases the fluid temperature as shown in Fig. 4. The response of the velocit and temperature to varing values of the non-newtonian parameter (γ) is illustrated in Fig. 5. An increase in the parameter γ leads to corresponding increases in those non-newtonian properties of the fluid, sa (visco)elasticit, that would result in increased resistance to flow and thus explain the slight reduction in fluid velocit with increasing non- Newtonian character as measured b the parameter γ, see Fig. 5. The reduced velocit in turn decreases the viscous heating source terms in the temperature equation and hence correspondingl decreases the fluid temperature as shown in Fig. 5. Since however the parameter γ onl enters the temperature equation implicitl through the velocit field, the effects of γ on the fluid temperature are not as noticeable as on the fluid velocit. The influence of the variable viscosit parameter on the velocit and temperature profiles is shown in Fig. 6. Increasing the parameter α reduces the fluid 934

5 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , Ha = Ha = 1 Ha = 5 Ha = Ha = Ha = 1 Ha = 5 Ha = Fig. 4. Effects of Hartmann number (Ha) on velocit and temperature γ =.1 γ =.2 γ =.25 γ = γ =.1 γ =.2 γ =.25 γ = Fig. 5. Effects of non-newtonian parameter (γ) on velocit and temperature. viscosit and hence correspondingl diminishes the fluid s resistance to flow. This necessaril leads to increased fluid velocit as illustrated in Fig. 6. The increased velocit in turn increases the viscous heating source terms in the temperature equation and hence correspondingl increases the fluid temperature as shown in Fig. 6. As with the non-newtonian parameter, the viscous effects are more pronounced in the velocit equation than in the temperature equation and this explains wh the effects of α on the fluid temperature are not as noticeable as in the fluid velocit. The effects of the chemical kinetics exponent m on the velocit and temperature profiles is shown in Fig. 7. Figure 7 shows that the internal heat generated in the fluid during a bimolecular tpe of exothermic chemical reaction (m =.5) is higher than that generated either under the Arrhenius (m = ) or Sensitised (m = 2) reaction tpes. This is so since an increase in the parameter m leads to corresponding increases in the strengths of the chemical reaction source terms in the temperature equation. This leads to increased fluid temperatures as shown in Fig. 7. The increased temperature in turn leads to a reduction in fluid viscosit and hence indirectl to increased fluid velocit as illustrated in Fig. 7. Since however the parameter m onl enters the velocit equation implicitl through the temperature/viscosit coupling, the effects of m on the fluid velocit at best look marginal and are not as pronounced as on the fluid temperature. The effects of the activation energ parameter ε on the velocit and temperature profiles is shown in Fig. 8. The parameter ε plas a more-or-less similar role (both Mathematicall and phsicall) to the parameter m described in the earlier Fig. 7 and hence its effects are similarl explained. The graphs of Fig. 8 correspond to the case of Bimolecular reactions (m =.5). It should however be noted that in the cases of Sensitized and Arrhenius reactions (in which m ) both temperature and velocit are expected to decrease with increasing ε, see Fig. 9. This is due to the fact that the function: ( ) (1 + ε) m exp, m,] (13) 1 + ε decreases as ε increases. Since this function represents source terms in the temperature equation, the fluid temperature expectedl decreases with increasing ε, so that the maximum 935

6 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , α = α =.1 α =.5 α = α = α =.1 α =.5 α = Fig. 6. Effects of variable viscosit parameter (α) on velocit and temperature..12 m = 2 m = m = m = 2 m = m = Fig. 7. Effects of parameter (m) on velocit and temperature ε = ε =.1 ε =.15 ε = ε = ε =.1 ε =.15 ε = Fig. 8. Effects of activation energ parameter (ε) on velocit and temperature. 936

7 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , ε = ε =.1 ε =.15 ε = ε = ε =.1 ε =.15 ε = (a) m =.15 (b) m = 2 Fig. 9. Effects of activation energ parameter (ε) on temperature =.1 = 1 = 3 = =.1 Bi = 1 2 = 3 Bi = Fig. 1. Effects of the Biot number ( ) on velocit and temperature Pr = 7.1 Pr = 1 Pr = 15 Pr = Pr = 7.1 Pr = 1 Pr = 15 Pr = Fig. 11. Effects of the Prandtl number (Pr) on velocit and temperature. 937

