Chemical Reaction and Thermal Diffusion Effects on Mass Transfer Flow through an Inclined Plate

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1 Open Jounal of Fluid Dynamics, 6, 6, 6-74 Published Online Mach 6 in SciRes. Chemical Reaction and Themal Diffusion Effects on Mass Tansfe Flow though an Inclined Plate Fajana Akte, Md. Manjiul Islam, Aiful Islam, Md. Shakhaoath Khan, Md. Saddam Hossain Mathematics Discipline, Khulna Univesity, Khulna, Bangladesh Discipline of Chemical Engineeing, Univesity of Newcastle, Callaghan, Austalia Received 6 Febuay 6; accepted 6 Mach 6; published 9 Mach 6 Copyight 6 by authos and Scientific Reseach Publishing Inc. This wok is licensed unde the Ceative Commons Attibution Intenational License (CC BY). Abstact A numeical investigation of bounday laye mass tansfe flow though an inclined plate with the effect of chemical eaction and themal diffusion is pesented in this study. The govening patial diffeential equations (PDE) ae tansfomed to a system of dimensionless non-simila coupled PDEs. The tansfomed, non-simila consevations equations (momentum balance equation, enegy balance equation and concentation balance equation) ae then solved using a numeical appoach known as explicit finite diffeence method (EFDM). Basically EFDM intoduced fo the unsteadiness in the momentum, tempeatue, and concentation fluid fields is based on the time dependent fluid velocity, tempeatue and concentation of the bounday suface. Duing the couse of discussion, it is found that the vaious paametes elated to the poblem influence the calculated esultant expessions. The computed numeical solution esults fo the velocity, tempeatue, and concentation distibution with the effect of vaious impotant dimensionless paametes (Gashof numbe, Modified Gashof numbe, Pandtl numbe, Schmidt numbe, Soet numbe, Dufou numbe, chemical eaction paamete and inclination paamete) enteing into the poblems ae citically analyzed and discussed gaphically. It can be seen that two physical phenomena chemical eaction and themal diffusion can geatly effect on the bounday laye fluid flows though an inclined plate. Keywods Chemical Reaction, Mass Tansfe, Inclined Plate, Soet Effects, Dufou Effects How to cite this pape: Akte, F., Islam, Md.M., Islam, A., Khan, Md.S. and Hossain, Md.S. (6) Chemical Reaction and Themal Diffusion Effects on Mass Tansfe Flow though an Inclined Plate. Open Jounal of Fluid Dynamics, 6,

