OSCILLATORY CHEMICALLY-REACTING MHD FREE CONVECTION HEAT AND MASS TRANSFER IN A POROUS MEDIUM WITH SORET AND DUFOUR EFFECTS: FINITE ELEMENT MODELING

Transcription

1 OSCILLATORY CHEMICALLY-REACTING MHD FREE CONVECTION HEAT AND MASS TRANSFER IN A POROUS MEDIUM WITH SORET AND DUFOUR EFFECTS: FINITE ELEMENT MODELING R. Bhargava, R. Sharma, and O.A. Bég 3 Mathmatics Dpartmnt, Indian Institut of Tchnology, Roork-47667, India. Mathmatics Dpartmnt, Indian Institut of Tchnology, Roork-47667, India. 3 Enginring Magntohydrodynamics and Hat Transfr Rsarch, Thrmofluids Group, Mchanical Enginring Dpartmnt, Shaf Building, Shffild Hallam Univrsity, Shffild, South Yorkshir, S WB, England, UK. docoanwarbg@hotmail.co.uk Rcivd August 8; accptd 8 Dcmbr 9 ABSTRACT Th ffcts of thrmo-diffusion (Sort ffct) and diffuso-thrmal gradints (Dufour ffct) on th unstady incomprssibl magnto-hydrodynamic (MHD) fr convction flow with mass transfr past a smi-infinit vrtical plat in a Darcian porous mdium in th prsnc of significant thrmal radiation, first ordr homognous chmical raction and viscous hating ar analyzd. Th govrning diffrntial quations ar non-dimnsionalizd using a similarity transformation rndring a systm of coupld, nonlinar partial diffrntial quations which ar solvd numrically using th robust, xtnsivly-validatd finit lmnt mthod. Dimnsionlss vlocity (U) is dcrasd with incrasing magntic paramtr (M). An incras in Eckrt numbr (Ec) causs gratr mchanical nrgy to b dissipatd as thrmal nrgy and nhancs fluid tmpraturs (θ). An incras in chmical raction paramtr (γ) incrass vlocity (U), tmpratur (θ) and also concntration valu (φ). Tmpraturs (θ) ar lvatd substantially with dcrasing Sort numbr (Sr) and simultanous incrasing Dufour numbr (Du). Concntration valus (φ) ar convrsly nhancd with incrasing Sort numbr (Sr) and a concurrnt dcras in Dufour numbr (Du). Both tmpratur and vlocity ar supprssd with a ris in hat absorption paramtr ( Φ ); On th othr hand an incras in thrmal radiation absorption paramtr (Q ) gnrats an incras in both vlocity and tmpratur filds. Incrasing Schmidt numbr (Sc) is found to caus a dcras in both tmpratur and concntration profils. Finally, th numrical valus of local skin friction, local rat of hat transfr paramtr and local mass transfr paramtr ar also prsntd in tabular form. Th prsnt problm has significant applications in chmical nginring matrials procssing, solar porous wafr absorbr systms and mtallurgy. Kywords: Porous mdium, oscillatory flow, MHD, fr convction, hat transfr, mass transfr, finit lmnts, Sort and Dufour ffcts, chmical nginring systms. INTRODUCTION Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

2 6 R. Bhargava t al. Boundary layr convction hat and mass transfr flows abound in many aras of chmical nginring. For xampl in th molcular vaporator (Lutisan t al ), combind hat and mass transfr xrts a significant rol in liquid film prformanc charactristics. Othr applications includ polymr procssing (Klinstrur and Wang 989), catalytic slab systms (Mihail and Todorscu 978), lctrochmical phnomna (Probstin 989) and adsorption procsss (Yang 99). Many physical phnomna also involv fr convction drivn by hat gnration or absorption as ncountrd in for xampl chmical ractor dsign and dissociating fluids. Hat gnration ffcts may altr th tmpratur distribution and thrfor, th particl dposition rat in such systms. Hat gnration may also b critical in nuclar ractor cors, lctronic chips, smi conductor wafrs and also fir dynamics (Bég 6). In th rcnt yars, th ffct of magntic fild on hat and mass transfr flows through a porous mdium has also stimulatd considrabl intrst owing to divrs applications in film vaporization in combustion chambrs, transpiration cooling of r-ntry vhicls, solar wafr absorbrs, manufactur of gls, magntic matrials procssing, astrophysical flows and hybrid MHD powr gnrators. Porous mdia abound in chmical nginring systms and magntic filds ar frquntly usd to control transport phnomna in lctrically-conducting flunt mdia. Iliuta and Larachi (Iliuta and Larachi 3) usd a Kozny-Carman porous hydrodynamic modl to study th isothrmal hydromagntic two-fluid flow in trickl bd ractors subjctd to a homognous transvrs magntic fild. Othr intrsting analyss of hydromagntic flow in porous mdia includ th studis by Al-Nimr and Hadr (Al-Nimr and Hadr 999) concrning opn-ndd vrtical porous channls, Iliuta and Larachi (Iliuta and Larachi 3) who invstigatd multi-phas porous mdia hydromagntics undr spatially uniform magntic-fild gradints in a procss intnsification contxt, Dahikar and Sonolikar (6) who considrd hydromagntic flows in a circulating fluidizd bd, Takhar and Bég (997) who studid numrically th ffcts of Hartmann numbr and inrtial porous drag on flat-plat hydromagntic boundary layr convction and Gindrau and Auriault (Gindrau and Auriault ) who drivd a modifid prmability tnsor for magntohydrodynamic flow in a Darcian porous mdium. Bég t. al. (Bég t. al. ) usd a numrical diffrnc mthod to analyz th two-dimnsional stady fr convction magnto-viscolastic flow in a Darcy-Brinkman porous mdium. Also vry rcntly Bég t al. (Bég t. al. 8) usd th ntwork thrmodynamic simulation approach to study th hydromagntic convction flow from an isothrmal sphr to a non-darcian porous mdium with hat gnration or absorption ffcts. Extnsiv studis hav also matrializd prtaining to th ffcts of chmical raction on coupld hat and mass transfr flows in porous mdia owing to potntial applications in drying tchnologis, distribution of tmpratur and moistur ovr agricultural filds and grovs of fruit trs, nrgy transfr in wt cooling towrs, flow in dsrt coolrs and th drying and/or burnout of procssing aids in th colloidal procssing of advancd cramic matrials. Gatica t al. (Gatica t al.989) studid fr convction boundary layr flow in a porous mdium with chmical raction ffcts. Stangl and Aksay (Stangl and Aksay 99) studid simultanous ractiv momntum, hat and mass transfr phnomna in disordrd porous mdia with applications in optimization of procssing conditions in th dsign of an improvd bindr rmoval procss. Souza t al. (Souza t al. 3) studid mass transfr in a packd-bd ractors including disprsion in th main fluid phas, intrnal diffusion of th ractant in th pors of th catalyst, and surfac raction insid th catalyst. Thy mployd volum avraging and Darcy s law for a spatially priodic porous mdium. Prud homm and Jasmin (Prud homm and Jasmin 6) studid biochmical fr convction flow in a porous mdium with intrnal hat gnration from microbial oxidation, as a simulation of a composting ractor, for Rayligh numbrs qual to.5 and 5. Silva t al. (Silva t al. 7) also usd a volum avraging transport modl to simulat two ractiv procsss in porous mdia charactrizing th porous mdium by diffrnt lngth scals. Zuco Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

