International Journal of Mathematical Archive-9(2), 2018, Available online through ISSN

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1 International Journal of Mathematical rchive-9() vailable online through ISSN RDITION ND CHEMICL RECTION EFFECTS ON MHD MIXED CONVECTIVE FLOW OF VERTICL SURFCE WITH OHMIC HETING ND VISCOUS DISSIPTION V. NGRJU K. HEMLTH Research Scholar Department of Mathematics Krishna Universit Machalipatnam India-5. Department of Mathematics V R Siddhartha Engineering College Vijaawada India-57. (Received On: 6--7; Revised & ccepted On: 8--8) BSTRCT We consider the radiation and chemical reaction effects on MHD mixed convective heat and mass transfer flow past a vertical surface under the influence of Ohmic and viscous dissipation. The governing sstem of partial differential equations is transformed to dimensionless equations using dimensionless variables. The dimensionless equations are then solved analticall using perturbation technique. The various parameters entering into the problem on the dimensionless velocit temperature and concentration fields within the boundar laer are discussed and the skinfriction coefficient the Nusselt number and Sherwood number are presented numericall in tabular form also explained qualitativel. Kewords: Mixed convective flow; MHD; Ohmic heating; viscous dissipation; Radiation; Chemical reaction.. INTRODUCTION The stud of boundar laer flow heat and mass transfer over an inclined plate has generated much interest from astrophsical renewable energ sstems and also hpersonic aerodnamics researchers for a number of decades. In recent ears MHD flow problems have come in view of its significant applications in industrial manufacturing processes such as plasma studies petroleum industries Magneto hdrodnamics power generator cooling of clear reactors and boundar laer control in aerodnamics. Man authors have studied the effects of magnetic fields on mixed natural and force convection heat and mass transfer problems. Heat transfer considerations arise due to chemical reactions and often due to the ver nature of the process. MHD free convection fluid-flows frequentl occur in natural world. Fluid passing through porous medium is of great interest nowadas and man researchers are attracted towards the applications in the fields of science and technolog namel in the area of agriculture engineering to know about ground water resources in technolog to stud the moment of natural gas oil and water through the oil reservoirs. The effects of mixed convection unstead stagnation point flow of a viscous fluid with a variable free stream velocit. The mass transfer effects on unstead flow past an accelerated vertical porous plate have been noticed b Das et al. []. Sattar [] has investigated the free convection and mass transfer flow past an infinite vertical porous plate with time dependent temperature and concentration. Chen et al. [] examined heat and mass transfer in MHD flow b natural convection from a permeable inclined surface with variable wall temperature and concentration. Masood et al. [4] used HM to formulate the MHD mixed convection Falkner Skan flow with convective boundar conditions. The MHD conjugative heat transfer problem from vertical surfaces embedded in saturated porous media was discussed b Duwairi and l-kablawi [5]. Nasir Uddin et al. [6] discovered the effect of conjugate heat and mass transfer on magneto hdrodnamic mixed convective flow past inclined plate in a porous medium. The skin friction in the MHD mixed convection stagnation point with mass transfer was studied b bdelkhalek [7].The number of investigations of natural convection flow with thermal radiation has increased greatl during the past few decades due to its importance in man practical situations. When natural convection flows occur at high temperature radiation effects on the fluid flow become significant. Radiation effects on the natural convection flow are important in the context of furnace design electric power generation thermo-nuclear fusion glass production casting and levitation Corresponding uthor: V. Nagaraju Research Scholar Department of Mathematics Krishna Universit Machalipatnam India-5. International Journal of Mathematical rchive- 9() Feb. 8 5