2 . Intoduction The effect of themal diffusion and chemical eaction on heat and mass tansfe fluid flow has a geat impotance to the engineeing community and to investigatos dealing with the poblems in many industial pocesses and technological fields which expeience not only tempeatue diffeence but also concentation diffeence. The concentation diffeence may sometimes poduce quantitative and qualitative changes to the ate of heat tansfe. Some industial applications ae found in food dying, food pocessing o wood and pape poduction. Besides, the chaacteistics of the flow of blood though ateies and veins ae of consideable medical inteest. In view of the above inteests, a seies of investigations have been made by diffeent scholas. Pea and Gebhat [] wee the fist authos to investigate and combine buoyancy effects of themal and mass diffusion on natual conviction flow. Soundalgeka and Ganesan [] pesented a finite diffeence analysis of unsteady natual convective flow past a semi-infinite plate with mass tansfe. Hydo-magnetic heat and mass tansfe in flow of a viscous incompessible fluid past an infinite vetical poous plate was analyzed by Singh et al. [3]. Using a finite diffeence technique, an unsteady fee convective heat and mass tansfe flow fom an infinite vetical plate with constant suction have been numeically investigated by Ambethka [4]. Recently Sivaiah et al. [5] studied heat and mass tansfe effects on MHD fee convective flow past a vetical poous plate. The poblem of mass tansfe flow though an inclined plate has geneated much inteest fom astophysical, enewable enegy system and also hypesonic aeodynamics eseaches fo a numbe of decades. Umemua and Law [6] developed a genealized fomulation fo the natual convection bounday laye flow ove a flat plate with abitay inclination. They found that the flow chaacteistics depend not only on the extent of inclination but also on the distance fom the leading edge. Chamkha and Khaled [7] investigated the poblem of coupled heat and mass tansfe by hydomagnetic fee convection fom an inclined plate in the pesence of intenal heat geneation o absoption, and similaity solutions wee pesented. Reddy and Reddy [8] pefomed an analysis to study the natual convection flow ove a pemeable inclined suface with vaiable tempeatue, momentum and concentation. Singh [9] studied heat and mass tansfe in MHD bounday laye flow past an inclined plate with viscous dissipation in poous medium. Micopola fluid behavio on MHD fee convection and mass tansfe with constant heat and mass fluxes is studied numeically by Ali et al. []. Recently Islam et al. [] have studied mass tansfe flow though an inclined plate with poous medium. In many tanspot pocesses existing in natue and in industial applications in which heat and mass tansfe is a consequence of buoyancy effects caused by diffusing of heat and chemical species. The study of such pocess is useful fo impoving a numbe of chemical technologies, such as polyme poduction, enhanced oil ecovey, undegound enegy tanspot, manufactuing of ceamic and food pocessing. In many hydometallugical and chemical technology industies, heat and mass tansfe is of consideable impotance due to chemical eaction effects. Basically chemical eaction takes place between an extenal mass and the fluid in many chemical engineeing pocesses. It can be designated as eithe a mixed o stable pocess, which be contingent on whethe it occus at a bounday o as a single-phase volume eaction. Chaudhay and Jha [] studied the effect of chemical eaction on MHD micopola fluid flow past a vetical plate in slip flow egime. Effects of mass tansfe with a chemical eaction on unsteady flow past an acceleated isothemal vetical plate ae investigated by Muthuchumaaswamy et al. [3]. Rajesh et al. [4] have discussed the effects of chemical eaction and adiation effects on MHD flow past an infinite vetical plate with vaiable tempeatue. The impotance of themal-diffusion and diffusion-themo effects fo vaious fluid flows has been studied by Ecket and Dake [5]. Olajuwon [6] examined convection heat and mass tansfe in a hydomagnetic flow of a second gade fluid past a semi-infinite steching sheet in the pesence of themal diffusion and themal adiation. Kuma et al. [7] have investigated themal diffusion and adiation effects on unsteady MHD flow though poous medium with vaiable tempeatue and mass diffusion in the pesence of heat souce o sink. Thee have been few studies on heat tansfe bounday laye flow in the liteatue howeve mass tansfe flow though an inclined plate with the effect of chemical eaction and themal diffusion effects still need moe attaction to the eseaches [8] [9]. The objective of this eseach is to investigate the fluid behavio with the effect of chemical eaction and themal diffusion on mass tansfe flow in an inclined plate. In this eseach the govening equations of the poblem contain a system of patial diffeential equations which ae tansfomed by usual tansfomation into a non-dimensional system of patial coupled non-linea diffeential equations. The obtained non-simila patial diffeential equations ae solved numeically by EFDM. The esults of this eseach ae discussed fo the diffe- 63

3 ent values of the well-known dimensionless paametes and ae shown gaphically.. Fomulation and Solution.. Mathematical Model of Flow By intoducing Catesian co-odinate system, the X-axis is chosen along the plate in the diection of the flow and the Y-axis is nomal to it. The physical configuation has been shown in Figue. Initially it has been consideed that the plate as well as the fluid is at the same tempeatue T( T ) and concentation level C( C ) eveywhee in the fluid is same. Also it is consideed that the fluid and the plate is at est afte that the plate is to be moving with a constant velocity U in its own plane and instantaneously at time t > the species concentation and the tempeatue of the plate ae aised to Cw ( > C ) and Tw ( > T ), which ae thee afte maintained constant, whee Cw, T w ae species concentation and tempeatue at the wall of the plate and C, T ae the concentation of the species fa away fom the plate. Within the famewok of the above stated assumptions with efeence to the genealized equations descibed befoe the equation elevant to the tansient two dimensional poblems ae govened by the following system of coupled non-linea diffeential equations using the Boussinesq and bounday laye appoximations. Continuity Equation u v + =. () x y Momentum Equation Enegy Equation Concentation Equation u u u u v gβt ( T T ) cosα gβc ( C C ) cosα υ u + + = + +. () t x y y T T T K T c + u + v = + D m t x y ρcp y y C C C C T + u + v = Dm + D T K C C C t x y y y. (3) ( ). (4) X U u v O Figue. The Physical model and coodinate system. Y 64