3 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr 7 t al. (Zuco t al. 8) rcntly xamind th influnc of chmical raction on th hydromagntic hat and mass transfr boundary layr flow from a horizontal cylindr in a Darcy-Forchhimr rgim using ntwork simulation. Bég t al. (Bég t al. 7) hav also analyzd th chmical raction rat ffcts on stady buoyancy-drivn dissipativ micropolar fr convctiv hat and mass transfr in a Darcian porous rgim. Effcts of chmical raction on fr convction flow of a polar fluid through a porous mdium in th prsnc of intrnal hat gnration ar xamind by Patil and Kulkarni (Patil and Kulkarni 8). Whn hat and mass transfr occur simultanously in a moving fluid, th rlations btwn th fluxs and th driving potntials may b of a mor intricat natur. An nrgy flux can b gnratd not only by tmpratur gradints but by composition gradints also. Th nrgy flux causd by a composition gradint is trmd th Dufour or diffusion-thrmo ffct. On th othr hand, mass fluxs can also b cratd by tmpratur gradints and this mbodis th Sort or thrmal-diffusion ffct. Such ffcts ar significant whn dnsity diffrncs xist in th flow rgim. For xampl, whn spcis ar introducd at a surfac in a fluid domain, with a diffrnt (lowr) dnsity than th surrounding fluid, both Sort (thrmo-diffusion) and Dufour (diffuso-thrmal) ffcts can bcom influntial. Sort and Dufour ffcts ar important for intrmdiat molcular wight gass in coupld hat and mass transfr in fluid binary systms, oftn ncountrd in chmical procss nginring and also in high-spd arodynamics. Dursunkaya and Work (Dursunkaya and Work 99) hav studid Sort/Dufour ffcts on transint and stady natural convction from vrtical surfac. Both fr and forcd convction boundary layr flows with Sort and Dufour hav bn addrssd by Abru t al. (Abru t al. 6). Rcntly Bég t al. (Bég t al. 8) usd th local nonsimilarity mthod with a shooting procdur to analyz mixd convctiv hat and mass transfr from an inclind plat with Sort/Duofur ffcts with applications in solar nrgy collctor systms. Bég t al. (Bég t al. 8) xtndd this study to includ chmical raction ffcts. Anghl t. al. (Anghl t. al. ) hav discussd th composit Dufour and Sort ffcts on fr convction boundary layr ovr a vrtical surfac mbddd in a Darcian porous mdium. Postlnicu (Postlnicu 4) has prsntd numrical solutions for th ffct of magntic fild on buoyancy-drivn hat and mass transfr from a vrtical surfac in a porous mdium including Sort and Dufour ffcts. Bég t al. (Bég t al. 8) obtaind finit lmnt solutions for coupld hat and mass transfr in a micropolar fluid-saturatd Darcy- Forchhimr porous mdium with Sort and Dufour ffcts. Rawat t al. (Rawat t al. 8) considrd th corrsponding problm for a Nwtonian fluid. Th abov studis nglctd th prsnc of thrmal radiation which bcoms important whn high tmpraturs ar rachd. Many chmical nginring and industrial procsss invok simultanous convctiv and radiativ hat transfr mods including combustion chambrs, glass production and mtallurgy. Makind (Makind 5) studid th fr convction flow with thrmal radiation and mass transfr past a moving vrtical porous plat. Sattar and Kalim (Sattar and Kalim 996) invstigatd th transint natural convction-radiation boundary layr flow along a vrtical porous plat. Chamkha t al. (Chamkha t al. ) usd th Blottnr diffrnc mthod to study laminar fr convction flow of air past a smi-infinit vrtical plat in th prsnc of chmical spcis concntration and thrmal radiation ffcts. Duwairi and Damsh (4) analyzd th combind ffcts of forcd and natural convction hat transfr in th prsnc of transvrs magntic fild form a vrtical surfacs with radiation hat transfr. Vry rcntly Ibrahim t al. (Ibrahim t al. 8) studid th ffcts of chmical raction and radiation absorption on transint hydromagntic natural convction flow with wall transpiration and hat sourc. Svral studis hav also dscribd thrmal radiation ffcts on convction flows in porous mdia. Takhar t al. (Takhar t al. 998) usd a nongray gas diffrntial modl to study numrically th radiation ffcts on convctiv boundary Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

4 8 R. Bhargava t al. layr flow in a Darcy-Brinkman-Forchhimr nonlinar porous mdium. Takhar t al. (Takhar t al. 3) latr mployd th Rossland diffusion radiativ algbraic modl to invstigat mixd radiation-convction flow in a non-darcy porous mdium. Sddk t al. (Sddk t al. 7) mor rcntly rportd on th ffct of chmical raction and variabl viscosity on hydromagntic mixd convction hat and mass transfr for Himnz flow through a Darcian porous mdia in th prsnc of radiation and magntic fild. Bég t al. (Bég t al. 8) usd th ntwork thrmodynamic simulation mthod to invstigat th transint fr convction hat and mass transfr boundary layr flow in a Darcy-Forchhimr isotropic porous mdium. Anothr ffct which bars grat importanc on hat transfr procsss is viscous dissipation. Th dtrmination of th tmpratur distribution whn th intrnal friction is not ngligibl is of grat significanc in diffrnt industrial filds, such as chmical and food procssing, oil xploration and bio-nginring. In such scnarios th ffcts of viscous dissipation must b includd in th nrgy quation. Svral authors hav rcntly studid numrically viscous dissipation ffcts in purly fluid and also porous mdia rgims. Zuco (Zuco 7) considrd th transint magntohydrodynamic natural convction boundary layr flow with suction, viscous dissipation and thrmal radiation ffcts. Bég t al. (Bég t al. 8) prsntd th first analysis of dissipativ third grad viscolastic flow in a Darcy-Forchhimr porous domain using finit lmnts. Bég t al. (Bég t al. 9) hav also studid numrically both viscous hating and Joul (Ohmic) dissipation ffcts on transint Hartmann-Coutt convctiv flow in a Darcian porous mdium channl also including Hall currnt and ionslip ffcts. In th prsnt papr, w xtnd th problm invstigatd in Ibrahim t al. (Ibrahim t al. 8) by including viscous dissipation, Darcian porous drag and also Sort and Dufour ffcts. Th xtndd consrvation quations ar solvd using th highly fficint finit lmnt mthod. Such a study constituts an important addition to numrical multi-physical fluid dynamics simulations and has not appard thusfar in th litratur. FORMULATION AND MATHEMATICAL MODELLING W considr th transint, incomprssibl, two-dimnsional, coupld, convctiv hat and mass transfr of a Nwtonian, viscous, lctrically-conducting and hat absorbing fluid along an infinit, porous, vrtical plat mbddd in an isotropic, homognous, Darcian porous mdium in th prsnc of thrmal radiation, viscous dissipation and homognous chmical raction with Sort and Dufour ffcts. Th x-axis is dirctd along th infinit plat and th y-axis is transvrs to this. A magntic fild B of uniform strngth is applid transvrsly to th dirction of th flow. For t, th plat starts moving impulsivly in its own plan with constant vlocity, u p, with plat tmpratur raisd tot and th concntration lvl at th plat raisd to C. CTh physical modl and gomtrical coordinats ar shown in Fig.. A hat sourc is placd within th flow to simulat possibl hat absorption ffcts. Undr th abov assumptions, th physical variabls ar functions of y and t only. With th usual boundary layr and Boussinsq approximations, th govrning quations may b writtn as follows: Continuity Equation: v = y () Momntum Consrvation Equation: Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