2 plasma phsics cosmic flights propulsion sstems solar power technolog spacecraft re-entr aerothermodnamics etc. It is worth noting that unlike convection/conduction the governing equations taking into account the effects of radiation become quite complicated. Hence man difficulties arise while solving such equations. However some reasonable approximations are proposed to solve the governing equations with radiative heat transfer. The influence of magnetic field on the flow of an electricall conducting viscous fluid with mass transfer and radiation absorption is also useful in planetar atmosphere research. Kinanjui et al. [8] investigated simultaneous heat and mass transfer in unstead free convection flow with radiation absorption past an impulsivel started infinite vertical porous plate subjected to a strong magnetic field. Suneetha [9] noticed the problem of radiation and mass transfer effects on MHD free convection flow past an impulsivel started isothermal vertical plate with dissipation. din and Kaa [] proposed the MHD mixed convective heat transfer about a semi-infinite inclined plate in the presence of magneto and thermal radiation effects. Israel Cooke et al. [] noticed the influence of viscous dissipation and radiation on unstead MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction. Ogulu [] examined the influence of radiation absorption on unstead free convection and mass transfer flow of a polar fluid in the presence of uniform magnetic field. Rajesh [] has investigated radiation effects on MHD free convection flow near a vertical plate with ramped wall temperature. Sivaiah et al. [4] noticed that the heat and mass transfer effects on MHD free convective flow past a vertical porous plate. Radiation effects on mixed convection along anisothermal vertical plate were investigated b Hossain and Takhar [5]. Raptis and Perdiki [6] discovered the effects of thermal radiation and free convection flow past a moving vertical plate. In man chemical engineering processes chemical reaction occurs between a foreign mass and the fluid in which the plate is moving. These processes take place in numerous industrial applications viz. polmer production manufacturing of ceramics or glassware and food processing. The effect of chemical reaction on unstead MHD flow through an impulsivel started semi-infinite vertical plate was examined b Muthucumaraswam et al. [7]. Sudhir Babu et al. [8] studied radiation and chemical reaction effects on an unstead MHD convection flow past a vertical moving porous plate embedded in a porous medium with viscous dissipation. Dulal Pal et al. [9] analzed Perturbation nalsis of unstead magnetohdrodnamic convective heat and mass transfer in a boundar laer slip flow past a vertical permeable plate with thermal radiation and chemical reaction. The effect of radiation and chemical reaction on transient MHD free convective flow over a vertical plate through porous media was formulated b Sandeep et al. []. Recentl Kumar et al. [] investigated effects of chemical reaction and radiation on MHD free convection flow past an exponentiall accelerated vertical plate. njali Devi and Kandasam [] have discovered the effects of chemical reaction heat and mass transfer on MHD flow past a semi-infinite plate. Ibrahim et al. [] have examined the effects of the chemical reaction and radiation absorption on the unstead MHD free convection flow past a semiinfinite vertical permeable moving plate with heat source and suction. Effects of chemical reaction and radiation absorption on MHD flow of dust viscoelastic fluid are analzed b Prakash et al. [4]. The chemical and radiation absorption effects on MHD convective heat and mass transfer flow past a semi-infinite vertical moving porous plate with time dependent suction was proposed b Ramana