8 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , λ =.1 λ =.5 λ =.8 λ = λ =.1 λ =.5 λ =.8 λ = Fig. 12. Effects of the reaction parameter (λ) on velocit and temperature Ω =.1 Ω =.5 Ω = 1 Ω = Ω =.1 Ω =.5 Ω = 1 Ω = Fig. 13. Effects of the viscous heating parameter (Ω) on velocit and temperature Gr =.1 Gr =.5 Gr = 1 Gr = Gr =.1 Gr =.5 Gr = 1 Gr = Fig. 14. Effects of the buoanc parameter (Gr) on velocit and temperature α = α =.1 α =.5 α =.6.15 α = α =.1 α =.5 α = C f Nu λ λ Fig. 15. Variation of C f and Nu with λ and α. 938

9 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , C f γ = γ =.1 γ =.2 γ =.3 C f Ha = Ha = 1 Ha = 5 Ha = λ λ Fig. 16. Variation of wall shear stress with λ, γ and Ha. temperature is recorded for ε = as shown in Fig. 9. The effects of the Biot number on the velocit and temperature profiles is illustrated in Fig. 1. As seen from the temperature boundar condition, Eq. (11), higher Biot numbers mean correspondingl higher degrees of convective cooling at the channel walls and hence lead to lower temperatures at the channel walls and hence also in the bulk fluid. The overall temperature profiles thus decrease with increasing Biot number as the bulk fluid continuall adjusts to the lower wall temperatures. The reduced temperatures correspondingl decrease the fluid viscosit and hence also marginall decreases the fluid velocit through the viscosit coupling. As noted earlier, such a coupling depends on other parameters as well (sa α) does not necessaril result in drastic changes in velocit profiles even though the temperature profiles show well pronounced changes, see Fig. 1. The effects of the Prandtl number Pr on the velocit and temperature profiles is illustrated in Fig. 11. Larger values of the Prandtl number correspondingl decrease the strength of the source terms in the temperature equation and hence in turn reduces the overall fluid temperature as clearl illustrated in Fig. 11. As pointed out earlier, the reduced temperature results in decreased fluid viscosit and hence reduced fluid velocit, see Fig. 11. The effects of the reaction parameter λ on the velocit and temperature profiles is illustrated in Fig. 12 respectivel. The reaction parameter λ plas a roughl opposite role to the Prandtl number just described. Increased values of λ lead to significant increases in the reaction and viscous heating source terms and hence significantl increase the fluid temperature as shown in Fig. 12 and also in the blow up figure Fig. 3. The significant temperature rise in response to the increased λ mean that the viscosit coupling to the velocit is no longer weak and hence significant reductions in the viscosit lead to appreciable increases in the fluid velocit. The effects of the viscous heating parameter Ω on the velocit and temperature profiles is illustrated in Fig. 13. The effects of Ω mirror those of λ, albeit on a smaller scale since Ω is not connected to the exponentiall increasing reaction source terms but onl to the viscous heating terms. The effects of the buoanc parameter Gr on the velocit and temperature profiles is illustrated in Fig. 14 respectivel. An increase in the buoanc parameter Gr leads to a corresponding increase in the fluid velocit and shown in Fig. 14. This results from the fact that the Buoanc terms act as source terms in the momentum relation. The increased velocit correspondingl increases the fluid temperature, albeit on a smaller scale, due to reasons alread given earlier. The wall shear stress (C f ) and wall heat transfer rate (Nu) dependance on the reaction parameter λ is illustrated in Fig. 15 for varing values of the viscosit variation parameter α. Figure 16 shows the wall shear stress dependance on λ for varing values of the non-newtonian parameter γ and and also for varing values of the Hartmann number Ha. In general, parameters that decrease (increase) the fluid velocit correspondingl decrease (increase) the wall shear stress respectivel. The wall heat transfer dependance on λ for varing values of γ and the wall heat transfer dependance on λ for varing values of Ha are both qualitativel similar to the wall heat transfer rate dependance on λ for varing values of α as shown in Fig. 15. As with the wall shear stress, parameters that decrease (increase) the fluid temperature correspondingl decrease (increase) the wall heat transfer. In fact, since both the parameters α and γ onl marginall increase the temperature, their effects on the wall heat transfer are also similarl marginal. All the results for the wall shear stress and the wall heat transfer were obtained at time t =

