Research Article Unsteady Heat and Mass Transfer of Chemically Reacting Micropolar Fluid in a Porous Channel with Hall and Ion Slip Currents

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1 Internatonal Scholarly Research Notces Volume 4 Artcle ID pages Research Artcle Unsteady Heat and Mass Transfer of Chemcally Reactng Mcropolar Flud n a Porous Channel wth Hall and Ion Slp Currents Odelu Ojjela and N. Naresh Kumar Department of Appled Mathematcs Defence Insttute of Advanced Technology (Deemed Unversty) Pune 45 Inda Correspondence should be addressed to Odelu Ojjela; odelu@yahoo.co.n Receved June 4; Accepted September 4; Publshed 9 October 4 Academc Edtor: Frédérc Lebon Copyrght 4 O. Ojjela and N. Naresh Kumar. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense whch permts unrestrcted use dstrbuton and reproducton n any medum provded the orgnal work s properly cted. Ths paper presents an ncompressble two-dmensonal heat and mass transfer of an electrcally conductng mcropolar flud flow n a porous medum between two parallel plates wth chemcal reacton Hall and on slp effects. Let there be perodc njecton or sucton at the lower and upper plates and the nonunform temperature and concentraton at the plates are varyng perodcally wth tme. The flow feld equatons are reduced to nonlnear ordnary dfferental equatons usng smlarty transformatons and then solved numercally by quaslnearzaton technque. The profles of velocty components mcrorotaton temperature dstrbuton and concentraton are studed for dfferent values of flud and geometrc parameters such as Hartmann number Hall and on slp parameters nverse Darcy parameter Prandtl number Schmdt number and chemcal reacton rate and shown n the form of graphs.. Introducton The flow and heat transfer of a flud through porous channels s of great mportance n both engneerng and scence. Applcatons of these were found n dfferent areas such as ol exploraton geothermal energy extractons the boundary layer control extruson of polymer fluds soldfcaton of lqud crystals coolng of a metallc plate n a bath MHD power generators and suspenson solutons. Many authors nvestgated the two-dmensonal ncompressble flud flow through porous channels theoretcally. Terrll and Shrestha [] consdered the lamnar ncompressble flow of an electrcally conductng vscous flud through a porous channel and gave a soluton for a large sucton Reynolds number and Hartmann number. Erngen [ ] ntated the theory of mcropolar fluds and ths theory consttutes a subclass of mcrofluds. Atta [4] consdered the problem on MHD flow of a dusty flud n a crcular ppe wth Hall and on slp and obtaned a seres soluton for reduced governng equatons. A perturbaton soluton was obtaned for heat and mass transfer n a rotatng vertcal channel wth Hall current by Ahmed and Zueco [5]. Eldabe and Ouaf [6] nvestgated the problem on MHD flow and heat and mass transfer of a mcropolar flud over a stretchng surface wth ohmc heatng. Magyar et al. [7] have studed Stokes frst problem for mcropolar flud and solved the problem analytcally by Laplace transforms and numercally by Valkó- Abate procedure. Bhattacharyya et al. [8] studed the effects of chemcal reacton on the boundary layer flow of vscous flud and a numercal soluton obtaned by shootng method. The problem of steady ncompressble MHD flow of a mcropolar flud between concentrc porous cylnders was dscussed by Srnvasacharya and Shferaw [9] who obtaned a numercal soluton by quaslnearzaton technque. The steady mcropolar flud flow between two vertcal porous parallel plates n thepresenceofmagnetcfeldwasconsderedbybhargava et al. [] and the governng dfferental equatons have been solved usng the quaslnearzaton method. Abdulazz et al. [] examned the MHD fully developed natural convecton flow of mcropolar flud between parallel vertcal plates and the reduced governng nonlnear equatons are solved by usng HAM method. Das [] studedtheeffectsofthermal