4 With the coesponding initial and bounday conditions ae At t = u =, v=, T T, C C eveywhee. (5) u =, v =, T T, C C at x = t > u = U, v =, T Tw, C Cwat y =. (6) u =, v =, T T, C C at y whee xy, ae Catesian co-odinate, uv, ae the components of flow velocity along xy, diections espectively. Hee g is the local acceleation due to gavity; υ is the kinetic viscosity; ρ is the density of the fluid, D m is the coefficient of mass diffusivity, D T is the coefficient of Themal diffusivity and K C be the chemical eaction paamete... Mathematical Fomulation Since the solution of the govening equations unde the initial and bounday conditions will be based on a finite diffeence method it is equied to make the said equations dimensionless. Fo this pupose it has been now intoduced the following dimensionless vaiables; xu yu u v X =, Y =, U =, V =, υ υ U U tu T T C C τ =, T = and C = υ T T C C w whee τ epesents the dimensionless time, X & Y be the dimensionless Catesian coodinates, whee U and V be the dimensionless velocity components, T and C is the dimensionless tempeatue and concentation espectively. Using the above elation, we obtain the following non-dimensional coupled patial diffeential equation, U V + =. (7) X Y whee, ( T T ) w 3 w U U U U + U + V = + GTcos cos α + GmC α. (8) τ X Y Y T T T T C + U + V = + D f τ X Y P Y Y C C C C T o τ X Y SC Y Y. (9) + U + V = + S KC. () G = υgβ (Gashof Numbe), T U ( Cw C ) Gm = υgβc (Modified Gashof Numbe), 3 U υρc p P = (Pandtl Numbe), K υ SC = (Schmidt Numbe), Dm DT ( Tw T ) So = (Soet Numbe), υ C C ( ) w 65

5 D K f ( w ) ( T T ) Dm C C = υ C w (Dufou Numbe), K υ = (Chemical Reaction Paamete). U Also the associate initial and bounday condition become At τ = U =, V =, T =, C = eveywhee. () τ > U =, V =, T =, C = at X = U =, V =, T =, C = at Y =. () U =, V =, T =, C = at Y.3. Numeical Solution The explicit finite diffeence method (EFDM) has been used to solve the govened second ode nonlinea coupled dimensionless patial diffeential equations with the coesponding initial and bounday conditions. To obtain a system of finite diffeence equations, the flow egion is divided into a gid o meshes of lines paallel to X and Y axes whee X-axis is taken along the plate and Y-axis is nomal to the plate. Hee it is consideed that X max (=4) i.e. X vaies fom to 4 and egading Y max (=5) as coesponding to Y i.e. Y vaies fom to 5. It is also consideed that m = 6 and n = 6 gid spacing in X and Y diections espectively. We have the constant mesh size along X diection, X = 6.67( X 4) and the constant mesh size along Y diection, Y =.4( Y 5) with the smalle time-step τ =.5 Let U, C and T denote the values of UC, and T at the end of a time-step espectively. Using the finite diffeence appoximations we obtain the following set of finite diffeence equations, Ui, j Ui, j Vi, j Vi, j + =. (3) X Y U, U, U, U,,,, i j i j i j i U j i j + U U i j,,,,, cos, cos i j + U + U i j i j + Ui j + Vi j = GT i j α + GmCi j α +. (4) τ X Y Y ( ) ( ) ( ) T T T T T T T T + T C C + C X Y P Y Y i, j i, j i, j i, j i, j+ i, j i, j+ i, j i, j i, j+ i, j i, j + Ui, j + Vi, j = + D f τ C C C C C C C C + C T T + T X Y S Y Y i, j i, j i, j i, j i, j+ i, j i, j+ i, j i, j i, j+ i, j i, j + Ui, j + Vi, j = + S o τ C And the initial and bounday conditions with finite diffeence scheme ae; U =, V =, T =, C =, U =, V =, T =, C =, U =, V =, T =, C =, i, j i, j i, j i, j n n n n, j, j, j, j n n n i, i, i, i, U =, V =, T =, C =, n n n n il, il, il, il, ( ) ( ). (5). (6) whee, L. Hee the subscipts iand j designate the gid points xand y coodinates espectively and the supescipt n epesents a value of time, τ = n τ whee, n =,,, 3,. Fom the initial condition, the values of U is known at τ =. Then at the end of the anytime-step τ, the new tempeatue T and concentation C, the new velocity U at all inteio nodal points may be obtained by successive application of tempeatue, concentation and momentum equation espectively. This pocess is epeated in time and povided the time-step is sufficiently small, hence UVT,, and C should eventually convegent values which appoximate the steady-state (7) 66

