MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel

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1 Global Jornal of Pre and Applied Mathematics. ISSN Volme 3, Nmber 7 (7), pp Research India Pblications MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel P. Grivi Redd T.S.Redd S.V.K.Varma 3 Reader in mathematics, S. B. V. R. Degree College, Badvel, A.P., India. Department of Mathematics, Global College of Engineering, Kadapa -56, A.P India 3 Department of Mathematics, S.V.Universit, Tirpati -575, A.P, India. Abstract This manscript is an investigation on the effects of slip condition, chemical reaction, radiation and nstead MHD periodic flow of a viscos, incompressible, electricall condcting flid throgh a poros medim in in two cases viz. Case I: Uniform plate Temperatre and Uniform Concentration and Case II: Constant heat and mass flx.. The governing eqations have been solved b pertrbation techniqe. The skin friction and rate of heat transfer and mass transfer are also derived. The effects of varios phsical parameters like magnetic parameter, Renolds nmber, Grashof nmber, modified Grashof nmber, permeabilit parameter, chemical reaction parameter, and Schmidt nmber are analzed thogh graphs. Shear stress increases with the increase in magnetic parameter or permeabilit parameter and it shows a reverse effect in the case of slip parameter or Schmidt nmber in both cases of the std. Kewords: MHD, Chemical reaction, Radiation, Periodic flow, Planer channel and Slip flow regime.. INTRODUCTION: The phenomenon of slip-flow regime has attracted the attention of a large nmber of scholars de to its wide ranging applications. The problem of the slip flow regime is ver important in this era of modern science, technolog and vast ranging

2 38 P. Grivi Redd, T.S.Redd and S.V.K.Varma indstrialization. In man practical applications, the particle adjacent to a solid srface no longer takes the velocit of the srface. The particle at the srface has a finite tangential velocit, it slips along the srface. The flow regime is called the slip flow regime and its effect cannot be neglected. The flid slippage phenomenon at the solid bondaries appear in man applications sch as micro channels or Nano channels and in applications where a thin film of light oils is attached to the moving plates or when the srface is coated with special coating sch as thick monolaer of hdrophobic mechanical device where a thin film of lbricant is attached to the srface slipping over one another or when the srfaces are coated with special coating to minimize the friction between them. The effect of the flid slip page at the wall for Cotte flow are considered b Marqes et al. [] nder stead state conditions and onl for gases. The closed form soltions for stead periodic and transient velocit field nder slip condition have been stdied b Khaled and Vafai [3]. The effect of slip condition on MHD stead flow in a channel with permeable bondaries has been discssed b Makinde and Osalsi [4]. Anil Kmar et.al.[5] discssed the pertrbation techniqe to nstead MHD periodic flow of viscos flid throgh a planer channel. The effect of slip condition on nstead MHD oscillator flow of a viscos flid in a planer channel is investigated b Mehmood and Ali [6]. Mageto-hdrodnamics is attracting the attention of the man athors de to its applications in geophsics and astrophsics, it is applied to std the stellar and solar strctres, interstellar matter, radio propagation throgh the ionosphere etc. In engineering in MHD pmps, MHD bearings etc. Since some flids can also emit and absorb thermal radiation, it is of interest to std the effect of magnetic field on the temperatre distribtion and heat transfer when the flid is not onl an electrical condctor bt also when it is capable of emitting and absorbing thermal radiation. This is of interest becase heat transfer b thermal radiation is becoming of greater importance when we are concerned with space applications and higher operating temperatres. Sondalgekar and Takhar [7] first, stdied the effect of radiation on the natral convection flow of a gas past a semi-infinite plate sing the Cogl-Vincentine- Gilles eqilibrim model. For the same gas Takhar et al. [8] investigated the effects of radiation on the MHD free convection flow past a semi-infinite vertical plate. Later, Hossain et al. [9] stdied the effect of radiation on free convection from a poros vertical plate. Mthcmarswam and Kmar [] stdied the thermal radiation effects on moving infinite vertical plate in presence of variable temperatre and Mass diffsion. An analtical soltion for nstead free convection in poros media has been stdied b Magari et al. []. Chamkha et al. [] stdied the effects of Hdro magnetic combined heat and mass transfer b natral convection from a permeable srface embedded in a flid satrated poros medim. Mzmdar and Deka [3] stdied MHD flow past an implsivel started infinite vertical plate in presence of thermal radiation. The growing need for chemical reactions in chemical and hdrometallrgical indstries reqire the std of heat and mass transfer with chemical reaction. The effect of chemical reaction on heat and mass transfer in a laminar bondar laer flow has been stdied nder different conditions b several athors [4-47].