2 Internatonal Scholarly Research Notces radaton and chemcal reacton on ncompressble flow of electrcally conductng mcropolar flud over an nclned plate. Pal et al. [] nvestgatedtheproblemofoscllatory mxed convecton-radaton of a mcropolar flud n a rotatng system wth Hall current and chemcal reacton effects and obtaned a soluton usng perturbaton method. Rahman and Al-Lawata [4] examned mcropolar flud flow over a stretchng sheet wth varable concentraton and solved numercally usng the shootng method. Bakr [5] consdered the steady as well as unsteady MHD mcropolar flud wth constant heat source and chemcal reacton effect n a rotatng frame of reference and solved by usng the perturbaton method. Patl and Kulkarn [6] studed the free convectve flow of a polar flud n a porous medum n the presence of an nternal heat generaton wth chemcal reacton effect. Themagnetomcropolarfludflowandheatandmasstransfer n a porous medum wth Hall and on slp effects were dscussed by Motsa and Shatey [7] who obtaned a numercal soluton usng the successve lnearzaton method. Olajuwon [8] studed the heat and mass transfer of an electrcally conductng mcropolar flud flow over a horzontal flat plate nthepresenceoftransversemagnetcfeldwththechemcal reacton and Hall effects and the problem s solved by usng Runge-Kutta method. Arman and Cakmak [9] analyzed the three basc vscous flows of mcropolar flud between parallel plates and solved analytcally. Modather et al. [] examned an ncompressble MHD flow and heat and mass transfer of a mcropolar flud over a vertcal plate n a porous medum. Km [] nvestgated an ncompressble mcropolar flud flow past a sem-nfnte plate n a porous medum and obtaned an analytcal soluton by perturbaton method. Reddy Gorla et al. [] obtaned a numercal soluton for the steady boundary layer flow and heat transfer of a mcropolar flud over a flat plate. The MHD flow of vscoelastc flud over a porous stretchng sheet was analyzed by Hymavath and Shanker [] and a numercal soluton was obtaned by usng the quaslnearzaton method. Bhatnagar et al. [4] dscussed the steady ncompressble lamnar flow of vscoelastc flud through a porous cylndrcal annulus and the reduced governng equatons are solved numercally usng quaslnearzaton method. Ths paper deals wth the unsteady two-dmensonal ncompressble MHD flow and heat transfer of a mcropolar flud n a porous medum between parallel plates wth chemcal reacton Hall and on slp effects. The flow s generated due to perodc sucton or njecton at the plates and the reduced flow feld equatons are solved numercally usng the quaslnearzaton technque. The effects of varous parameters on velocty components temperature dstrbuton mcrorotaton and concentraton are studed and shown graphcally.. Formulaton of the Problem Consder an ncompressble electrcally conductng mcropolar flud flow through a porous medum between two parallel plates at y=and y=h(fgure 8). The lower and upper plates are subjected to perodc njecton and sucton of the forms Real (V e Ωt )andreal(v e Ωt ) respectvely. Assume that the temperature and concentraton at the lower and upper plates are T e Ωt T e Ωt and C e Ωt C e Ωt respectvely. The equatons governng the flow and heat and mass transfernthepresenceofamagnetcfeldandntheabsence of body forces and body couples are gven by Sherclff [5]: q = () ρ[ q t (q )q] = gradpk