Redd et al. [5]. Sudheer Babu et al. [6] investigated the effects of thermal radiation and chemical reaction on MHD convective flow of a polar fluid past a moving vertical plate with viscous dissipation. Saritha and Sata Naraana [7] investigated thermal diffusion and chemical reaction effects on unstead MHD free convection flow past a semi-infinite vertical permeable moving plate. For the problem of coupled heat and mass transfer in MHD free convection the effect of Ohmic heating has not been considered in the above investigations. However it is more realistic to include this effect to explore the impact of the magnetic field on the thermal transport in the boundar laer. With this awareness the effect of Ohmic heating on the MHD free convection heat transfer has been examined on a Newtonian fluid b Hossain [8]. Chaudhar et al. [9] have discovered the radiation effect with simultaneous thermal and mass diffusion in MHD mixed convection flow from a vertical surface with Ohmic heating. The effect of Ohmic heating on the MHD free convection heat transfer has been studied on a Newtonian fluid b Hossain []. The problem of combined heat and mass transfer of an electricall conducting fluid in MHD natural convection adjacent to a vertical surface with Ohmic heating was analzed b Chen []. The effect of Ohmic heating and viscous dissipation on MHD mixed convection heat and mass transfer about a vertical plate are analzed b din and Kaa []. Babu and Redd [] have discovered the mass transfer effects on MHD mixed convective flow from a vertical surface with ohmic heating and viscous dissipation. Sibanda and Makinde [4] proposed the effects of magnetic fields on heat transfer on a rotating disk in a porous medium with Ohmic heating and viscous dissipation. For the problem of coupled heat and mass transfer in MHD free convection the effect of both viscous dissipation and Ohmic heating are not studied in the above investigations. However it is more realistic to include these two effects to explore the impact of the magnetic field on the thermal transport in the boundar laer. With this awareness the effect of Ohmic heating on the MHD free convection heat transfer has been studied for a Newtonian fluid b Hossain [5]. 8 IJM. ll Rights Reserved 5

3 . FORMTION OF THE PROBLEM We consider the mixed convection flow of an incompressible electricall conducting viscous fluid such that x - axis which is taken along the plate in upward direction and - axis is normal to it. transverse constant magnetic field is applied in the direction -axis. Since the motion is two dimensional and length of the plate is large therefore all the phsical variables are independent of x. Let u and v be the components of velocit in x and directions respectivel taken along and perpendicular to the plate. The governing equations of continuit momentum and energ for a flow of an electricall conducting radiation is given b v p P is independent of u u ϑu ρ v µ ρgβ( T T ) ρgβ ( C C ) σb u K v - v (constant) () () + + qr ρ T T u Cv p k B u µ + + σ C C v D R ( C C ) Here g is the due to gravit T the temperature of the fluid near the plate T the free stream temperature c -the concentration β -the coefficient of thermal expansion k the thermal conductivit P the pressure C p the specific heat of constant pressure B the magnetic field coefficient µ the viscosit of the fluid r () (4) (5) q the radiative heat flux ρ the densit σ the magnetic Permeabilit of the fluid v -the constant suction velocit v the kinematic viscosit and D molecular diffusivit. The radiative heat flux is given b where q r 4( ) T T I debλ Kλw dλ dt w Kλ w is the absorption coefficient at wall and eb λ is Planck s function. (6) The boundar conditions are u T T w C C (7) w u T T C C Introducing the following non dimensional quantities are u u v v T T C C θ C ϑ T T C C w v w w µ c p Pr ϑ Sc k D gβϑ( T T ) gβϑ( C ) Gr w C σb Gm ϑ v K ρ M K v ρv ϑ v E C T Tα p ( ) w F 4I ϑ ρcv p ϑr Kr v (8) 8 IJM. ll Rights Reserved 54