3 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel 383 In this paper the inflence of slip on MHD periodic flow of viscos flid in presence of radiation and chemical reaction in a planer channel is discssed in two cases viz., Case (I) with niform temperatre and concentration, Case (II) with heat and mass flx have been stdied. The copled non-linear partial differential eqations are solved sing a pertrbation scheme. And a series of soltions are expressed in terms and hperbolic sine and cosine fnctions for velocit, temperatre and concentration fields for both cases of this std.. NOMENCLATURE : * : Velocit along x-axis k : Permeabilit of poros medim C * : Concentration N : Nsselt nmber C : Flid concentration at the bondar = P e : Peclet nmber C w : Flid concentration at the bondar = a T * : Flid temperatre T : Flid temperatre at the bondar = Sc : Schmidt nmber T w : Flid temperatre at the bondar = a D a : Darc nmber P * : Flid pressre t * : Time : Kinematic viscosit coefficient B : Electromagnetic indction H : Intensit of magnetic field C p : Specific heat at constant pressre : Thermal condctivit q * : The radiative heat flx D : Mass diffsivit E : Eckert nmber G m : Modified Grashof nmber G r : Thermal Grashof nmber g : Gravitation de to acceleration M : Magnetic parameter K c : Non- dimensional rate of chemical reaction * K c : Rate of chemical reaction R e :Renolds nmber n : Freqenc of the periodic flow : Non- dimensional velocit F : Radiation parameter q r : Radiative flx k : Non- dimensional permeabilit coefficient of a poros medim GREEK SYMBOLS : Thermal condctivit ρ : Flid densit ν : Kinematic viscosit θ : Dimensionless temperatre σ : Electrical condctivit μ : Dnamic viscosit T : Coefficient of volme expansion e : Condctvit of the flid : The mean radiation absorption coefficient τ : Non-dimensional shear stress C : Coefficient of volme expansion with concentration

4 384 P. Grivi Redd, T.S.Redd and S.V.K.Varma 3. FORMULATION OF THE PROBLEM: Here we consider an nstead MHD flow of viscos, incompressible, electricall noncondcting flid in a planer channel in the presence of transverse applied magnetic * field. The x -axis is taken along the plate in vertical pward direction against to the gravitational field and the * -axis is taken normal to it. Initiall, one plate is heated at constant temperatre T w and another one is heated to the temperatre T. A transverse magnetic field of niform strength B is applied normal to the direction of the flow. The flid has small electrical condctivit and the electromagnetic force prodced is ver small, so the indced magnetic field is neglected and there is no applied voltage which implies the absence of an electrical field. Viscos and Joles dissipations are neglected in energ eqation. The flid considered here is gra, absorbing/emitting radiation bt a non-scattering medim. There is a first order chemical reaction between the diffsing species and the flid. Then nder sal Bossinesq s approximation the fll developed flow is governed b the following set of eqations as described b Redd et al. [5, 6]: * * * P B * * * * * * * * g ( * * T T T ) gc( C C) t x k () T T q * * * * t CP CP C t * * C * * * D ( ) KC C C * * * * The relevant bondar conditions are given as follows () (3) Case I: Uniform plate Temperatre and Concentration: * * U * * * * * U h * T T W C C W at * * * * * * U T T C C at a (4) Case II: Constant heat and mass flx: * * * * * U T q C q * U h, W * at * * k * * * * * * U T T C C at a (5)

5 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel 385 Consider the flid which is oppticall thin with a relativel low denisit and radiative heat flx is given b q * * * 4 ( T T * ) (6) Where is the the mean radiation absorption coefficient On introdcing the following non dimensional qantities, Case I: * * * * * * * x t v T T C C ap x,,, t,, C, P, * * a a v a T T C C v * * * * W 4 a g ( T T ) a g ( C C ) a K F G G D K v v a * * * * * T W C W, r, m, a * KC va va vacp a B C, C, e, e, K S R P M (7) v D Case II: * * * * * * x t v T T * * x,,, t, C C C, a a v a qa * aqw k 3 3 * 4 a gta q gca q ap W F, Gr, Gm, P K v v v * KC va va vacp a B C, C, e, e, W k a * *, Da K S R P M (8) v D The non-dimensional form of the governing eqations (3.) to (3.3) redce to P e ( ) r m R K M G G C t x Pe S F t () C C K S C t () C C C (9)