curl l (μk ) curl q μk qj B k ρj [ l t (q ) l] = k l k curl q γcurl curl l () ρc [ T t (q ) T] = k T μd : D k (curl (q) l) γ l: l μk k () q J σ (4) [ C t (q )C]=D C k (C C ). (5) The coeffcents μ k α β γ n the above equatons are related by the nequaltes μ k k αβγ γ β. (6) Neglectng the dsplacement currents the Maxwell equatons andthegeneralzedohm slaware B= B=μ J E= B t J=σ(Eq B) βe (J B) βeβ B B (J B) B where B = B k b b s nduced magnetc feld βe s the Hall parameter β s the on slp parameter and μ s magnetc permeablty. For mxed sucton case that s V V followng Terrl and Shreshta [] we take the velocty mcrorotaton temperaturedstrbutonandconcentratonas (7) q=u V j l=n k (8)

3 Internatonal Scholarly Research Notces where u (x t) =( U a V x h )f () e Ωt V (x t) =V f () e Ωt N (x t) = h (U a V x h )g() eωt T (x t) =(T (μ k )V ρhc C (x t) =(C (φ () ( U av x h ) φ ())) e Ωt n A hυ [G () ( U av x h ) G ()]) e Ωt where =y/h U s the average entrance velocty a= (V /V )andf() g() φ () φ () G () andg () are functonsof to be determned. The boundary condtons of the velocty components mcrorotaton temperature and concentraton are u (x t) = V (x t) =V e Ωt N (x t) = T(x t) =T e Ωt C (x t) =C e Ωt at = u (x t) = V (x t) =V e Ωt N (x t) = T(x t) =T e Ωt C (x t) =C e Ωt at =. (9) () Substtutng (9) nto () () (4) and (5) and then comparng the real parts on both sdes we get φ φ Re (ff f f ) cos ψ =Rg (R) f V Ha αe βe αe f (R) D f J (fg f g) cos ψ= s (g f )g Re Pr ( 4 s R f (R) Pr g Ha (R)(αe βe ) f D f fφ ) cos ψ= φ Re Pr (f φ fφ f R R (R) (f g) Ha (R)(αe βe ) f D f ) cos ψ Re R s g cos ψ= G = G Kr G Sc Re fg cos ψ G = Kr G Sc Re (fg f G ) cos ψ () where prme denotes the dfferentaton wth respect to and αe = βeβ. From (9) the dmensonless forms of temperature and concentraton are T = C = T T e Ωt (T T )e Ωt =E(φ ζ φ ) C C e Ωt (C C )e Ωt = Sh (G ζ G ) () where E=(μk )V /ρhc(t T ) s the Eckert number and ζ = ((U /av ) (x/h)) s the dmensonless axal varable. The boundary condtons ()ntermsoff g φ φ G and G are f () = a f() = f () = f () = g () = g() = φ () = φ () = E φ () = φ () = G () = G () = Sh G () = G () =.. Soluton of the Problem () The nonlnear dfferental equatons () are converted nto the system of frst order dfferental equatons by the followng substtuton: (f f f f gg φ φ φ φ G G G G ) =(x x x x 4 x 5 x 6 x 7 x 8 x 9 x x x x x 4 )

4 4 Internatonal Scholarly Research Notces dx =x dx =x dx 4 = R ( Re (x x 4 x x ) cos ψ R(s (x x 5 ) dx =x 4 J (x x 6 x x 5 ) cos ψ) Ha αe βe αex D x ) dx 5 =x 6 dx 6 =s (x x 5 )J (x x 6 x x 5 ) cos ψ dx 7 =x 8 dx 8 = x 9 dx dx dx 4 Re Pr ( 4 R x s Pr (R) x 5 = Re Pr Ha (R) (αe βe ) x D x x x 8 ) cos ψ dx 9 =x (x x 9 x x R x R (R) (x 4x 5 4x x 5 ) Ha (R)(αe βe ) x D x ) cos ψ Re R s x 6 cos ψ dx =x = x Krx Sc Re x x cos ψ dx =x 4 = Krx Sc Re (x x 4 x x ) cos ψ. (4) The boundary condtons () n terms of x x x x 4 x 5 x 6 x 7 x 8 x 9 x x x x x 4 are x () = a x () = x 5 () = x 7 () = x 9 () = x () = x () = x () = x () = x 5 () = x 7 () = E x 9 () = x () = Sh x () =. (5) The system of equatons (4) ssolvednumercallysubject to the boundary condtons (5) usng the quaslnearzaton method gven by Bellman and Kalaba [6]. Let (x n =...4) be an approxmate current soluton and (x n =...4) be an mproved soluton of (4). Usng Taylor s seres expanson about the current soluton by neglectng the second and hgher order dervatve terms the coupled frst order system (4)slnearzedas dx n dx n 4 = =xn dx n =xn dx n =xn 4 ( Re (xn x n 4 R xn 4 x n xn x n xn xn ) cos ψ R(s (x n x n 5 ) dx n 6 J (x n x n 6 xn xn 6 x n x n 5 x n xn 5 )) cos ψ) R (Re (xn xn 4 xn xn ) cos ψ RJ (x n xn 6 xn xn 5 ) cos ψ) Ha αe (R) (αe βe ) xn D x n dx n 5 =s (x n x n 5 ) =xn 6 J (x n x n 6 xn xn 6 x n xn 5 x n x n 5 ) cos ψ J (x n xn 6 xn xn 5 ) cos ψ dx n 7 =xn 8