4 The non-dimensional form of the governing Eqs. () - (5) reduce to u + u M + u [ GrT + GmC] K T + PrT F PrT + Pr E( u ) + Pr EMu () C + ScC ScKrC () where Gr is the Grashof number Gm the modified Grashof number Pr - the Prandtl number F- the radiation parameter Sc- the Schmidt number E-the Eckert number M-the magnetic parameter Kr-the chemical reaction. (9) The corresponding boundar conditions in dimension less form are reduced to u T C at u T C as () The phsical variable u T and C can expand in the power of Eckert number. This can be possible phsicall as E for the flow of an incompressible fluid is alwas less than unit. It can be interpreted phsicall due to the Ohmic dissipation which is improved on the main flow.. SOLUTION OF THE PROBLEM To reduce the above sstem of partial differential equations to the sstem of ordinar differential equations in a dimension less from we ma represent the translational velocit temperature and concentration as u( ) u( ) + Eu( ) + O( E ) T ( ) T( ) + ET( ) + O( E ) C( ) C( ) + EC( ) + O( E ) () Using equation () in equation (9) - () and equating the Coefficient of like power of E we have u + u nu GrT GmC (4) T + PrT FPrT C + ScC ScKrC u + u n u GrT GmC ( ) T + PrT E PrT + Pr u + Pr Mu C + ScC ScKrC The corresponding boundar conditions are u u T T C C at () u u T T C C as Solving equations (5)-(9) with the help of () we get u ( ) e e e m m m 4 (5) (6) (7) (8) (9) () ( m m ) ( m m ) ( m m ) ( ) m 6 m 5 m 4 m m 4 4 u e e e e e e e e T e m ( ) ( ) ( ) m 5 m4 m m m+ m4 m+ m m+ m4 T e e 4 e 5 e 6 + e 7 e 8 + e 9 C e m C () () (4) (5) (6) 8 IJM. ll Rights Reserved 55

5 Substituting equations ()-(6) in Equation () we obtain the velocit Temperature and concentration distribution in the boundar laer as follows: m 6 m 5 m 4 m e 8 e + e + e m ( ) 4 m m u t e e e + E m ( m4+ m) ( m+ m) ( m+ m4) + e 4 e 5 + e 6 e 7 m 5 m4 m m e e 4 e 5 e m 6 T( t ) ( e ) + E ( m+ m4) ( m+ m) ( m+ m4) + e 7 e 8 + e 9 C t e m ( ) (9) The skin-friction Nusselt number and Sherwood number are important phsical parameters for this tpe of boundar laer flow. (7) (8) Skin Friction: The non-dimensional skin friction at the plate is given b: u τ τ ( ) m + m m m m m4+ m + m + E + ( m + m4 ) 5 ( m + m ) 6 + ( m4 + m ) 7 Nusselt Number: The non- dimensional form of the rate of heat transfer in terms of Nusselt number at the plate is given b: T Nu m 5 + m m 5 + m 6 Nu m E ( m m4) 7 ( m m) 8 ( m4 m) Sherwood Number: The non- dimensional form of the rate of heat transfer in terms of Sherwood number at the plate is given b: C Sh Sh m () () () RESULTS ND DISCUSSION The sstem of ordinar differential equations (4) - (9) with boundar conditions () is solved analticall b emploing the perturbation technique. The solutions are obtained for the velocit fields from () - () temperature fields from () - (4) and concentration fields are given b (5) - (6). To assess the phsical depth of the problem the effects of various parameters like Grashof number Gr modified Grashof number Gm magnetic parameter M porosit parameter K Prandtl number Pr Eckert number E Schmidt number Sc Chemical reaction Kr Radiation parameter Fon velocit distribution temperature distribution and concentration distribution are studied in figures -8 while keeping the other parameters as constants. The variation in Skin friction the rate of mass transfer in the form of Sherwood number studied through the following default parameter values are adopted Gr5. Gm. M. K. Pr.7 E. Sc. Kr. F.. Figure. depicts the velocit profile u against for different values of Grashof number Gr and modified Grashof number Gm. From this we observe that as Grashof number Gr increases velocit field u increases modified Grashof number Gm increases and velocit field u increases. The effects of the magnetic parameter M and porosit parameter K on the dimensionless velocit is shown in the Figure. It is obvious that an increase in magnetic parameter M results in a decrease in the velocit. For different values of porosit parameter K it can be seen that the velocit profile increase with the increase of permeabilit parameter K. 8 IJM. ll Rights Reserved 56