5 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr u u u σ Bo ν + v = ν u β g T T β g C C t y y ρ k p ( ) ( ) () 9 Enrgy Consrvation Equation: ν Q DmkT ( ) ( ) p ρ p s p T T T u C + v = α + T T + Q C C + t y y c y c c c y (3) Spcis Diffusion Equation: C C C D k T + v = D m k C C + t y y T y m T l ( ) m (4) In quation () th first trm on th lft hand sid is th tmporal vlocity gradint, th scond is th convctiv inrtial (acclration) trm. On th right hand sid of quation (), th first trm simulats th viscous shar ffcts (momntum diffusion), th first trm in th squar brackts rprsnts th Lorntzian magntohydrodynamic rtarding forc (acting transvrs to th magntic fild), th scond trm in th squar brackts is th Darcian linar porous drag forc, th pnultimat trm on th right hand sid of quation () is th thrmal buoyancy forc and th final trm dsignats spcis buoyancy. In quation (3) th first trm on th lft is th tmporal thrmal gradint, th scond bing th thrmal convctiv trm. On th right hand sid of quation (3), th first trm rprsnts thrmal diffusion, th scond is th viscous dissipation trm (du to intrnal friction in th fluid), th third trm is th hat sourc trm, th fourth is th thrmal radiation sourc trm and th final trm rprsnts Dufour concntration gradint ffcts on th tmpratur fild. In quation (4) th first trm on th lft hand sid signifis th tmporal concntration gradint, and th scond trm is th convctiv mass transfr trm. On th right hand sid of quation (4), th first trm is th mass (spcis) diffusion trm, th scond is th chmical raction ffct trm and th final trm rprsnts th contribution from Sort tmpratur gradints on th concntration fild. Th corrsponding boundary conditions on th vrtical surfac and in th fr stram can b dfind now as follows: nt y = : u = u, v = v( t), T = T + ε ( T T ), C = C + ε ( C C ) p w w w w y : u =, T T, C C (5) whr u is th x -dirction vlocity, v is th y -dirction vlocity, u p is th plat translational vlocity, t is th tim, ρ is fluid dnsity, ν is kinmatic viscosity, σ is th fluid lctrical conductivity, B is th magntic fild intnsity, k p is th prmability of th porous mdium,α is th thrmal diffusivity, Qis th hat absorption cofficint, Q is th radiation absorption cofficint, g is gravity, kl is th chmical raction paramtr, β and β ar th thrmal and concntration xpansion cofficints rspctivly, Dm is cofficint of mass diffusivity, cp is spcific hat at constant prssur, csis th concntration suscptibility, Tm is man fluid tmpratur, kt is thrmal diffusion ratio, T, C ar th dimnsional tmpratur and nt Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

6 R. Bhargava t al. concntration rspctivly, T w, C w ar th wall tmpratur and concntration rspctivly, T andc ar th fr stram tmpratur and concntration rspctivly and n dnots th frquncy of oscillations. Th last trms on th right-hand sid of th nrgy quation (3) and concntration quation (4) signify rspctivly, th Dufour or diffusion-thrmo ffct and th Sort or thrmo-diffusion ffct. W assum that th solution of th quation () taks th following form: nt v = v( t) = V ( + ε A ) (6) whr A is a ral positiv constant, ε is constant ( ε < ) and V is scal of suction vlocity at th plat surfac. 3 TRANSFORMATION OF MODEL To obtain th non-dimnsional form of th govrning quations, w now introduc th following dimnsionlss variabls: u v V y V t u nν T T C C U, V, Y, T, U, N, θ = = = = = = =, φ = V V V V T T C C p p ν ν w w (7) whr U is dimnsionlss x-dirction vlocity, V is dimnsionlss y-dirction vlocity, Y is dimnsionlss y coordinat (normal to th plat), T is dimnsionlss tim, Up is dimnsionlss plat translational vlocity, N is dimnsionlss frquncy of oscillation, θ is dimnsionlss tmpratur function and φ is th dimnsionlss concntration function. Implmnting th transformations (7) into th quations () to (4), w obtain th following dimnsionlss coupld partial diffrntial quations: U NT U U ( + ε A ) = M U Grθ Gmφ T Y Y K (8) θ NT U ( ε A ) θ θ Ec + = + θ Q Φ + φ + Du φ T Y Pr Y Y Y (9) T Y Sc Y y φ NT ( + ε A ) φ = φ γφ + Sr θ () σ Bν whr M = is th magntic fild paramtr simulating th rlativ ffcts of th ρv k pv magntic drag and th viscous hydrodynamic forc, K = is th prmability paramtr ν which rprsnts th hydraulic conductivity of th fluid prcolating in th porous mdium, ν gβ ( Tw T ) Gr = is th Grashof numbr which simulats th rlativ influnc of thrmal 3 V Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

7 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr ( ) * ν gβ Cw C buoyancy to viscous hydrodynamic forcs, Gm = is th solutal (spcis) 3 V Grashof numbr which mbodis th ratio of spcis buoyancy to viscous hydrodynamic ν forcs, Pr = is th Prandtl numbr which rprsnts th ratio of momntum to thrmal α V diffusivity, Ec = is th Eckrt numbr which simulats th rlationship btwn cp ( Tw T ) th kintic nrgy of th flow and nthalpy, and is usd to charactriz Qν Q ( Cw C ) ν dissipation, Φ = is th hat sourc paramtr, Q = is th radiation ρc V T T V p ( ) ν absorption paramtr, Sc = is th Schmidt numbr which dfins th ratio of th shar Dm componnt for diffusivity viscosity/dnsity to th diffusivity for mass transfr i.. th ratio of k momntum diffusivity (viscosity) and mass diffusivity, l ν γ = is th chmical raction V ( ) DmkT Cw C DmkT Tw T paramtr, Du = is th Dufour numbr and Sr = cscp ( Tw T ) νtm Cw C numbr. Th corrsponding transformd boundary conditions ar: NT Y = : U = U, θ = + ε, φ = + ε p NT w ( ) ( ) is th Sort Y : U, θ, φ () In for xampl th dsign of chmical nginring systms, th following paramtrs ar usful to comput. Th skin friction cofficint which signifis th surfac shar strss is dfind as: C f τ u =, τ w = µ = ρv U () ρ y w uw y= i.. C = 4 U () () f Th local Nusslt numbr which mbodis th ratio of convctiv to conductiv hat transfr across (normal to) th boundary and is a quantification of th surfac tmpratur gradint (hat transfr rat at th wall) is dfind as: x T Nu ( x) =, ( T T ) y w y= i.. Nu ( x) = θ () (3) R x Finally th local Shrwood numbr which ncapsulats th ratio of convctiv to diffusiv mass transport and simulats th surfac mass transfr rat, is dfind as: x C Sh( x) =, thn ( C C ) y w y= Sh( x) = φ () (4) R x Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