6 386 P. Grivi Redd, T.S.Redd and S.V.K.Varma Where K a * D K,The corresponding bondar conditions are a Case I: h,, C at,, C at () Case II: C h,, at,, C at (3) For prel periodic flow, let the Pressre gradient be of the form, is a constant and is the freqenc of oscillations P e x it where 4. SOLUTION OF THE PROBLEM: In order to solve the governing eqations (9) to () sbject to the bondar conditions () and (3), the following pertrbation techniqe is sed. The governing eqations (9) to () are expanded in small pertrbation parameter ( ). Let t e o (4) it (, ) o( ) ( ) ( )... (5) it (, t) o( ) e ( ) o( )... C t c e c o (6) it (, ) o( ) ( ) ( )... Using eqations (4) to (6) into the eqations (9) to () and we get Zero-th order terms: m G G C (7) '' r m F (8) '' C ' m C (9) 5

7 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel 387 First order terms: m G G C () '' / r m '' m () 3 C ' m C () 4 The corresponding bondar conditions are Case I: d d h, h,,, C, C d d at,,,, C, C at (3) Case II: d d C C h, h,,,, d d at,,,, C, C, at (4) Solving eqations (7) to () nder the bondar conditions (3) and (4), the following soltions are obtained. Case I: ( ) c cosh( m ) c sinh m k (cosh N k sinh N) k (cosh m k sinh m ) (5) o ( ) cosh F k sinh F (6) C ( ) cosh m k sinh m (7) 5 5 ( ) c cosh m c sinh m k (8) ( ) (9) C ( ) (3) Finall the expressions for velocit, Temperatre and concentration are given b (, t) cosh F k sinh F (3) C(, t) cosh m5 k sinh m5 (3)

8 388 P. Grivi Redd, T.S.Redd and S.V.K.Varma c cosh m c sinh m k3(cosh F k sinh F) (, t) k4(cosh m5 k sinh m5 ) it e ( c3 cosh( m ) c4 sinh( m ) k6) (33) Case II: ( ) c 5 cosh( m ) c 6 sinh m k 8 (cosh N k sinh N) k 9 (cosh m 5 k sinh m 5 ) (34) ( ) cosh F k sinh F (35) kf C ( ) cosh m k sinh m (36) 5 5 km ( ) c cosh m c sinh m k (37) ( ) (38) C ( ) (39) Finall the expressions for velocit, Temperatre and concentration are given b (, t) cosh F k sinh F (4) kf C(, t) cosh m5 k sinh m5 (4) km 5 (, t) c 5 cosh( m ) c 6 sinh m k 8 (cosh N k sinh N) i 9 (cosh 5 sinh 5 ) t k m k m e ( c 3 cosh( m ) c 4 sinh( m ) k 6 ) (4) 5. NUSSELT NUMBER: From temperatre field, the rate of heat transfer in terms of Nsselt nmber is given in non-dimensional form as follows: For Case (I): N k F (43)

9 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel 389 For Case (II): N (44) 6. SHERWOOD NUMBER: From concentration field, the rate of mass transfer in terms of Sherwood nmber is given in non-dimensional form as follows: C For Case (I): Sh k m 5 (45) C For Case (II) : Sh (46) 7. SKIN-FRICTION: From velocit field, the rate of change of velocit at the plate in terms of Skin-friction is given in non-dimensional form as follows: For Case(I): it [ c m ( k k 3 F k k 4 m 5 )] [ e ( c 4 m )] (47) it For Case (II): [ c 6 m ( k k 8 F k k 9 m 5 )] [ e ( c 4 m )] (48) 8. RESULTS AND DISCUSSIONS: Case (I): In order to point ot the effects of varios parameters on flow characteristics, the following discssion is set ot. The vales of Prandtl nmber are chosen Pr = 7 (water) and Pr=.7 (air). The vale of the Schmidt nmber is chosen to represent the presence of species b Carbon dioxide (.). Profiles for velocit, temperatre and concentration fields in addition to skin-friction are presented in figres -8 for Gr> and Gm>. Figre shows the effect of wall slip h on the velocit field. From this figre it is observed that velocit increases as the slip parameter h increases. Velocit profiles for different vales of time t, Magnetic parameter M, Renolds nmber Re, and permeabilit parameter k are presented in figres from -5 respectivel. It is observed that velocit decreases with increasing vales of time t or magnetic parameter M or Renolds nmber Re or permeabilit