6 solution of the poblem. 3. Results and Discussion To discuss the effects of the associated non-dimensional paametes on the flow vaiables, the numeical solutions of the poblem ae obtained by the EFDM numeical methodology with the help of a compute pogamming language Compaq Visual Fotan. In ode to analyze the physical solution of the model, we have computed the steady numeical values of the non-dimensional Velocity U, Tempeatue T, and Concentation C within the bounday laye fo diffeent values of Inclination ( α ), Gash of numbe ( G ) ( P ), Schmidt numbe ( S c ), Soet numbe ( S ), and Chemical eaction paamete ( K ), Pandtl numbe o. To obtain the steadystate solutions of the computation, the calculations have been caied out up to non-dimensional time τ = to 6. Hence the velocity, tempeatue and concentation pofile ae dawn fo dimensionless time τ =, 3 and 6. In the Figues - x-axis taken as dimensionless co-odinate vaiable (Y) which is vaies to 5 and y-axis taken as fluid velocity, tempeatue and concentation espectively. The effect of the inclination on the velocity field is pesented in Figue. It is obseved that the dimensionless velocity distibution of fluid deceases with the ise of inclination α (=45, 5 and 6 ) fo diffeent time steps. Theefoe angle of inclination has a geat impact on the velocity of fluids. Figue 3 epesents the evolution of dimensionless velocity with Y fo diffeent values of dimensionless times (,3 and 6) τ = and Gashof numbe as atio of the buoyancy to viscous foce acting on a fluid was consideed as.,. and.5. An incease in Gashof numbe clealy enhances pofiles of velocity, theefoe induces a obust speeding up in the flow. A distinct velocity shoot aises fo all pofiles nea the bounday suface (Y = ) and this is emphasized with inceasing Gashof numbe. With inceasing Gashof numbe the themal buoyancy foce is augmented which aids in momentum change in the bounday laye. Also the bounday laye thickness of velocity is theefoe inceased as well. With supeio intevene of time, the velocity was found to be consideably enhanced. The same effect on the velocity cuve is found in Figue 4 that is the velocity deceases with incease of Pandtl numbe P (=.7,. and 7.). This is because Pandtl numbe implies the elative influence of momentum diffusion to themal diffusion in the peiphey laye egime. When P =, the momentum diffusion ate supasses themal diffusion ate. As a esult the velocity in the fluid egime will be deceased with a ise in P. The dimensionless velocity distibution of fluid was found to be stongly educed fo the deceasing effect of 6 5 Time 3 6 Line Patten Velocity 4 3 α = 45, 5, 6 Figue. Velocity pofiles fo G m =., G =.5, S o =., P =.7, S c =.96, D f =., K =.. 67

7 6 5 Time 3 6 Line Patten Velocity 4 3 G =.,.,.5 Figue 3. Velocity pofiles fo G m =., α = 45, S o =., P =.7, S c =.96, D f =., K = Time Line Patten 3 6 Velocity 4 3 P =.7,., 7. Figue 4. Velocity pofiles fo G m =., α = 45, G =.5, S o =., S c =.96, D f =., K =.. 68

8 Velocity Time Line Patten 3 6 K =.6,.,. Figue 5. Velocity pofiles fo G m =., α = 45, G =.5, S o =., P =.7, S c =.96, D f =... Time Line Patten Tempeatue P =.7,., Figue 6. Tempeatue pofiles fo G m =., α = 45, G =.5, S o =., S c =.96, D f =., K =.. 69

9 Time 3 6 Line Patten Tempeatue K =.6,.,..3.. Figue 7. Tempeatue pofiles fo G m =., α = 45, G =.5, S o =., P =.7, S c =.96, D f = Time 3 6 Line Patten Concentation Sc =.96,., Figue 8. Concentation pofiles fo α = 45, G =.5, S c =.96, S o =., P =.7, K =., D f =.. 7

10 Time Line Patten Concentation So =.5,.8,.. 5 Figue 9. Concentation pofiles fo G m =., α = 45, G =.5, S c =.96, P =.7, K =., D f =.. Time Line Patten Concentation K =.6,., Figue. Concentation pofiles fo G m =., α = 45, G =.5, S o =., P =.7, S c =.96, D f =.. 7