8 R. Bhargava t al. V x whr R x = is th local Rynolds numbr. ν 4 NUMERICAL SOLUTION BY THE FINITE ELEMENT METHOD Th quations ar solvd using th finit lmnt mthod (FEM) as dscribd by Rddy (Rddy 6). Th authors hav implmntd this xcllnt mthod in a wid spctrum of magntohydrodynamic and also numrous non-magntic transport phnomna problms of intrst in nrgy, chmical and biomchanical nginring systms. For xampl Naroua t al. (Naroua t al. 7) studid th influnc of hat sourc and Hall and ionslip currnts on rotating unstady plasma flow. Takhar t al. (Takhar t al. 7) analysd third grad hydrodynamic flow in a Darcy-Forchhimr porous mdium. Bhargava t al. (Bhargava t al. 7) invstigatd coupld hat and mass transfr in micropolar boundary layr flow from a nonlinar strtching sht. Furthr studis utilizing th prsnt finit lmnt approach includ Bég t al. (Bég t al. 8) who studid hat transfr in biomagntic micropolar flow in Darcy-Forchhimr porous mdia, and Bhargava t al. (Bhargava t al. 8) who considrd micropolar hat and mass transfr from a cylindr. Th prsnt finit lmnt cod has thrfor bn xtnsivly validatd with othr numrical schms including finit diffrnc solvrs, asymptotic mthods and ntwork numrical simulation (Bhargava t al. 7, Bhargava t al. 8). It is basd on a consrvativ approach and dtaild convrgnc tsts hav bn conductd to guarant monotonicity. Furthr dtails ar providd latr in th papr. FEM is xtrmly ffctiv in solving nonlinar multipl dgr partial and ordinary diffrntial quation systms. Th fundamntal stps comprising th mthod ar as follows: Stp : Discrtization of th domain into lmnts: Th whol domain is dividd into finit numbr of sub-domains, a procss known as discrtization of th domain. Each sub-domain is trmd a finit lmnt. Th collction of lmnts is dsignatd th finit lmnt msh. Stp : Drivation of th lmnt quations: Th drivation of finit lmnt quations i.. algbraic quations among th unknown paramtrs of th finit lmnt approximation, involvs th following thr stps: a. Construct th variational formulation of th diffrntial quation. b. Assum th form of th approximat solution ovr a typical finit lmnt. c. Driv th finit lmnt quations by substituting th approximat solution into variational formulation. Ths stps rsults in a matrix quation of th form K { c } = { F } lmnt modl of th original quation. Stp 3: Assmbly of lmnt quations:, which dfins th finit Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

9 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr 3 Th algbraic quations so obtaind ar assmbld by imposing th intr-lmnt continuity conditions. This yilds a larg numbr of algbraic quations, constituting th global finit lmnt modl, which govrns th whol flow domain. Stp 4: Impositions of boundary conditions: Th physical boundary conditions dfind in quation () ar imposd on th assmbld quations. Stp 5: Solution of th assmbld quations: Th final matrix quation can b solvd by a dirct or indirct (itrativ) mthod. For computational purposs, th coordinat y is varid from to y max = 8, whr y max rprsnts infinity i.. xtrnal to th momntum, nrgy and concntration boundary layrs. Th whol domain is dividd into a st of 8 lin lmnts of qual width., ach lmnt bing twonodd. Monotonic convrgnc is achivd. 4.. Variational Formulation Th variational formulation associatd with quations (8) to () ovr a typical two-nodd linar lmnt (, Y + ) is givn by U NT U U Y + w ( + ε A ) + M + U Grθ Gmφ dy = T Y Y K (5) NT U ε θ φ T Y Pr Y Y Y Y + w θ θ θ φ ( + A ) Ec + Φ Q Du dy = (6) NT 3 ε γφ T Y Sc Y Y + w φ ( A ) φ φ Sr θ + + dy = (7) whr w, w and w 3 ar arbitrary tst functions and may b viwd as th variation in U, θ and φ rspctivly. Aftr rducing th ordr of intgration and non-linarity, w arriv at th following systm of quations: U NT U w U w ( + ε A ) w + T Y Y Y U dy w = (8) Y Y Y + M + wu Grwθ Gmwφ + K + Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

10 4 R. Bhargava t al. θ NT θ w θ w ( + ε A ) w + T Y Pr u Y + U U w θ φ Ecw + Φwθ Q wφ dy Duw Y Y + = (9) Pr Y Y Y + + w φ Du u Y φ NT φ w3 φ 3 ( ) 3 Y w + ε A w + T Y Sc Y Y w dy 3 φ θ Srw 3 w Y 3 θ + Sc Y Y = () + + γ w3φ + Sr Y Y + 4. Finit-Elmnt Formulation Th finit-lmnt modl may b obtaind from quations (8) to () by substituting finit lmnt approximations of th form: j j, j j, j j j= j= j= () U = U ψ θ = θ ψ φ = φ ψ with w w w ψ ( i ) = = 3 = i =,, whr U j, θ j and φ j ar th vlocity, tmpratur and concntration rspctivly at j th nod of typical th lmnt (, Y + ) and functions for this typical lmnt (, Y + ) and ar takn as: Ψi ar th shap Y Y, Y ψ = + ψ Y = Y Y Y Y () Th finit lmnt modl of th quations for th lmnt thus formd is givn by { } { θ } { φ } { } { θ } { φ } K K K U M M M U b K K K + M M M = b K K K M M M b { } { } { } whr { K mn, M mn } and {{ U },{ θ },{ φ },{ U }, { θ },{ } { }} φ and b m ( m, n =,,3) ar th st of matrics of ordr and rspctivly and (dash) indicats d/dy. Ths matrics ar dfind as follows: Ψ NT j Ψ Ψ j K ( ) i = + ε A Ψ i dy + dy + M + ( Ψi Ψ j ) dy Y Y Y K Y (3) Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