6 Figure &6 show the dimensionless velocit and temperature profiles for different values of Prandtl number Pr. It can be seen that both velocit and the temperature profiles decrease with the increase of Prandtl number Pr. In figure 4 it exhibits the velocit profiles for various value of chemical reaction parameter Kr Schmidt number Sc. From this figure it is observed that velocit decreases with increases in chemical reaction Kr and the velocit increases with a decrease in Schmidt number Sc. Figure 8 illustrates the effects of chemical reaction Kr Schmidt number Sc on the concentration profile. It is seen from these figures that there is a stead decrease in the concentration with the increase in chemical reaction Kr and there is a decrease in the velocit with the increase in Schmidt number Sc. Effects of Eckert number E and Radiation parameter F on velocit and temperature profiles are studied from figure 5&7. From these figures it is observed that velocit and temperature profiles decrease as Radiation parameter F increases and it can be seen that both velocit and the temperature profiles increase with the increase of Eckert number E M.K.Pr.7F. E.Sc.Kr..4 u... Curve Gr Gm Figure-: Velocit profiles for different values of Gr and Gm Gr5.Gm.Pr.7 F.E.Sc.Kr. u.. Curve M K Figure-: Velocit profiles for different values of K and M 8 IJM. ll Rights Reserved 57

7 .7.6 Gr5.Gm.M.K. Pr.7F.E..5.4 u... Curve Sc Kr Figure-: Velocit profiles for different values of Scand Kr.8 u Gr5.Gm.M.K. F.E.Sc.Kr... Curve Pr Figure-4: Velocit profile for different values of Pr Gr5.Gm.M.K. Pr.7Sc.Kr.. u.. Curve F E Figure-5: Velocit profiles for different values of F and E 8 IJM. ll Rights Reserved 58

8 .9.8 Gr5.Gm.M.K. F.E.Sc.Kr..7.6 T Curve Pr Figure-6: Temperature profile for different values of Pr.9.8 Gr5.Gm.M.K. Pr.7Sc.Kr..7.6 T Curve F E Figure-7: Temperature profiles for different values of F and E.9.8 Gr5.Gm.M.K. Pr.7F.E..7.6 C Curve Sc Kr Figure-8: Concentration profiles for different values of Scand Kr 8 IJM. ll Rights Reserved 59

9 Table-: Numerical values of the Skin-friction (Cf) Nusselt number (Nu) and Sherwood number (Sh) for Gr Gc M K. Gr Gc M K Cf Nu Sh Table-: Numerical values of the Skin-friction (Cf) Nusselt number (Nu) and Sherwood number (Sh) for Pr F E. Pr F E Cf Nu Sh Table-: Numerical values of the Skin-friction (Cf) Nusselt number (Nu) and Sherwood number (Sh) for Sc Kr. Sc Kr Cf Nu Sh Tables () () and () show the numerical values of the skin friction (Cf) Nusselt number (Nu) and Shear wood number (Sh). The effects of where Grashof number Gr modified Grashof number Gm magnetic parameter M porosit parameter K Prandtl number Pr Radiation parameter F Eckert number E Schmidt number Sc and Chemical reaction Kr on the skin-friction(cf) Nusselt number(nu) Sherwood number (Sh)are shown in Tables to. From Table it is noticed that as the skin-friction increases as Grashof number Gr modified Grashof number Gc porosit parameter K increases. Magnetic parameter M increases the skin-friction decreases. The Nusselt number decreases as Grashof number Gr modified Grashof number Gc porosit parameter K increases. Magnetic parameter M increases the Nusselt number also increases. 8 IJM. ll Rights Reserved 6

10 From the table presents it is observed that the Prandtl number Pr Radiation parameter F increases skin-friction decreases Nusselt number increases.eckert number E increases and the skin-friction increase Nusselt number decreases. From the table presents it is observed that the Schmidt number Sc increases the skin-friction and the Sherwood number also increases. Chemical reaction Kr increases the skin-friction and Nusselt number decreases but Sherwood number increases. CONCLUSION In this paper we discussed the radiation and chemical reaction effects on MHD flow of a vertical surface in presence of ohmic heating and viscous dissipation has been studied. The governing equations are solved for the velocit field temperature and