11 chemical eaction paamete K (=.6,. and.) found on Figue 5. Figue 6 shows that the steady state tempeatue of fluid deceases with the incease of Pandtl numbe P (=.7,. and 7.). This is because the themal bounday laye thickness deceases due to incease in this paamete. With inceasing peiod, tempeatues ae yet again detected to be stongly enhanced though the bounday suface. In Figue 7 it was obseved that the fluid tempeatue emain unchanged with the inceasing value of chemical eaction paamete K (=.6,. and.). A deceasing effect of dimensionless mass distibution cuves ae obseved fo inceasing Schmidt numbe S C in Figue 8 that is fluid concentation deceases with incease of Schmidt numbe S C (=.96,. and.6). Theefoe the atio of diffusivity of momentum i.e., viscosity and diffusivity of mass has a emakably effects on fluid concentation. And the Schmidt numbe can be used to descibe fluid flows in which thee ae instantaneous pocesses of momentum and mass diffusion convection. The unchanged effect of Soet numbe So on concentation pofiles ae obseved in Figue 9 whee the mass distibution of fluid is emain same with inceasing value of Soet numbe S o. The atio of themos-diffusion and diffusion coefficient was assumed to be.5,.8 and.. In Figue it can be seen that the fluid concentation emains unchanged with the inceasing value of chemical eaction paamete K (=.6,. and.). Fom the above figues we also obseved that fluid behavio is vaies with dimensionless time τ. With the ise of dimensionless time τ the fluid velocity, tempeatue and concentation ae gadually ises. 4. Conclusions Mixed (combined) convection bounday laye flows ae of geat attention because of thei innumeable industial, engineeing and scientific applications in heat and mass tansfe. In this pape, a bounday laye analysis fo mixed convection heat and mass tansfe fluid flow though an inclined plate in the pesence of chemical eaction and themal diffusion is consideed. The govening of bounday laye flow such as momentum, enegy and mass balance is then numeically analyzed by explicit finite diffeence method (EFDM). The esults ae pesented gaphically with the effect of vaious dimensionless paametes. The significant findings obtained fom the gaphical obsevation ae listed below: The velocity distibution deceases with the incease of inclined angle ( ) The velocity distibution inceases with the incease of Gashof numbe ( ) The velocity distibution deceases with the incease of Pandtl numbe ( ) The Concentation distibution deceases with the incease of Schmidt numbe ( ) c The Concentation distibutions emain unchanged with the incease of Soet numbe ( ) o α. G. P, also the tempeatue distibution deceases with the incease of Pandtl numbe ( P ). S. S. The velocity distibution and concentation distibution both ae deceased with the incease of chemical eaction paamete ( K ) while the tempeatue distibution emains unchanged with the vaiety of Chemi- K. cal eaction paamete ( ) Acknowledgements Thee of authos (FA, MMI and MSH) of this eseach pape ae thankful to thei M.Sc. (Applied Mathematics) thesis supeviso AI fo his endless suppot and valuable suggestion duing the pepaation of the thesis as well as manuscipt. All authos ae also gateful to Mathematics Discipline, Khulna Univesity, Bangladesh fo outstanding facilities of Math laboatoy and limitless coopeation of faculty membes. Refeences [] Pea, L. and Gebhat, B. (97) The Natue of Vetical Natual Convection Flows Resulting fom the Combined Buoyancy Effects of Themal and Mass Diffusion. Intenational Jounal of Heat and Mass Tansfe, 5, [] Soundalgeka, V.M. and Ganesan, P. (98) Finite-Diffeence Analysis of Tansient Fee Convection with Mass Tansfe on an Isothemal Vetical Flat Plate. Intenational Jounal of Engineeing Science, 9, [3] Singh, A.K., Singh, A.K. and Singh, N.P. (3) Heat and Mass Tansfe in MHD Flow of a Viscous Fluid Past a Vetical Plate unde Oscillatoy Suction Velocity. Indian Jounal of Pue and Applied Mathematics, 34, [4] Ambethka, V. (8) Numeical Solutions of Heat and Mass Tansfe Effects of an Unsteady MHD Fee Convective 7