11 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr = Ψi Ψ j, = Ψi Ψ j K Gr dy K Gm dy + 3 i j M = Ψ Ψ dy, M = M = + U Ψ j K = EcΨi dy Y Y Ψ NT j Ψ Ψ j K ( ) i = + ε A Ψ i dy + dy + Φ ( Ψi Ψ j ) dy Y Pr Y Y + Ψ K Q Du dy 3 Ψi j = Ψi Ψ j +, Y Y + 3 = =, = Ψi Ψ j M M M dy Ψ Ψ i j K =, K = Sr dy Y Y Ψ 33 NT j Ψ Ψ j K ( ) i = + ε A Ψ i dy + dy + γ ( Ψi Ψ j ) dy Y Sc Y Y = =, = Ψi Ψ j M M M dy + + U θ φ i = Ψ i, i i i Y = Ψ + Ψ Y Pr Y Y And b b Du b φ θ = Ψ + SrΨ Sc Y Y i i i + (4) whr U = U i Ψ i. Th whol domain is dividd in to a st of 8 intrvals of qual i = lngth,.. At ach nod 3 functions ar to b valuatd; hnc aftr assmbly of th lmnts, w obtain a st of 43 quations. Th systm of quations aftr assmbly of th lmnts, ar nonlinar and consquntly an itrativ schm is mployd to solv th matrix systm. Th systm is linarizd by incorporating known functionu, which ar solvd using th Gauss limination mthod maintaining an accuracy of.5. 5 VALIDATION AND MONOTONIC CONVERGENCE Bnchmarking of th sourc FEM cod has bn prformd against finit diffrnc mthods. Excllnt agrmnt was found. Dtails hav bn omittd howvr for brvity. Th prsnt program has bn adaptd for ovr 5 diffrnt nonlinar boundary valu problms by th authors and is thrfor xtrmly rliabl. Prvious validations hav bn prformd Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

12 6 R. Bhargava t al. rigorously and compard with publishd rsults in th litratur. Th radr is rfrrd to Agarwal t al. (Agarwal t al. 99), Takhar t al. (Takhar t al. 998), Takhar t al. (Takhar t al. ), Takhar t al. (Takhar t al. ), Bhargava t al. (Bhargava t al. 3) and Bhargava t al. (Bhargava t al. 4). Th lin lmnts mployd achivd rapid convrgnc. A monotonic convrgnc critrion was also stablishd for which th twonods lin ( rod-typ ) lmnts mployd wr slctd to nsur that th msh was compatibl. Whn monotonic convrgnc is achivd th accuracy of th solution rsults in a continuous incras with furthr rfinmnt of th finit lmnt msh. As such msh rfinmnt is xcutd by dlinating a priori utilizd lmnts into two or mor lmnts, rsulting in mbdding in th nw msh. Effctivly, as documntd by Bath (Bath 996) th nw spac of finit lmnt intrpolation functions ncapsulats th prviously utilizd spac and with msh rfinmnt th dimnsion of th finit lmnt solution spac is nhancd continuously to mbody th xact solution. Excllnt convrgnc was achivd in th prsnt study. 6 RESULT AND DISCUSSION W ar primarily intrstd in xamining th influnc of thrmal-diffusion and diffusionthrmo ffcts i.. Sort numbr (Sr) and Dufour numbr (Du) on th flow variabls. Additionally w hav computd th influnc of th magntic paramtr (M), chmical raction paramtr (γ), Eckrt numbr (Ec), Schmidt numbr (Sc), hat absorption paramtr ( Φ ) and radiation absorption paramtr (Q ). Th valus of othr paramtrs ar takn to b fixd as follows: A =.5, ε =., N (dimnsionlss frquncy of oscillations) =., plat translational vlocity (U p ) =.5, Prandtl numbr (Pr) =.7 (air), Schmidt numbr (Sc) =. (hydrogn at 5 dgrs Clsius and atmosphr prssur, following Gbhart and Pra (97)), thrmal Grashof numbr (Gr) = 5, solutal Grashof numbr (Gm) = 5, prmability paramtr ( K ) =.5 and T (dimnsionlss tim) =. Th prmability in all th figurs plottd is st at.5 which corrsponds to a highly porous rgim i.. wak Darcian bulk drag associatd with th mdium fibrs. Such a situation may accuratly simulat th proprtis of foams or loosly arrangd arrays of particls in a filtration matrial rgim. Sc =. physically corrsponds to hydrogn gas diffusing in air. Such data thrfor corrsponds to hydrogn gas diffusing in air prcolating in a highly prmabl isotropic, homognous porous mdium undr th action of wak thrmal and spcis buoyancy forcs. Gnrally wak magntic fild (.g. M =.3 in most of our graphs) is also studid, which ngats th nd to considr Hall currnt or ionslip currnt ffcts, as indicatd by Sutton (965). Th valus of Sr and Du hav bn slctd to nsur that th product Sr Du is constant, assuming that th man tmpratur is constant. In th prsnt analysis for consrvation of spac w hav xcludd plots for th ffcts of Gr, Gm, Pr and K. Th variation of skin friction, local hat transfr paramtr and local mass transfr paramtr with rspct to th Sort numbr (Sr) and Dufour numbr (Du), Eckrt numbr (Ec), chmical raction paramtr (γ), hat absorption paramtr ( Φ ) and radiation absorption paramtr (Q ) ar prsntd in tabls to 3. Tabl indicats that th skin friction cofficint, U () dcrass with a dcras in Sort numbr (Sr) and an incras in Dufour numbr (Du). Th rat of hat transfr, θ (), incrass as Sr dcrass from. to. and.5; thraftr howvr it dcrass with a Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

13 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr 7 subsqunt lowring in Sr from. through to th last valu of.. Th rat of mass transfr, φ (), incrass continuously with a dcras in Sr i.. th maximum mass transfr rat corrsponds to th minimum Sr valu of. (and th maximum Du valu of.75). Tabl : Distribution of skin friction { U () }, th rat of hat transfr { θ () } and th rat of mass transfr { φ () } with diffrnt valus of Sort numbr (Sr) (or Dufour numbr Du) for M=.3, Φ =, Q =, Sc=., γ =, Ec =.. Sr Du U () θ () φ () Tabl indicats that skin friction incrass with an incras in Eckrt numbr (Ec) but dcrass with a ris in chmical raction paramtr (γ). Hat transfr rat howvr dcrass with a ris in Ec but incrass with incrasing γ. Incrasing Eckrt numbr implis mor thrmal nrgy is addd to th fluid so that hat is conductd from th plat into th fluid i.. causing a dcras in hat transfr at th wall. Th rat of mass transfr { φ () } is incrasd both with a ris in Ec and γ. Tabl : Distribution of skin friction { U () }, th rat of hat transfr { θ () } and th rat of mass transfr { φ () } with diffrnt valus of Eckrt numbr (Ec) and Chmical raction paramtr (γ). M =.3, Φ =, Q =, Sc =., γ = Du=.6, Sr =. M=.3, Ec =., Q =, Sc =. Du =.6, Sr =., Φ = Ec U () θ () φ () γ U () θ () φ () Tabl 3 indicats that skin friction, U (), dcrass with an incras in hat absorption paramtr (Φ ) but is nhancd with an incras in th radiation absorption paramtr (Q ). Hat transfr rat, θ (), howvr is strongly boostd with an incras in hat absorption paramtr (Φ ) but is considrably dcrasd with an incras in th radiation absorption paramtr (Q ). Th rat of mass transfr { φ () } is markdly rducd with an incras in hat absorption paramtr (Φ ) but is substantially boostd with an incras in th radiation absorption paramtr (Q ). Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