10 38 P. Grivi Redd, T.S.Redd and S.V.K.Varma parameter k. The effects of thermal Grashof nmber Gr, modified Grashof nmber Gm, radiation parameter F, Schmidt nmber Sc and chemical reaction parameter Kc are shown in figres from 6- respectivel. It is observed that velocit increases with increasing vales of thermal Grashof nmber Gr or modified Grashof nmber Gm while decreases with increasing vales radiation parameter F or Schmidt parameter Sc or chemical reaction parameter Kc. Temperatre profiles for different vales of radiation parameter F and Schmidt parameter Scare shown in figres and respectivel. From these figres it is seen that velocit increases with the decrease in radiation parameter F or Schmidt parameter Sc. Concentration profiles for different vales of chemical reaction parameter Kc are presented in figre 3. It is noticed that concentration increases a chemical reaction parameter Kc decreases. A variation in shear stress τ is shown in figres 4-8 for different vales of wall slip parameter h, magnetic parameter M, permeabilit parameter k, radiation parameter F and chemical reaction parameter Kc. It is fond that skin friction decreases with the increase in wall slip parameter h or magnetic parameter M or permeabilit parameter k or radiation parameter F or chemical reaction parameter Kc. Case (II): In order to know the inflence of different phsical parameters, viz., wall slip parameter h, magnetic parameter M, permeabilit parameter k, radiation parameter F, chemical reaction parameter Kc, Renolds nmber Re, thermal Grashof nmber Gr, modified Grashof nmber Gm, Schmidt nmber Sc and time t on the phsical flow field, comptations carried ot for velocit, temperatre, concentration and skin-friction and the are represented throgh figres 9-36 for Gr> and Gm>. The vales of Prandtl nmber are chosen Pr = 7 (water) and Pr=.7 (air). The vale of the Schmidt nmber is chosen to represent the presence of species b Carbon dioxide (.). Figre 9represents the effect of wall slip h on the velocit field. From this figre it is observed that velocit increases as an increase in slip parameter h. Velocit profiles for different vales of time t, Magnetic parameter M, Renolds nmber Re, and permeabilit parameter k are presented in figres from -3 respectivel. It is noticed that as t or M or Re or k increases velocit also increases. The effects of thermal Grashof nmber Gr, modified Grashof nmber Gm, radiation parameter F, Schmidt nmber Sc and chemical reaction parameter Kc are shown in figres from 4-8 respectivel. It is fond that velocit increases with increasing vales of thermal Grashof nmber Gr or modified Grashof nmber Gm while it a reverse effect with increasing vales radiation parameter F or Schmidt parameter Sc or chemical reaction parameter Kc. Temperatre profiles for different vales of radiation parameter F is shown in figres 9. From this figre it is seen that velocit increases with the decrease in radiation parameter F. Concentration profiles for different vales of Schmidt parameter Sc and chemical reaction parameter Kcare presented in figres 3 and 3 respectivel. It is observed that concentration increases with decrease in Schmidt parameter Sc or chemical reaction parameter Kc. Shear stress is presented against time t for different vales of wall slip parameter h, magnetic parameter M, permeabilit parameter k, radiation parameter F and chemical reaction parameter Kc in figres 3-36

11 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel 38 respectivel. It is observed that Shear stress decreases with the increase in wall slip parameter h or magnetic parameter M or permeabilit parameter k or radiation parameter F or chemical reaction parameter Kc. GRAPHS: Case I: t =..5 h=.,.5,.,.5..5 Figre.. Effect of wall slip h and time t on velocit field when S c =, G r =, R e =, P e =.7, k =, F =, ω =, =, K c =, M = M =.,.5,., Re =, 3, 5, Figre.. Effect of Magnetic parameter M and Renolds Nmber Re on velocit field when S c =, G r =, P e =.7, k =, F =, ω =, =, K c =, h =,t=