12 Flow Past an Infinite Vetical Plate with Constant Suction. Jounal of Naval Achitectue and Maine Engineeing, 5, [5] Sivaiah, M., Nagaajan, A.S. and Reddy, P.S. (9) Heat and Mass Tansfe Effects on MHD Fee Convective Flow past a Vetical Poous Plate. The ICFAI Univesity Jounal of Computational Mathematics,, 4-. [6] Umemua, A. and Law, C. K. (99) Natual Convection Bounday Laye Flow ove a Heatedplate with Abitay Inclination. Jounal of Fluid Mechanics, 9, [7] Chamka, A. and Khaled, A.R.A. () Simultaneously Heat and Mass Tansfe in Fee Convection. Industial Engineeing Chemical, 49, [8] Reddy, M.G. and Reddy, N.B. () Mass Tansfe and Heat Geneation Effects on MHD Fee Convection Flow past an Inclined Vetical Suface in a Poous Medium. Jounal of Applied Fluid Mechanics, 4, 7-. [9] Singh, P.K. () Heat and Mass Tansfe in MHD Bounday Laye Flow past an Inclined Plate with Viscous Dissipation in Poous Medium. Intenational Jounal of Scientific & Engineeing Reseach, 3, [] Ali, L.E., Islam, A. and Islam, N. (5) Investigate Micopola Fluid Behavio on MHD Fee Convection and Mass Tansfe with Constant Heat and Mass Fluxes by Finite Diffeence Method. Ameican Jounal of Applied Mathematics, 3, [] Islam, M., Akte, F. and Islam, A. (5) Mass Tansfe Flow though an Inclined Plate with Poous Medium. Ameican Jounal of Applied Mathematics, 3, [] Chaudhay, R.C. and Jha, A.K. (8) Effects of Chemical Reactions on MHD Micopola Fluid Flow past a Vetical Plate in Slip-Flow Regime. Applied Mathematics and Mechanics, 9, [3] Muthuchumaaswamy, R. (9) Fist Ode Chemical Reaction on Exponentially Acceleated Isothemal Vetical Plate with Mass Diffusion. Annals Faculty of Engineeing, 7, [4] Rajesh, V. and Vama, S.V.K. (9) Chemical Reaction and Radiation Effects on MHD Flow Past an Infinite Vetical Plate with Vaiable Tempeatue. Fa East Jounal of Mathematical Sciences, 3, [5] Ecket, E.R.G. and Dake, R.M. (97) Analysis of Heat and Mass Tansfe. McGaw-Hill Book Company, New Yok. [6] Olajuwon, B.I. () Convection Heat and Mass Tansfe in a Hydomagnetic Flow of a Second Gade Fluid in the Pesence of Themal Radiation and Themal Diffusion. Intenational Communications in Heat and Mass Tansfe, 38, [7] Kuma, A.G.V., Goud, Y.R. and Vama, S.V.K. () Themal Diffusion and Radiation Effects on Unsteady MHD Flow though Poous Medium with Vaiable Tempeatue and Mass Diffusion in the Pesence of Heat Souce/Sink. Advances in Applied Science Reseach, 3, [8] Fedows, M., Khan, M.S., Bég, O.A. and Alam, M.M. (3) Numeical Study of Tansient Magneto Hydodynamicadiative Fee Convection Nanofluid Flow fom a Stetching Pemeable Suface. Jounal of Pocess Mechanical Engineeing, -6. [9] Khan, M.S., Kaim, I., Ali, L.E. and Islam, A. () MHD Fee Convection Bounday Laye Unsteady Flow of a Nanofluid along a Stetching Sheet with Themal Radiation and Viscous Dissipation Effects. Intenational Nano Lettes,,

13 Nomenclatue C p : Specific heat at constant pessue C: Concentation D m : Mass diffusivity D T : Themal diffusivity g: Local acceleation due to gavity k: Themal conductivity T: Fluid tempeatue T : Dimensionless fluid tempeatue U: Dimensionless pimay velocity V: Dimensionless seconday velocity u, v: Velocity components along x and y axes espectively Y: Dimensionless coodinate vaiable Dimensionless paametes D f : Dufou numbe G : Gashof numbe G m : Modified Gashof numbe K : Chemical eaction paamete P : Pandtl numbe S C : Schmidt numbe S o : Soet numbe Geek Symbols β T : Themal expansion coefficient β C : Mass expansion coefficient ρ: Density of the fluid µ: Dynamic viscosity of the fluid υ: Kinematic viscosity of the fluid α: Angle of inclination : Diffeential opeato δ: Bounday laye thickness Subscipts w: Condition of the wall : Condition of the fee steam Abbeviations EFDM: Explicit finite diffeence method MHD: Magneto hydodynamics 74