14 8 R. Bhargava t al. Tabl 3: Distribution of skin friction { U () }, th rat of hat transfr { θ () } and th rat of mass transfr { φ () } with diffrnt valus of hat absorption paramtr ( Φ ) and radiation absorption paramtr (Q ). M =.3, Ec =., Q =, Sc =., γ = Du=.6, Sr =. M=.3, Ec =., γ =, Sc =. Du =.6, Sr =., Φ = Φ U () θ () φ () Q U () θ () φ () Figurs to 4 dpict th spatial distribution through th boundary layr of vlocity, tmpratur and concntration functions at a fixd tim, T =. In figurs to 4 th ffct of chmical raction on th flow variabls is shown. In figur vlocity, U, is clarly boostd with strongr chmical raction i.. as th chmical raction paramtr, γ, incrass from through 5, to 5 (vry high rat), profils ar liftd continuously throughout th boundary layr, transvrs to th plat. A distinct vlocity scalation occurs nar th wall aftr which profils dcay smoothly to th stationary valu in th fr stram. Chmical raction thrfor boosts momntum transfr i.. acclrats th flow. A similar rspons for th non-magntic cas has bn documntd by Chamkha t al. (Chamkha t al. ) in th prsnc of thrmal radiation and latr for th magntohydrodynamic cas (without porous mdia, viscous hating and Sort/Dufour ffcts) by Ibrahim t al. (Ibrahim t al. 8). Tmpratur (θ) and concntration (φ) ar likwis incrasd in figurs 3 and 4, rspctivly, with an incras in th chmical raction paramtr (γ) although profils in both ths cass dscnd from a maximum at th wall (plat surfac) to zro in th frstram i.. th profils ar monotonic dcays. Th prsnc of chmical raction is thrfor assistiv to momntum, hat and mass transfr procsss in th rgim. x, u u p Fluid saturatd Darcian porous mdium v( t) T, C B T, C Smi- infinit Vrtical Porous Plat g y, v Figur : Flow Configuration and coordinat systm Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

15 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr 9 Figur Figur 3 Figur 4 Figurs to 4: Effct of chmical raction paramtr on th dimnsionlss vlocity, tmpratur and concntration for air (Pr =.7) with M =.3, K=.5, Gr = 5, Gm = 5, Φ =, Q =, Ec =., Du =.6, Sr =. and Sc =. for T =. Figurs 5 to 6 illustrat th variation of vlocity, U, and tmpratur function, θ, for various valus of th radiation absorption paramtr (Q ). It is immdiatly apparnt that vlocity (U) as wll as tmpratur (θ) clarly incras as Q riss from to 3. Vlocity rachs a maximum in clos proximity to th wall and thn falls gradually to zro at th dg of th boundary layr. Inspction of Fig. 6 shows that for a small valu of Q (Q <) tmpratur profil continuously dcrass from th wall, whil for highr valus of Q it incrass attaining a maximum nar th plat boundary and thn dcrass. As such thr is a noticabl tmpratur ovrshoot with Q > sinc considrabl thrmal nrgy is impartd via th prsnc of a thrmal radiation sourc to th fluid causing an lvation in tmpraturs nar th wall. Figur 5 Figur 6 Figurs 5 to 6: Effct of radiation absorption paramtr on th dimnsionlss vlocity and tmpratur for air (Pr =.7) with M =.3, K =.5, Gr = 5, Gm = 5, Φ =, γ =, Ec =., Du =.6, Sr =. and Sc =. for T =. Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

16 3 R. Bhargava t al. Figurs 7 to 8 illustrat th variation of vlocity and tmpratur functions with th ffcts of hat absorption paramtr, Φ. U valus ar clarly rducd with incrasing Φ; again an ovrshoot is computd clos to th plat both in th prsnc and absnc of hat absorption. Hat absorption howvr supprsss th ovrshoot. For a non-zro valu of Φ, tmpratur profil, θ, continuously dcrass whil for Φ = a tmpratur ovrshoot occurs vry clos to th wall with tmpratur dcaying continuously thraftr to zro in th fr stram. Figur7 Figur 8 Figurs 7 to 8: Effct of hat absorption paramtr on th dimnsionlss vlocity and tmpratur for air (Pr=.7) with M =.3, K =.5, Gr = 5, Gm = 5, Q =, γ =, Ec =., Du =.6, Sr =. and Sc =. for T = Figurs 9 to show th tmpratur and concntration distributions with collctiv variation in Sort numbr, Sr, and Dufour numbr, Du. Sr rprsnts th ffct of tmpratur gradints on mass (spcis) diffusion. Du simulats th ffct of concntration gradints on thrmal nrgy flux in th flow domain. W obsrv from figur 9 that a ris in Du from.6 to.75 boosts th influnc of spcis gradints on th tmpratur fild, so that θ valus ar clarly nhancd i.. th fluid in th porous mdium is hatd. Th Sr valus fall from. to.8 ovr this rang (th product of Sr and Du must stay constant i...6). Tmpratur continuously dcrass as w mov into th boundary layr. In figur, φ (concntration function) in th Darcian flow is incrasd as Sr incrass from.8 to., i.. mass transfr is boostd as a rsult of th contribution of tmpratur gradints. Ths rsults concur with th trnds in Anghl t al. (Anghl t al ) who considrd th non-magntic Darcian cas and also Postlnicu (Postlnicu 4) who considrd th magntohydrodynamic Darcian cas. Figur 9 Figur Figurs 9 to : Effct of Sort and Dufour numbr (product stays constant i.. at.6) on th dimnsionlss tmpratur and concntration for air (Pr=.7) with M =.3, K =.5, Gr = 5, Gm = 5, Q =, Φ =, γ =, Ec =. and Sc =. for T= Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

17 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr 3 Figur shows that an incras in Eckrt numbr, Ec, from (no viscous hating) through.5, to.5 (vry high viscous hating) clarly boosts tmpraturs in th porous rgim. Eckrt numbr signifis th quantity of mchanical nrgy convrtd via intrnal friction to thrmal nrgy i.. hat dissipation. Incrasing Ec valus will thrfor caus an incras in thrmal nrgy contributing to th flow and will hat th rgim. For all non-zro valus of Ec th tmpratur ovrshoot nar th wall is distinct; this ovrshoot migrats marginally furthr into th boundary layr with an incras in Ec. Figur : Effct of Eckrt numbr on th dimnsionlss tmpratur for air (Pr=.7) with M=.3, K=.5, Gr=5, Gm=5, Φ =, Q =, Du=.6, Sc=., Sr=. and = for T= Figurs to 3 illustrat th tmpratur and concntration fild distributions with transvrs coordinat for diffrnt Schmidt numbr, Sc. An incras in Sc causs a considrabl rduction in tmpratur, θ, in Fig.. A much gratr rduction is obsrvd in concntration valus, φ, in Fig. 3. An incras in Sc will supprss concntration in th boundary layr rgim. Highr Sc will imply a dcras of molcular diffusivity (D) causing a rduction in concntration boundary layr thicknss. Lowr Sc will rsult in highr concntrations i.. gratr molcular (spcis) diffusivity causing an incras in concntration boundary layr thicknss. For th highst valu of Sc =., th momntum and concntration boundary layr thicknsss ar of th sam valu approximatly i.. both spcis and momntum will diffus at th sam rat in th boundary layr. Figur Figur 3 Figurs to 3: Effct of Schmidt numbr on th dimnsionlss tmpratur and concntration for air (Pr=.7) with M=.3, K=.5, Gr =5, Gm = 5, Φ =, Q =, Ec =., Du =.6, Sr =. and γ = for T= Finally figur 4 shows th variation of vlocity function with th magntic fild paramtr, M. Th prsnc of magntic fild in an lctrically-conducting flow crats a drag-lik forc calld th Lorntz forc. This typ of rsistiv forc tnds to slow down th motion of th fluid in th boundary layr i.. dclrats th flow, as shown in figurs 3 whr vlocity (U) clarly dcras as M riss from (lctrically non-conducting cas) through to th maximum magntic fild corrsponding to M =. Th rlativ influnc of magntic fild on Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