concentration b using perturbation technique in terms of dimensionless parameters. In the analsis of the flow the following conclusions are made: The velocit increases with an increase in Grashof number Gr and modified Grashof number Gc. The velocit decreases with an increase in the magnetic parameter M. The velocit increases with an increase in the porosit parameter K. Increase the Prandtl number Pr decreases the velocit and the temperature field. Increase the Radiation parameter F decreases the velocit and the temperature field. n increase in the Eckert number E leads to increase in the velocit and the temperature field. The velocit as well as concentration decreases with an increase in the Schmidt number Scand Chemical reaction Kr. REFERENCES. S. S. Das S. K. Sahoo and G. C. Dash Numerical Solution of Mass Transfer Effects on Unstead Flow Past an ccelerated Vertical Porous Plate with Suction Bulletin of Malsie. Math.Sci. Soc. 9() M.. Sattar Free Convection and Mass Transfer Flow Through a Porous Medium Past and Infinite Porous Plate with Time Dependent Temperature and Concentration Int. J. Pure and ppl. Math C.H. Chen Yunlin and Taiwan Heat and mass transfer in MHD flow b natural convection from a permeable inclined surface with variable wall temperature and concentration cta Mechanica vol. 7 4 pp Masood K li R Shahzad. MHD Falkner-skan flow with mixed convection and convective boundar conditions. Walaiak J SciTechnol ; : Chen C.H. (4) Heat and mass transfer in MHD flow b natural convection from a permeable inclined surface with variable wall temperature and concentration cta MechanicaVol.7 p Md. Nasiruddin M.. lim and M. M. K. Chowdhur Effect of Conjugate Heat and Mass Transfer on MHD Mixed Convective Flow past Inclined Porous Plate in Porous Medium International Journal of Mechanical Industrial Science and Engineering vol. 8 no. 4 pp bdelkhalek MM (6) The skin friction in the MHD mixed convection stagnation point with mass transfer. Int. Communications in Heat and Mass Transfer : Kinanjui M Kwanza JK and Uppal SM Energ Conversion and Management 4(8): Suneetha S Bhasker Redd N and Ram Chandra Prasad V Thermal Science 9 Vol. No. pp O. din and.kaa MHD mixed convective heat transfer flow about an inclined plate Heat Mass Transfer vol pp Israel cooke C. Ogulu. Omubo Pepple V.M. (). The influence of viscous dissipation and radiation on unstead MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction. Int. J. Heat Mass Transfer Ogulu The influence of radiation absorption on unstead free convection and mass transfer flow of a polar fluid in the presence of a uniform magnetic field International Journal of Heat and Mass Transfer vol. 48 no. -4 pp Rajesh V.. Radiation effects on MHD free convection flow near a vertical plate with ramped temperature mixed free-forced convection and mass transfer Int. J. ppl. Math and Mech. 6 (): Sivaiah M Nagarajan Redd PS. Heat and mass transfer effects on MHD free convective flow past a vertical porous plate. The ICFI Universit Journal of Computational Mathematics. 9; (): Hossain M.. and Takhar H.S Radiation effects on mixed convection along a vertical plate with uniform surface temperature. Heat and MassTransfer. : Raptis. and Perdikis C Radiation and free convection flow past a moving plate. Int. J. of p. Mech.nd Engg. 4: IJM. ll Rights Reserved 6

11 7. Muthucumaraswam. R Maheswari. J and Pandurangan. J (8): Unstead MHD flow past an impulsivel started semi-infinite vertical plate in the presence of chemical reaction Mathematics 4() p SudhirBabu M Satanaraana P V Sankar Redd and Uma Maheshwara Redd D dvances in pplied Science Research (5): Dulal Pal and BabulalTalukdar (). Perturbation analsis of unstead magneto hdrodnamic convective heat and mass transfer in a boundar laer slip flow past a vertical permeable plate with thermal radiation and chemical reaction. Communications in Nonlinear Science and Numerical Simulation pp N. Sandeep. Vijaa Bhaskar Redd and V. Sugunamma Effect of radiation and chemical reaction on Transient MHD