5 Internatonal Scholarly Research Notces 5 dx n 8 = Re Pr ( 8xn xn R s x n 5 xn 5 Pr (R) Ha x n xn (R)(αe βe ) D x n xn x n xn 8 x n 8 xn ) cos ψ Re Pr ( 4xn xn R s x n 5 xn 5 Pr (R) Ha x n xn (R) (αe βe ) dx n = Re Pr D x n xn xn xn 8 ) cos ψ xn 9 dx n 9 =xn (x n xn 9 x n x n 9 xn xn x n xn xn R R (R) x n (x n xn 8x n 5 xn 5 4x n xn 5 4x n x n 5 ) Ha x n xn (R)(αe βe ) D x n x n ) cos ψ Re R s x n 6 xn 6 cos ψ Re Pr (x n xn 9 xn xn xn xn R R (R) (x n xn 4xn 5 xn 5 4xn xn 5 ) Ha x n xn (R)(αe βe ) D x n xn ) cos ψ Re R s x n 6 xn 6 cos ψ dx n =xn dx n dx n 4 = xn Kr x n Sc Re (x n x n xn xn x n xn ) cos ψ dx n = Krxn Sc Re =xn 4 (x n x n 4 xn xn 4 x n x n x n xn x n xn 4 xn xn ) cos ψ. (6) To solve for (x n =...4) the soluton to seven separate ntal value problems denoted by x h () x h () x h () x h4 () x h5 () x h6 () x h7 () (whch are the solutons of the homogeneous system correspondng to (6)) and x p () (whch s the partcular soluton of (6)) wth the followng ntal condtons s obtaned by usng the 4th order Runge-Kutta method: x p 6 x h () = xh () = for = x h 4 () = xh () = for =4 x h 6 () = xh () = for =6 x h4 8 () = xh4 () = for =8 x h5 () = xh5 () = for = x h6 () = xh6 () = for = x h7 4 () = xh7 () = for = 4 x p () = a x p () =xp 7 () =xp =x p () =xp 8 () =xp 4 () =xp () =xp 5 () = () =xp 9 () =xp () () =xp () =xp 4 () =. (7) By usng the prncple of superposton the general soluton can be wrtten as x n () =C x h C 5 x h5 () C x h () C 6 x h6 () C x h () C 7 x h7 () C 4 x h4 () ()x p () (8) where C C C C 4 C 5 C 6 and C 7 are the unknown constants and are determned by consderng the boundary condtons at =.Thssoluton(x n =...4) s then compared wth soluton at the prevous step (x n =... 4) and further teraton s performed f the convergence has not been acheved.

6 6 Internatonal Scholarly Research Notces Axal velocty.... (a) Radal velocty (b).5 5 Mcrorotaton Temperature Ha = Ha = Ha =6 Ha =9 Ha = Ha = Ha =6 Ha =9 (c) (d) Fgure : Effect of Ha on (a) axal velocty (b) radal velocty (c) mcrorotaton and (d) temperature. For Kr =Re= a =. J = βe =.5 β =.5Sc=.Pr=7 R= ψ = s = s = D =. 4. Results and Dscussons To understand the flow characterstcs n a better way the numercal results of the axal velocty u radal velocty V mcrorotaton N temperature dstrbuton T and concentraton C arecalculatedcorrecttosxplacesofdecmal for varous values of nondmensonal chemcal reacton parameter Kr on slp parameter β Hall parameter βe nverse Darcy parameter D Hartmann number Ha Prandtl number Pr and Schmdt number Sc n the doman [ ]. The effect of Ha on velocty components mcrorotaton and temperature s presented n Fgures (a) to (d). It s observed that as Ha ncreases the axal velocty also ncreases towards the center of the plates and then decreases because of the Lorentz force whch tends to resst the flow and theradalveloctymcrorotatonandtemperaturedstrbuton are ncreasng from the lower plate to the upper plate. Fgures (a) to (d) gve the effect of βe on velocty components mcrorotaton and temperature dstrbuton. It can be analyzed that as βe ncreases the axal velocty also ncreases n the center of the channel whereas the temperature decreases towards the upper plate. However the radal velocty and mcrorotaton are decreasng up to the centerofthechannelandthenncrease.thssbecauseof the fact that the βe decreases the resstve force mposed by magnetc feld. The effect of β on velocty components mcrorotaton and temperature dstrbuton has been presented n Fgures(a) to (d) respectvely and these profles follow the same trend of βe. TheeffectofD on velocty components mcrorotaton and temperature dstrbuton s shownnfgures4(a) to 4(d).ItcanbeobservedthatasD ncreases the radal velocty and mcrorotaton are ncreasng up to the center of the channel and then decrease towards the upper plate and the axal velocty gves the maxmum