12 38 P. Grivi Redd, T.S.Redd and S.V.K.Varma k =.5,.,.5, Gr =.5,.,.5,...5. Figre.3. Effect of permeabilit parameter k and Grashof nmber G r on velocit field when S c =, R e =, P e =.7, M =, F =, ω =, =, K c =, h = F =.5,.,.5,..4. Gm =.5,.,.5, Figre.4. Effect of Modified Grashof Nmber G m and Radiation parameter F on velocit field when S c =, G r =, P e =.7, k =, M =, ω =, =, K c =, h =,t=

13 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel Sc =.,.78,.,..9.8 Kc =.5,.,.5, Figre.5. Effect of Schmidt Nmber Sc and chemical reaction parameter K c on velocit field when G m =, G r =, P e =.7, k =, F =, ω =, =, h =, t=, M= F =.5,.,.5,..8.7 Sc =.,.78,., C Figre.6. Effect of Radiation parameter F on Temperatre field and Schmidt nmber S on concentration field when G m =, G r =, P e =.7, k =, ω = c, =, h =, t=, M=

14 384 P. Grivi Redd, T.S.Redd and S.V.K.Varma h =.5,.,.5,..7.6 Kc =.5,.,.5,..8.7 C t Figre.7. Effect of the chemical reaction parameter K c on concentration field and k on skin friction when G m =, G r =, P e =.7, k =, F =, ω =, =, t=, M= M =.,.5,.,.5.8 k =.5,.,.5, t.6 t Figre.8. Effects of M and k on Skin-friction F =.5,.,.5,..6 Kc =.5,.,.5, t. t Figre.9. Effects of F and KC Skin-friction

15 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel 385 Case II: t =..5 h =.,.5,.,.5..5 Figre.. Effect of wall slip h and time t on velocit field when S c =, G r =, R e =, P e =.7, k =, F =, ω =, =, K c =, M = M =.,.5,.,.5 5 Re =, 3, 5, Figre.. Effect of Magnetic parameter M and Renolds Nmber Re on velocit field when S c =, G r =, P e =.7, k =, F =, ω =, =, K c =, h =,t= k =.5,.,.5, Gr =.5,.,.5,...5. Figre.3. Effect of permeabilit parameter k and Grashof nmber G r on velocit field when S c =, R e =, P e =.7, M =, F =, ω =, =, K c =, h =

16 386 P. Grivi Redd, T.S.Redd and S.V.K.Varma F =.5,.,.5, Gm =.5,.,.5, Figre.4. Effect of Modified Grashof Nmber G m and Radiation parameter F on velocit field when S c =, G r =, P e =.7, k =, M =, ω =, =, K c =, h =,t= Kc =.5,.,.5,. 5 Sc =.,.78,., Figre.5. Effect of Schmidt Nmber Sc and chemical reaction parameter K c on velocit field when G m =, G r =, P e =.7, k =, F =, ω =, =, h =, t=, M= F =.5,.,.5,..7.6 Sc =.,.78,.,..5 C Figre.6. Effect of Radiation parameter F on Temperatre field and Schmidt nmber S on concentration field c

17 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel h=.,.5,.,.5 C Kc =.5,.,.5, t Figre.7. Effect of the chemical reaction parameter h on Skin-friction K c on concentration field and M=.5,.,.5, k =.5,.,.5, t t 8 Figre.33. Skin-friction for different vales of magnetic parameter M and permeabilit parameter k 7 6 F =.5,.,.5,. 6 5 Kc =.5,.,.5, t t Figre.35. Skin-friction for different vales of radiation parameter F and Kc