18 3 R. Bhargava t al. th fluid tmpratur as wll as on concntration has also bn invstigatd; th rsults ar omittd hrin for brvity, although it has bn found in consistncy with othr publishd studis.g. Sddk t al. (Sddk t al. 7) and Zuco (Zuco 7), that th tmpratur as wll as concntration is slightly incrasd as magntic fild paramtr, M, incrass. Hnc magntic fild hats th fluid and aids in spcis diffusion in th porous rgim. W furthr not that for th cas M =, magntic and viscous forcs will hav th sam ordr of magnitud. As such th Hartmann boundary layr will b formd whn M = and this boundary layr will dcras with incras in M i.. will b lss for M =, as confirmd by Sutton (965). Figur 4: Effct of Magntic paramtr M on th dimnsionlss vlocity for air (Pr =.7) with Ec=., K=.5, Gr = 5, Gm= 5, Φ =, Q =, Sc =., Du =.6, Sr =. and γ = for T= 7 CONCLUSIONS A finit lmnt solution has bn dvlopd for th oscillatory chmically-racting, dissipativ, hydromagntic convction hat and mass transfr in a Darcian porous mdium with hat absorption and thrmal radiation ffcts. Th dimnsionlss solutions hav shown that:. An incras in th chmical raction paramtr, γ dcras th vlocity, tmpratur and concntration valus in th porous rgim.. Incrasing th radiation absorption paramtr, Q, incras th vlocity and tmpratur i.. acclrat and hats th flow throughout th ntir porous rgim. 3. Incrasing th hat absorption paramtr,φ, rducs both vlocity and tmpratur i.. rtards and cools th flow in th porous rgim. Thrfor a dsird tmpratur can b maintaind by controlling th hat absorption ffct in practical chmical nginring applications. 4. Incrasing Dufour numbr, Du (and simultanously rducing Sort numbr, Sr) incrass th tmpratur in th porous mdium 5. Incrasing Sort numbr, Sr, (and simultanously rducing th Dufour numbr, Du) incrass th concntration valus in th porous rgim i.. nhancs spcis diffusion. 6. Incrasing th magntic paramtr (M) dcrass th vlocity in th rgim. Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

19 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr Incrasing Eckrt numbr (Ec) hats th porous rgim i.. incrass tmpratur 8. Incrasing Schmidt numbr (Sc) rducs both tmpratur and concntration valus in th porous rgim. 9. Th skin friction cofficint, U () dcrass with a dcras in Sort numbr (Sr) i.. incras in Dufour numbr (Du), dcrass with a ris in chmical raction paramtr (γ), dcrass with an incras in hat absorption paramtr (Φ ) but incrass with an incras in Eckrt numbr (Ec) and an incras in th radiation absorption paramtr (Q ).. Th rat of hat transfr, θ (), incrass initially as Sr dcrass and thn dcrass with a subsqunt lowring in Sr, dcrass with a ris in Ec, strongly dcrass with an incras in th radiation absorption paramtr (Q ), but incrass with largr chmical raction paramtr valus (γ) and also with an incras in hat absorption paramtr (Φ ).. Th rat of mass transfr, φ (), incrass continuously with a dcras in Sr i.. (and an incras in Du), and is also incrasd with a ris in Ec, γ and also with an incras in th radiation absorption paramtr (Q ); howvr it is dcrasd with an incras in hat absorption paramtr (Φ ) 8 ACKNOWLEDGMENTS On of th authors (R. Sharma) would lik to thank Ministry of Human Rsourc Dvlopmnt (MHRD), Govrnmnt of India, for its financial support through th award of a rsarch grant. Th authors ar also gratful to th rviwrs for thir commnts which hav hlpd to improv th articl. REFERENCES Abru CRA, MF Alfradiqu and A Silva Tlls (6). Boundary layr flows with Dufour and Sort ffcts: I: Forcd and natural convction. Chmical Enginring Scinc, 6, 3, pp Agarwal RS, R Bhargava, and AVS Balaji (99). Finit lmnt solution of nonstady thrdimnsional micropolar fluid flow at a stagnation-point. Intrnational Journal of Enginring Scinc, 8, 8, pp Al-Nimr MA and Hadr MA (999). MHD fr convction flow in opn-ndd vrtical porous channls. Chmical Enginring Scinc, 54 (), pp Anghl M, HS Takhar, and I Pop (). Dufour and Sort ffcts on fr convction boundary layr ovr a vrtical surfac mbddd in a porous mdium. Studia Univrsitatis- Bolyai, Mathmatica, XLV, 4, pp. -. Bég OA (6). Hat Transfr Modlling, Tchnical Rport, Lds Mtropolitan Univrsity, Lds, UK. Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

20 34 R. Bhargava t al. Bég OA, HS Takhar, M Kumari, and G Nath (). Computational fluid dynamics modling of buoyancy-inducd viscolastic flow in a porous mdium with magntic fild ffcts. Int. J. Applid Mchanics and Enginring, 6 (), pp Bég OA, R Bhargava, S Rawat, HS Takhar and TA Bég (7). A study of buoyancy-drivn dissipativ micropolar fr convction hat and mass transfr in a Darcian porous mdium with chmical raction, Nonlinar Analysis: Modling and Control J., (), pp Bég OA, R Bhargava, S Rawat, HS Takhar and K Halim (8). Computational modling of biomagntic micropolar blood flow and hat transfr in a two-dimnsional non-darcian porous mdium, Mccanica, 43 (4), pp Bég OA, J Zuco, TA Bég, HS Takhar and E Kahya (8). NSM analysis of tim-dpndnt nonlinar buoyancy-drivn doubl-diffusiv radiativ convction flow in non-darcy gological porous mdia, Acta Mchanica, in prss. Bég OA, HS Takhar, R Bharagava, Rawat S, and Prasad VR (8). Numrical study of hat transfr of a third grad viscolastic fluid in non-darcy porous mdia with thrmophysical ffcts, Physica Scripta: Proc. Royal Swdish Acadmy of Scincs, 77, pp. -. Bég OA, Bharagava R, Rawat S and Kahya E (8). Numrical study of micropolar convctiv hat and mass transfr in a non-darcy porous rgim with Sort and Dufour diffusion ffcts. Emirats Journal for Enginring Rsarch,3 (), pp Bég OA, Zuco J, Bhargava R, and Takhar HS (8). Magntohydrodynamic convction flow from a sphr to a non-darcian porous mdium with hat gnration or absorption ffcts: ntwork simulation. Int. J. Thrmal Scincs, in prss. Bég OA, AY Bakir and V Prasad (8). Laminar mixd convction in hat and mass transfr in boundary layr flow along an inclind plat with Sort and Dufour ffcts, Math. Computr Modlling J., undr rviw. Bég OA, AY Bakir and V Prasad (8). Chmically-racting mixd convctiv hat and mass transfr along inclind and vrtical plats with Sort and Dufour ffcts: Numrical Solutions, Int. J. Applid Mathmatics and Mchanics, accptd, Dcmbr. Bég OA, J Zuco and HS Takhar (9). Unstady magntohydrodynamic Hartmann-Coutt flow and hat transfr in a Darcian channl with Hall currnt, ionslip, viscous and Joul hating ffcts: ntwork numrical solutions, Communications Nonlinar Scinc Numrical Simulation, 4 (4), pp Bhargava R, Kumar L and HS Takhar (3). Finit lmnt solution of mixd convction micropolar flow drivn by a porous strtching sht, Intrnational Journal of Enginring Scinc, 4 (8), pp Bhargava R, RS Agarwal, L Kumar and HS Takhar (4). Finit lmnt study of mixd convction micropolar flow in a vrtical circular pip with variabl surfac conditions. Intrnational Journal of Enginring Scinc, 4 (), pp Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