7 Internatonal Scholarly Research Notces 7 Axal velocty.4. Radal velocty (a).784. (b) Mcrorotaton Temperature βe = βe =. βe =.5 βe = 5 βe = βe =. βe =.5 βe = 5 (c) (d) Fgure : Effect of βe on (a) axal velocty (b) radal velocty (c) mcrorotaton and (d) temperature. For Kr =Re=a=. J = β =.5Sc= Pr=7 R= ψ =. s = s =Ha= D =. effect at the center of the plates. However the temperature ncreases from lower plate to upper plate. The effects of Kr and Sc on concentraton are shown n Fgures 5 and 6 respectvely.fromfgure 5 t s understood that as Kr ncreases the concentraton decreases because the chemcal reacton ncreases the rate of nterfacal mass transfer. It s depcted from Fgure 6 that the concentraton s decreasng as Sc ncreases and ths causes the concentraton buoyancy effect. Fgure 7 showstheeffectofprontemperature. It s notced that as Pr ncreases the temperature dstrbuton also ncreases towards the upper plate and ths s due to the fact that as Pr ncreases the thckness of the boundary layer decreases and ths gves the larger temperature values. 5. Conclusons In the present paper the effects of chemcal reacton and Hall and on slp on unsteady two-dmensonal lamnar flow of an electrcally conductng mcropolar flud through a porous medum between parallel plates wth heat and mass transfer are consdered. The reduced nonlnear ordnary dfferental equatons are solved by usng the quaslnearzaton method. The results are presented n the form of graphs for varous values of flud and geometrc parameters and from these the followng s concluded. ()Thechemcalreactonratereducestheconcentraton whle the Prandtl number enhances the temperature.

8 8 Internatonal Scholarly Research Notces Axal velocty (a) Radal velocty (b) Mcrorotaton... β = β = (c) β = 6 β = Temperature β = β = (d) β = 6 β = Fgure : Effect of β on (a) axal velocty (b) radal velocty (c) mcrorotaton and (d) temperature. For Kr =Re= a =. J = βe =.5Sc=.Pr=7 R= ψ = s = s =Ha=5 D =. () The effects of Hall parameter and on slp parameter on velocty components mcrorotaton and temperature are smlar. () The Hartmann number and nverse Darcy s parameter have the same result for velocty components mcrorotaton and temperature. Nomenclature h: Dstance between parallel plates V :Suctonvelocty V :Injectonvelocty a: Injecton sucton rato (V /V ) p: Flud pressure q: Veloctyvector c: Specfc heat at constant temperature l: Mcrorotaton vector N: Mcrorotaton component E: Eckertnumber(μ k )V /ρhc(t T ) k: Thermalconductvty k : Vscosty parameter k : Permeablty parameter u: Veloctycomponentnx-drecton V: Veloctycomponentny-drecton U : Average entrance velocty Pr: Prandtl number μc/k Re: Sucton Reynolds number ρv h/μ j: Gyraton parameter J: Current densty J : Nondmensonal gyraton parameter ρjhv /γ B: Totalmagnetcfeld b: Inducedmagnetcfeld B : Magnetc flux densty

9 Internatonal Scholarly Research Notces 9 Axal velocty (a) Radal velocty (b) Mcrorotaton Temperature D = D =5 D = D =5 D = D =5 D = D =5 (c) (d) Fgure 4: Effect of D on (a) axal velocty (b) radal velocty (c) mcrorotaton and (d) temperature. For Kr =Re= a =. J = βe =.5 β = Sc= Pr=7 R= ψ =. s = s =Ha=..7 Concentraton Kr = Kr = Kr = Kr = Fgure 5: Effect of Kr on concentraton. For Re = a =. J = βe =.5 β = Sc=. Pr=7 R= ψ = s = s = Ha =5andD =5.