10 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , CONCLUSION e computationall investigate the Buoanc effects on unstead MHD flow of a reactive third-grade fluid with asmmetric convective cooling. e observe that there is a transient increase in both fluid velocit and temperature with an increase in the Buoanc, reaction strength, viscous heating and fluid viscosit parameter (which decreases the viscosit). A transient decrease in both fluid velocit and temperature is observed with an increase in the non-newtonian character and magnetic field strength. The possible finite time blow-up of solutions means that the reaction strength needs to be carefull controlled. e also notice that due to the nature of the coupling of the source terms, the fluid velocit and temperature either both increase or both decrease together. Parameters that increase an such source terms would increase these quantities and similarl those that decrease the source terms correspondingl decrease the flow quantities. REFERENCES Branover, H. (1978). MHD flow in Ducts. Keter Publ. House Jerusalem Ltd.: ile and Israel Univ. Press. Chinoka, T. (28). Computational dnamics of a thermall decomposable viscoelastic lubricant under shear. Transactions of ASME Journal of Fluids Engineering 13(12), 12121(1 7). Chinoka, T. (211). Suction-injection control of shear banding in non-isothermal and exothermic channel flow of johnsonsegalman liquids. Transactions of ASME Journal of Fluids Engineering 133(7), 7125(1 12). Chinoka, T., S. Goqo, and B. Olajuwon (213). Computational analsis of gravit driven flow of a variable viscosit viscoelastic fluid down an inclined plane. Computers & Fluids 84(15), Chinoka, T. and O. Makinde (21). Computational dnamics of unstead flow of a variable viscosit reactive fluid in a porous pipe. Mechanics Research Communications 37, Chinoka, T. and O. Makinde (211). Analsis of transient generalized couette flow of a reactive variable viscosit thirdgrade liquid with asmmetric convective cooling. Mathematical and Computer Modelling 54(1-2), Chinoka, T., Y. Renard, M. Renard, and D. Khismatullin (25). Twodimensional stud of drop deformation under simple shear for oldrod-b liquids. Journal of Non-Newtonian Fluid Mechanics 31, Fosdick, R. and K. Rajagopal (198). Thermodnamics and stabilit of fluids of third grade. Proc. Ro. Soc. London A339, 351. Gebhart, B. and L. Pera (1971). The nature of vertical natural convection flows resulting from the combined buoanc effects of thermal and mass diffusion. Int. J. Heat Mass Transfer 14, Hartmann, J. (1937). Theor of laminar flow of an electricall conducting liquid in a homogeneous magnetic field, hgdnamics i. Math. Fs. Med. 15, Ireka, I. and T. Chinoka (213). Nonisothermal flow of a johnsonsegalman liquid in a lubricated pipe with wall slip. Journal of Non-Newtonian Fluid Mechanics 192, Loganathan, P. and N. Stepha (213). Chemical reaction and mass transfer effects on flow of micropolar fluid past a continuousl moving porous plate with variable viscosit. Journal of Applied Fluid Mechanics 6(4), Makinde, O. (27). Thermal stabilit of a reactive third grade fluid in a clindrical pipe: An exploitation of hermite-padé approximation technique. Applied Mathematics and Computation 189, Makinde, O. and T. Chinoka (21). Numerical investigation of transient heat transfer to hdromagnetic channel flow with radiative heat and convective cooling. Communications in Nonlinear Science and Numerical Simulations 15, Makinde, O. and T. Chinoka (211). Numerical stud of unstead hdromagnetic generalized couette flow of a reactive third-grade fluid with asmmetric convective cooling. Computers & Mathematics with Applications 61(4), Makinde, O. and P. Olanrewaju (21). Buoanc effects on thermal boundar laer over a vertical plate with a convective surface boundar condition. Transaction of ASME Journal of Fluid Engineering 132, 4452(1 4). 94

11 T. Chinoka and O.D. Makinde / JAFM, Vol. 8, No. 4, pp , 215. Makinde, O. D. and A. Aziz (21). Mhd mixed convection from a vertical plate embedded in a porous medium with a convective boundar condition. International Journal of Thermal Sciences 49, Ostrach, S. (1953). An analsis of laminar free convection flow and heat transfer along a flat plate parallel to the direction of the generating bod force. In NACA Report Prasad, K., K. Vajravelu, P. Datti, and B. Raju (213). Mhd flow and heat transfer in a power-law liquid film at a porous surface in the presence of thermal radiation. Journal of Applied Fluid Mechanics 6(3), Prasad, K., K. Vajravelu, and A. Sujatha (213). Influence of internal heat generation/absorption, thermal radiation, magnetic field, variable fluid propert and viscous dissipation on heat transfer characteristics of a maxwell fluid over a stretching sheet. Journal of Applied Fluid Mechanics 6(2), Redd, T., M. Raju, and S. Varma (213). Unstead mhd radiative and chemicall reactive free convection flow near a moving vertical plate in porous medium. Journal of Applied Fluid Mechanics 6(3), Shercliff, J. (1965). A Textbook of Magnetohdrodnamics. London, UK.: Pergoman Press. Shit, G. and S. Majee (214). Hdromagnetic flow over an inclined non-linear stretching sheet with variable viscosit in the presence of thermal radiation and chemical reaction. Journal of Applied Fluid Mechanics 7(2),