21 Oscillatory Chmically-Raction MHD Fr Convction Hat and Mass Transfr 35 Bhargava R, S Sharma, HS Takhar, OA Bég and P Bhargava (7). Numrical solutions for micropolar transport phnomna ovr a nonlinar strtching sht, Nonlinar Analysis: Modling and Control Journal, (), pp Bharagava R, OA Bég, S Rawat, and J Zuco (8). Numrical modlling of micropolar hydrodynamics, hat and mass transfr in axisymmtric stagnation flow with variabl thrmal conductivity and Rynolds numbr ffcts. Mathmatical and Computr Modlling Journal, undr rviw. Chamkha AJ, HS Takhar and VM Soundalgkar (). Radiation ffcts on fr convction flow past a smi-infinit vrtical plat with mass transfr. Chmical Enginring Journal, 84 (3), pp Dahikar SK and RL Sonolikar (6). Influnc of magntic fild on th fluidization charactristics of circulating fluidizd bd. Chmical Enginring Journal, 7 (3), pp Dursunkaya Z and WM Work (99). Diffusion-thrmo and thrmal-diffusion ffcts in transint and stady natural convction from vrtical surfac. Intrnational Journal of Hat and Mass Transfr, 35, pp Duwairi HM and Damsh RA (4). MHD-buoyancy aiding and opposing flows with viscous dissipation ffcts from radiat vrtical surfacs, Canadian Journal of Chmical Enginring, 8, pp Gindrau C and Auriault JL (). Magntohydrodynamic flows in porous mdia. J. Fluid Mchanics, 466, pp Gatica JE, Viljon HJ, and Hlavack V (989). Intraction btwn chmical raction and natural convction in porous mdia. Chmical Enginring Scinc, 44 (9), pp Gbhart B and L Pra (97). Th natur of vrtical natural convction flows rsulting from th combind buoyancy ffcts of thrmal and mass diffusion, Intrnational Journal of Hat and Mass Transfr, 4, pp Ibrahim FS, AM Elaiw and AA Bakr (8). Effct of th chmical raction and radiation absorption on th unstady MHD fr convction flow past a smi infinit vrtical prmabl moving plat with hat sourc and suction. Communications Nonlinar Scinc Numrical Simulation, 3 (6), pp Iliuta I and Larachi F (3). Magntohydrodynamics of trickl bd ractors: Mchanistic modl, xprimntal validation and simulations, Chmical Enginring Scinc, 58 (), pp Iliuta I and Larachi F (3). Two-phas flow in porous mdia undr spatially uniform magntic-fild gradints: novl way to procss intnsification. Canadian Journal of Chmical Enginring, 8, pp Lutisan J, Cvngros J and Miroslav M (). Hat and mass transfr in th vaporating film of a molcular vaporator. Chmical Enginring Journal, 85 (-3), pp Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.

22 36 R. Bhargava t al. Klinstrur C and TY Wang (989). Mixd convction hat and surfac mass transfr btwn powr-law fluids and rotating prmabl bodis, Chmical Enginring Scinc, 44(), pp Makind OD (5). Fr convction flow with thrmal radiation and mass transfr past a moving vrtical porous plat. Intrnational Communications in Hat and Mass Transfr, 3, pp Mihail R and C Todorscu (978). Catalytic raction in a porous solid subjct to a boundary layr flow. Chmical Enginring Scinc, 33(), pp Naroua H, HS Takhar, PC Ram, TA Bég, OA Bég, and R Bhargava (7). Transint rotating hydromagntic partially-ionizd hat-gnrating gas dynamic flow with Hall/Ionslip currnt ffcts: finit lmnt analysis. Intrnational Journal of Fluid Mchanics Rsarch, 34 (6), pp Patil PM and PS Kulkarni (8). Effcts of chmical raction on fr convction flow of a polar fluid through a porous mdium in th prsnc of intrnal hat gnration. Intrnational Journal of Thrmal Scincs, 47, pp Postlnicu A (4). Influnc of a magntic fild on hat and mass transfr by natural convction from vrtical sufacs in porous mdia considring Sort and Dufour ffcts. Intrnational Journal of Hat and Mass Transfr, 47, pp Probstin RF (989). Physico-Chmical Hydrodynamics: An Introduction, Buttrworth- Hinmann Sris in Chmical Enginring, Boston, USA. Prud homm M and S Jasmin (6). Invrs solution for a biochmical hat sourc in a porous mdium in th prsnc of natural convction, Chmical Enginring Scinc, 6 (5), pp Rawat S, R Bhargava, OA Bég, and TA Bég (8). Numrical simulation of Nwtonian hat and mass transfr in a non-darcian porous rgim with Sort/Dufour thrmal/diffusion ffcts, Applid Thrmal Enginring, undr rviw. Rddy JN (6). An Introduction to th Finit Elmnt Mthod, McGraw-Hill Book Company, Nw York, 3 rd Edition. Sattar MDA. and Kalim MDH (996). Unstady fr-convction intraction with thrmal radiation in a boundary layr flow past a vrtical porous plat. Journal of Mathmatical and Physical Scincs, 3, pp Sddk MA, AA Darwish and MS Abdlmguid (7). Effcts of chmical raction and variabl viscosity on hydromagntic mixd convction hat and mass transfr for Himnz flow through porous mdia with radiation, Communications in Nonlinar Scinc and Numrical Simulation, (), pp Silva EA t al. (7). Prdiction of ffctiv diffusivity tnsors for bulk diffusion with chmical ractions in porous mdia, Brazilian Journal of Chmical Enginring,.4 (), pp Int. J. of Appl. Math and Mch. 5 (6): 5-37, 9.