10 Internatonal Scholarly Research Notces Concentraton Sc =. Sc = Sc = Sc = Fgure 6: Effect of Sc on concentraton. For Kr =Re= a =. J = βe =.5 β = Pr =7 R= ψ = s = s = Ha =5andD =5. 6 D: Rate of deformaton tensor E: Electrcfeld Ha: Hartmann number B h σ/μ D : Inverse Darcy parameter h /k R: Nondmensonal vscosty parameter k /μ s : Nondmensonal mcropolar parameter k h /γ s : Nondmensonal mcropolar parameter γc/h k T: Temperature T : Temperature at the lower plate T : Temperature at the upper plate T : Dmensonless temperature (T T e Ωt )/(T T )e Ωt C : Dmensonless concentraton (C C e Ωt )/(C C )e Ωt C: Concentraton D : Massdffusvty k : Chemcalreactonrate Kr: Nondmensonal chemcal reacton parameter k h /D Sc: Modfed Schmdt number υ/d. Temperature 5 4. Pr = Pr = Pr =5 Pr =7 Greek Letters : Dmensonless y coordnate y/h γ: Gyrovscosty parameter ζ: Dmensonless axal varable ((U /av ) (x/h)) ρ: Flud densty μ: Fludvscosty μ : Magnetc permeablty σ: Electrc conductvty ψ: Nondmensonal frequency parameter Ωt β: Ionslpparameter βe: Hall parameter αe: Hall and on slp parameters ββe. Fgure 7: Effect of Pr on temperature. For Kr =Re= a =. J = βe = β = Sc = R= ψ =. s = s = Ha =andd =. Z B Y h V e Ωt T e Ωt C e Ωt V e Ωt T e Ωt C e Ωt X Fgure 8: The schematc dagram of flud flow between two porous parallel plates. Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper. Acknowledgments The authors are thankful to the referees and Professor J. V. Ramanamurthy NIT Warangal for ther valuable suggestons whch led to the mprovement of the paper. References [] R. M. Terrll and G. M. Shrestha Lamnar flow through parallel and unformly porous walls of dfferent permeablty Zetschrft für angewandte Mathematk und Physk ZAMP vol. 6 pp [] A. C. Erngen Smple mcrofluds Internatonal Engneerng Scencevol.pp

11 Internatonal Scholarly Research Notces [] A. C. Erngen Theory of mcropolar fluds Mathematcs and Mechancsvol.6pp [4] H. A. Atta Analytcal soluton for flow of a dusty flud n a crcular ppe wth hall and on slp effects Chemcal Engneerng Communcatonsvol.94no.pp [5] S. Ahmed and J. Zueco Modelng of heat and mass transfer n a rotatng vertcal porous channel wth hall current Chemcal Engneerng Communcatons vol.98no.pp [6] N. T. Eldabe and M. E. Ouaf Chebyshev fnte dfference method for heat and mass transfer n a hydromagnetc flow of a mcropolar flud past a stretchng surface wth Ohmc heatng and vscous dsspaton Appled Mathematcs and Computatonvol.77no.pp [7] E. Magyar I. Pop and P. P. Valkó Stokes frst problem for mcropolar fluds Flud Dynamcs Research vol.4no. Artcle ID 55 pp. 5. [8] K. Bhattacharyya M. S. Uddn G. C. Layek and W. Al Pk Dffuson of chemcally reactve speces n boundary layer flow over a porous plate n porous medum Chemcal Engneerng Communcatonsvol.no.pp.7 7. [9] D. Srnvasacharya and M. Shferaw Numercal soluton to the MHD flow of mcropolar flud between two concentrc porous cylnders Internatonal Appled Mathematcs and Mechancsvol.4no.pp [] R.BhargavaL.KumarandH.S.Takhar Numercalsolutonof free convecton MHD mcropolar flud flow between two parallel porous vertcal plates Internatonal Engneerng Scencevol.4no.pp. 6. [] O. Abdulazz N. F. Noor and I. Hashm Homotopy analyss method for fully developed MHD mcropolar flud flow between vertcal porous plates Internatonal Journal for Numercal Methods n