18 388 P. Grivi Redd, T.S.Redd and S.V.K.Varma 9. CONCLUSIONS: In this paper we have stdied the effects of slip condition, chemical reaction and radiation on MHD free convection periodic flow throgh a satrated poros medim bonded b a vertical srface in a planer channel in two cases viz. Case I: Uniform plate Temperatre and Concentration and Case II: Constant heat and mass flx. In the analsis of the flow the following conclsions are made.. In case (I) and (II) of the std, the velocit increases with an increase in slip parameter h, Grashof nmber Gr, modified Grashof nmber Gm, Schmidt nmber Sc, and Radiation parameter F, and it shows a reverse effect in the case of magnetic parameter M, time t, permeabilit parameter k or Renolds nmber Re.. Temperatre decreases with the increase in radiation parameter F for Case (I) and (II). 3. Concentration decreases with an increase in chemical reaction parameter kc or Schmidt nmber Sc in Case (I) as well as in Case (II) of the problem. 4. Shear stress increases with the increase in magnetic parameter M or permeabilit parameter k and it shows a reverse effect in the case of slip parameter h or Schmidt nmber Sc in both cases of the std. REFERENCES [] C. Derek, D. C. Trethewa and C. D. Meinhart., Apparent flid slip at hdrophobic micro channels walls, Phsics of Flids, 4, pp- L9-L,. [] W. Jr. Marqes, G. M. Kremer and F. M. Shapiro., Cotte flow with slip and jmp bondar Conditions, Continmm Mech. Thermodnam.,, ,. [3] A.R. Khaled and K. Vafai., The effect of the slip condition on Stokes and Coette Flows de to an oscillator wall: exact soltions, International Jornal of Nonlinear Mechanics, 39, pp , 4. [4] O. D.Makinde and E. Osalsi., MHD stead flow in a channel with slip at the permeable bondaries, Romanian Jornal of Phsics, 5, pp , 6. [5] C. L. Anil kmar, Varshne and Sajjanlal., Pertrbation techniqe to nstead periodic Flow of viscos flid throgh a planer channel, Jornal of Enng and Technolog Vol.(4), pp.73-8,. [6] A. Mehmood. and A. Ali., The effect of slip condition on nstead MHD oscillator flow of a viscos flid in a planer channel, Romanian Jornal of Phsics, 5, pp. 85-9, 7. [7] V. M.. Sondalgekar and H. S. Takhar., Radioactive convective flow past a semi-infinite vertical plate, Modeling Measre and Cont., 5, 3-4, 99. [8] H. S. Tahkar, S. R. Gorla and V. M. Sondalgekar., Radiation effects on MHD free convection flow of a radiating gas past a semi-infinite vertical plate, Int. J. Nmerical Methods heat flid flow, 6, 77-83, 996.

19 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel 389 [9] A. M. Hossain and M. A. Alim and D. A. S. Rees., Effect of radiation on free convection from a poros a vertical plate, Int.J.Heat Mass Transfer, 4, 8-9, 999. [] R. Mthcmarswam and G. S. Kmar., Heat and Mass Transfer effects on moving vertical plate in the presence of thermal radiation, Theoret. Appl. Mach., 3(), 35-46, 4. [] E. Magari, I. Pop and B. Keller., Analtical soltions for nstead free convection flow throgh a poros media, J.Eng. math., 48, 93-4, 4. [] A. J. Chamaka., Hdro magnetic combined heat and mass transfer b natral convection from a permeable srface embedded in a flid satrated poros medim, Int. J. Nm.Methods for heat and flid flow, (5), ,. [3] M. K. Mzmdar and R.K. Deka., MHD flow past an implsivel started infinite vertical plate in presence of thermal radiation, Rom. Jorn. Phs., Vol. 5, No pp , 7. [4] E. L. Cssler., Diffsion Mass transfer in Flid sstems, Cambridge Universit press, London, 988. [5] A. J. Chamkha., MHD flow of a niforml stretched vertical permeable srface in the presence of heat generation/absorption and a chemical reaction, Int.Commn. Heat mass transfer, 3(4), 43-4, 3. [6] R. Mthcmarswam and P. Ganesan., Effect of chemical reaction and injection on flow characteristics in an nstead pward motion of an isothermal plate, J.Appl.mech.Tech. Phs., 4, No.4, ,. [7] R. Mthcmarswam., Effects of chemical reaction on a moving isothermal vertical srface with sction, Acta Mech.55, 65-7,. [8] K. Manivannan, R. Mthcmarswam and V. Thangaraj., Radiation and chemical reaction effects on isothermal vertical oscillating plate with variable mass diffsion, Thermal science, Vol.3, No., pp.55-6, 9. [9] P. R. Sharma, N. Kmar and P. Sharma., Inflence of chemical reaction and radiation on nstead MHD free convection flow and mass transfer throgh viscos incompressible flid past a heated vertical plate immersed in poros medim in the presence of heat sorce, Appl. Math. Sciences, Vol.5, No.46, 49-6,. [] N. Ananda Redd, S. V. K. Varma and M. C. Raj, Thermo diffsion and chemical effects with simltaneos thermal and mass diffsion in MHD mixed convection flow with Ohmic heating., Jornal of Naval Architectre and Marine Engineering, Vol. 6, No., 9, DOI: 39/jname.v6i.376. [] M.C.Raj, S. V. K. Varma, Unstead MHD free convection oscillator Coette flow throgh a poros medim with periodic wall temperatre, Jornal on Ftre Engineering and Technolog, Jl, Vol.6, No.4. Pp.7-. [] M. C. Raj, S.V. K.Varma, N. A. Redd., MHD Thermal diffsion Natral convection flow between heated inclined plates in poros medim, Jornal on Ftre Engineering and Technolog.Vol.6, No., pp.45-48,