free convective flow over a vertical plate through porous media Chemical and Process Engineering Research vol. pp Kumar.G. V. Varma S.V. K. and Rammohan. Chemical reaction and radiation effects on MHD free convective flow past an exponentiall accelerated vertical plate with variable temperature and variable mass diffusion nnals of Facult Engineering Hunedoara Tome X Fascicule pp: S. P. njali Devi and R. Kandasam Effects of chemical reaction heat and mass transfer on MHD flow past a semi-infinite plate Zeitschrift fur ngewandtemathematik und Mechanik vol. 8 no. pp F.S. Ibrahim.M. Elaiw and.. Bakr Effects of chemical reaction and radiation absorption on the unstead MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction Communications in Nonlinear Science and Numerical Simulation Vol. no. 6 8 pp Prakash. J Vijaa Kumar.G Madhavi M and Varma S. V. K (4). Effects of Chemical Reaction and Radiation bsorption on MHD Flow of Dust Visco elastic Fluid pplications and pplied Mathematics: n international Journal Vol.9 No. pp G. V. Ramana Redd G. Sudershan Redd and K. Jaarami Redd: Chemical and radiation absorption effects on MHD convective heat and mass transfer flow past a semi-infinite vertical moving porous plate with time dependent suction International Journal Of Mathematical rchive () pp SudheerBabu M Satanaraana P V Sankar Redd T and Umamaheswara Redd D (): Effects of thermal radiation and chemical reaction on MHD convective flow of a polar fluid past a moving vertical plate with viscous dissipation Elixir ppl. Math. 4:pp Saritha S and Sata Naraana P V(): Thermal diffusion and chemical reaction effects on unstead MHD free convection flow past a semi-infinite vertical permeable moving plate sian Journal of Current Engineering and Maths : pp M.. Hossain The effect of Ohmic heating on the MHD free convection heat transfer has been examined for a Newtonian fluid Int. J. Heat and Mass transfer 5 P R. C. Chaudhar Bhupendra Kumar Sharma and bha Kumar Jha Radiation effect with simultaneous thermal and mass diffusion in MHD mixed convection flow from a vertical surface with Ohmic heating Rim. Journ.Phs. vol. 5 Nos.7-8 pp M.. Hossain (99) Viscous and Ohmic heating effects on MHD free convection flow with variable plate temperature Int. J. Heat and Mass transfer 5 p Chien-Hsin-Chen (4) Combined heat and mass transfer in MHD free convection from a vertical surface with Ohmic heating and viscous dissipation Int. J. Engineering Science 4 p O. din and. Kaa (9) the effect of Ohmic heating and viscous dissipation on MHD mixed convection heat and mass transfer about a vertical plate ppl. Math. Model. () pp Babu V. S. H. and Redd G. V. R..Mass transfer effects on MHD mixed convective flow from a vertical surface with ohmic heating and viscous dissipation. dvances in ppl. Sci. Res. () (4) P. Sibanda and O. D. Makinde On stead MHD flow and heat transfer past a rotating disk in a porous medium with ohmic heating and viscous dissipation International Journal of Numerical Methods for Heat and Fluid Flow vol. no. pp Hossain M. Viscous and Ohmic heating effects on MHD free convection flow with variable plate temperature Int. J. Heat and Mass transfer 5 (99) pp PPENDIX m m 5 Sc + Sc + 4KrSc Pr+ Pr + 4E Pr ( + Pr ) m m 6 Sc + Sc + 4KrSc + + 4n m M n M + 4m Pr m EPr K Pr+ Pr + 4F Pr 8 IJM. ll Rights Reserved 6 m Gr Gm m + + 4n 4 + m mn m mn ( m + Pr M) m Pr m EPr 4 ( Pr ) m M 4m Pr m EPr

12 ( mm 4 + Pr M) ( 4 ) Pr ( 4 ) ( mm 4 + Pr M) ( + ) Pr ( + ) m + m m + m EPr 7 m m m m EPr Gr 4 4m4 m4 n 7 Gr 5 4m m n ( mm + Pr M) ( ) Pr ( ) m + m m + m EPr Gr m m n Gr 6 4 4m mn 6 ( m + m ) ( m + m ) n ( ) ( ) 5 Gr 4 4 ( ) ( ) 7 Gr 9 m + m m + m n 4 4 Gr 8 m + m m + m n Source of support: Nil Conflict of interest: None Declared. [Cop right 8. This is an Open ccess article distributed under the terms of the International Journal of Mathematical rchive (IJM) which permits unrestricted use distribution and reproduction in an medium provided the original work is properl cited.] 8 IJM. ll Rights Reserved 6