Engneerng vol.78no.7pp [] K. Das Slp effects on heat and mass transfer n MHD mcropolar flud flow over an nclned plate wth thermal radaton and chemcal reacton Internatonal Journal for Numercal Methods n Fludsvol.7no.pp.96. [] D.PalB.TalukdarI.S.ShvakumaraandK.Vajravelu Effects of hall current and chemcal reacton on oscllatory mxed convecton-radaton of a mcropolar flud n a rotatng system Chemcal Engneerng Communcatonsvol.99no.8pp [4] M. M. Rahman and M. Al-Lawata Effects of hgher order chemcal reacton on mcropolar flud flow on a power law permeable stretched sheet wth varable concentraton n a porous medum Canadan Chemcal Engneerng vol.88no.pp.. [5] A. A. Bakr Effects of chemcal reacton on MHD free convecton and mass transfer flow of a mcropolar flud wth oscllatory plate velocty and constant heat source n a rotatng frame of reference Communcatons n Nonlnear Scence and Numercal Smulaton vol. 6 no. pp [6] P. M. Patl and P. S. Kulkarn Effects of chemcal reacton on free convectve flow of a polar flud through a porous medum n the presence of nternal heat generaton Internatonal Journal of Thermal Scencesvol.47no.8pp [7] S. S. Motsa and S. Shatey The effects of chemcal reacton hall and on-slp currents on MHD mcropolar flud flow wth thermal dffusvty usng a novel numercal technque Journal of Appled Mathematcs vol.artcleid6895pages. [8] B. I. Olajuwon Heat and mass transfer n a hydromagnetc flow of a mcropolar power law flud wth hall effect n the presence of chemcal reacton and thermal dffuson the Serban Socety for Computatonal Mechancs vol.6no.pp [9] T. Arman and A. S. Cakmak Some basc vscous flows n mcropolar fluds Rheologca Acta vol.7no.pp [] M.ModatherA.M.RashadandA.J.Chamkha Ananalytcal study of MHD heat and mass transfer oscllatory flow of a mcropolar flud over a vertcal permeable plate n a porous medum Turksh Engneerng and Envronmental Scencesvol.no.4pp [] Y. J. Km Unsteady conveton flow of mcropolar fluds past a vertcal porous plate embedded n a porous medum Acta Mechancavol.48no. 4pp.5 6. [] R. S. Reddy Gorla R. Pender and J. Eppch Heat transfer n mcropolar boundary layer flow over a flat plate Internatonal Engneerng Scencevol.no.7pp [] T. Hymavath and B. Shanker A quaslnearzaton approach to heat transfer n MHD vsco-elastc flud flow Appled Mathematcs and Computaton vol.5no.6pp [4] R. Bhatnagar H. W. Vayo and D. Okunbor Applcaton of quaslnearzaton to vscoelastc flow through a porous annulus Internatonal Non-Lnear Mechancsvol.9 no. pp [5] J. A. Sherclff A Textbook of Magnetohydrodyamcs Pergamon Press Oxford UK 965. [6] R. E. Bellman and R. E. Kalaba Quaslnearzaton and Boundary-value Problems Elsever New York NY USA 965.

12 Advances n Operatons Research Volume 4 Advances n Decson Scences Volume 4 Appled Mathematcs Algebra Volume 4 Probablty and Statstcs Volume 4 The Scentfc World Journal Volume 4 Internatonal Dfferental Equatons Volume 4 Volume 4 Submt your manuscrpts at Internatonal Advances n Combnatorcs Mathematcal Physcs Volume 4 Complex Analyss Volume 4 Internatonal Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Mathematcs Volume 4 Volume 4 Volume 4 Volume 4 Dscrete Mathematcs Volume 4 Dscrete Dynamcs n Nature and Socety Functon Spaces Abstract and Appled Analyss Volume 4 Volume 4 Volume 4 Internatonal Stochastc Analyss Optmzaton Volume 4 Volume 4