20 383 P. Grivi Redd, T.S.Redd and S.V.K.Varma [3] T. S. Redd, S. V. K. Varma & M. C. Raj; Chemical reaction and radiation effects on nstead MHD free convection flow near a moving vertical plate, Jornal on Ftre Engineering & Technolog, Vol. 7,No. 4, -, Ma - Jl. [4] M. C. Raj, S.V.K Varma, N. Ananda Redd., Radiation and mass transfer effects on a free convection flow throgh a poros medim bonded b a vertical srface, Jornal of Ftre Engineering and Technolog, Vol. 7, No:, pp. 7-. [5] T. S. Redd, M. C. Raj & S. V. K. Varma, Chemical reaction and radiation effects on MHD free convection flow throgh a poros medim bonded b a vertical srface with constant heat and mass flx, Jornal of comptational and Applied research in Mechanical Engineering, Vol. 3 (), 53-6, 3. [6] T. S. Redd, M. C. Raj & S. V. K. Varma, Unstead MHD radiative and chemicall reactive free convection flow near a moving vertical plate in poros medim, JAFM, Vol.6, no.3, pp , 3. [7] M. C. Raj, S. V. K. Varma, R. R. K. Rao, Unstead MHD free convection and chemicall reactive flow past an infinite vertical poros plate, i-manager Jornal of Ftre Engineering and Technolog, Vol.8 (3), 3, [8] M. C. Raj, N. Ananda Redd, S. V. K. Varma, Analtical std of MHD free convictive, dissipative bondar laer flow past a poros vertical srface in the presence of thermal radiation, chemical reaction and constant sction, Ain Shams Engineering Jornal, Vol. 5 (4), 4, DOI:.6/j.asej ISSN: [9] M. C. Raj, S.V.K.Varma, B. Seshaiah, Heat transfer effects on a viscos dissipative flid flow past a vertical plate in the presence of indced magnetic field, Ain Shams Engineering Jornal, Vol.6, No., 5, doi:.6/j.asej [3] M. C. Raj, and S.V. K.Varma, Soret effects de to natral convection in a non-newtonian flid flow in poros medim with heat and mass transfer, Jornal of Naval architectre and Marine Engineering, Vol. (), 4, pp [3] B. Mamtha, M. C. Raj, S.V.K.Varma, Thermal diffsion effect on MHD mixed convection nstead flow of a micro polar flid past a semi-infinite vertical poros plate with radiation and mass transfer, International Jornal of Engineering Research in Africa, Vol. 3 (5) pp -37. [3] B. Mamatha, S.V.K.Varma, M. C. Raj, Unstead MHD mixed convection, radiative bondar laer flow of a micro polar flid past a semi-infinite vertical poros plate with sction, International Jornal of Applied Science and Engineering, 5, 3,: [33] S. Harinath Redd, M. C. Raj, E. Keshava Redd, Unstead MHD free convection flow of a Kvshinski flid past a vertical poros plate in the presence of chemical reaction and heat sorce/sink, International Jornal of Engineering Research in Africa Vol. 4 (5) pp [34] V. Ravikmar, M.C. Raj, G.S.S. Raj., Theoretical investigation of an nstead MHD free convection heat and mass transfer flow of a non-

21 MHD Flow of a Newtonian Flid Throgh a Poros Medim in Planer Channel 383 Newtonian flid flow past a permeable moving vertical plate in the presence of thermal diffsion and heat sink, International Jornal of Engineering Research in Africa Vol. 6(5), 9-9, doi:.48/ [35] M. Umamaheswar, M. C. Raj, S.V.K. Varma, Analsis of MHD transient free convection flow of a Newtonian flid past an infinite vertical poros plate, Frontiers in Heat and Mass Transfer (FHMT) 6-8 (5). DOI: [36] M.Umamaheswar, M. C. Raj and S. V. K. Varma, MHD convective heat and mass transfer flow of a Newtonian flid past a vertical poros plate with chemical reaction, radiation absorption and thermal diffsion, International Jornal of Engineering Research in Africa Vol. 9 (6), 37-56, doi:.48/ [37] P. Chandraredd, M. C. Raj, G. S. S. Raj, Magnetohdrodnamic Convective Doble Diffsive Laminar Bondar Laer Flow Past an Accelerated Vertical Plate, International Jornal of Engineering Research in Africa, Vol. (6), 8-9, doi:.48/ [38] M. C. Raj, A. J. Chamkha, J.Philip, S.V.K.Varma, Soret effect de to mixed convection on nstead magnetohdrodnamic flow past a semi-infinite vertical permeable moving plate in presence of thermal radiation, heat absorption and homogenos chemical reaction, International Jornal of Applied and Comptational Mathematics(6). DOI.7/s x [39] S.H. Redd, M. C. Raj, E. Keshava Redd, Radiation absorption and chemical reaction effects on MHD flow of heat generating Casson flid past Oscillating vertical poros plate, Frontiers in Heat and Mass Transfer (FHMT), 7, (6). DOI:.598/hmt.7. [4] M. C. Raj, S. V. K.Varma, A. J. Chamkha, Unstead free convection flow past a periodicall accelerated vertical plate with Newtonian heating, International Jornal of Nmerical Methods for Heat and Flid flow, 6 Vol. 6 No. 7, 6, pp DOI.8/HFF [4] P. Chandra Redd, M. C. Raj and G. S. S. Raj, S.V.K. Varma., Free convective magneto-nanoflid flow past a moving vertical plate in the presence of radiation and thermal Diffsion, Frontiers in Heat and Mass Transfer (FHMT), 7, 8 (6). DOI:.598/hmt.7.8 [4] P.Chandraredd, M.C.Raj, G.S.S.Raj, Free convective heat and mass transfer flow of heat generating nanoflid past a vertical moving poros plate in condcting field, Special Isse: Theoretical and Mathematical Modeling of Flow and Transport in Poros Media-Special Topics & Reviews in Poros Media - An International Jornal, Vol.7 No., 6, 6-8. DOI:.65/SpecialTopicsRevPorosMedia [43] C. Veeresh., S. V. K. Varma., A.G. V. Kmar., M. Umamaheswar and M. C. Raj, Jole heating and thermal diffsion effects on MHD radiative and convective Casson flid flow past an oscillating semi-infinite vertical poros plate, Frontiers in Heat and Mass Transfer, 8, (7) DOI:.598/hmt.8.

22 383 P. Grivi Redd, T.S.Redd and S.V.K.Varma [44] L. Ramamohan Redd, M. C. Raj, G. S. S. Raj and S. M. Ibrahim, Chemical reaction and thermal radiation effects on MHD micropolar flid past a stretching sheet embedded in a non-darcian poros medim, Jornal of Comptational and Applied Research in Mechanical Engineering, Vol. 6. No., pp. 7-46, 7. DOI:.6/jcarme.7.58 [45] N.A.Redd, T.S.Redd, M. C. Raj, S.V.K.Varma, The effects of slip condition and chemical reaction on nstead MHD periodic flow of a viscos flid throgh satrated poros medim in a planner channel, Proceedings of the International Conference on Mathematics in Engineering &Bsiness Management, March 9-, 6-, [46] N. Ananda Redd,S.Vijaa Kmar Varma and M. C. Raj, Three dimensional MHD flow with heat and mass transfer throgh a poros medim with periodic permeabilit and chemical reaction, Proceedings of MARTEC, The International Conference on Marine Technolog - December, pp [47] L. Rama Mohan Redd, M. C. Raj, G. S. S. Raj., Unstead MHD free convection flow of a visco-elastic flid past a vertical poros plate in the presence of thermal radiation, radiation absorption, heat generation/absorption and chemical reaction, International Jornal of Applied Science and Engineering, Vol.4 (), 6,