Heat and Mass Transfer on the Unsteady MHD Flow of Chemically Reacting Micropolar Fluid with Radiation and Joule Heating

Transcription

1 Inernaional Jornal of heoreical and Alied Mahemaics 7; (): - h://.scienceblishinggro.com//iam doi:.68/.iam.7. Hea and Mass ransfer on he nsead MHD Flo of hemicall Reacing Microolar Flid ih Radiaion and Jole Heaing Shamshddin MD. Dearmen of Mahemaics aagdei ollege of Engineering Warangal India address: shammahs@gmail.com o cie his aricle: Shamshddin MD. Hea and Mass ransfer on he nsead MHD Flo of hemicall Reacing Microolar Flid ih Radiaion and Jole Heaing. Inernaional Jornal of heoreical and Alied Mahemaics. ol. o doi:.68/.iam.7. Receied: March 7; Acceed: Aril 7; Pblished: Ma 7 Absrac: o exlore nmerical simlaion of ransor in rheological maerials rocessing in he crren aer a finie elemen comaional solion is resened for magneohdrodnamic (MHD) incomressible radiaie and chemicallreacing microolar flid flo hea and mass ransfer adacen o a erical oros lae embedded in a saraed homogenos oros medim. Rosseland s diffsion aroximaion is sed o describe he radiaie hea flx in he energ eqaion. A Darc model is emloed for he oros medim. he homogeneos chemical reacion of firs order is acconed for in he mass diffsion eqaion. he nmerical solions of he ssem of non-linear arial differenial eqaions hich are rendered ino non-dimensional form are obained sing a Galerkin formlaion ih a eighed residal scheme. he imac of Eringen coling nmber radiaion-condcion nmber chemical reacion arameer lae moing eloci arameer magneic arameer hermal Grashof nmber secies (solal) Grashof nmber ermeabili arameer Ecker nmber on linear eloci micro-roaion emerare and concenraion rofiles. Frhermore he inflence of seleced hermo-hsical arameers on fricion facor srface hea ransfer and mass ransfer rae is also ablaed. he finie elemen solions are erified ih solions from seeral limiing cases in he lierare. Ineresing feares in he flo are idenified and inerreed. Keords: Radiaie Hea ransfer hemical Reacion Jole Dissiaion Boanc Microolar Flid FEM. Inrodcion he flo of non-eonian flid feare idel in an exensie range of echnological alicaions inclding food rocessing lasic fabricaion bioechnolog and ain emlsion manfacre. o simlae he comlex shear sresssrain characerisics of sch flids nmeros mahemaical models hae been deeloed. Researchers in his area ere iniiaed b Eringen [] inrodced he microflid model and laer simlified his model o microolar flids hich can describe sohisicaed henomena inclding cole sresses bod coles and exhibi graor moions hich canno be analzed ih simler non-eonian models and conined o no in arios case sdies. Man recen asecs of microolar hdrodnamics are docmened in Eringen [] and Lkasiascz []. Exensie discssion of oher alicaions in chemical and mechanical engineering are aailable in he aricles of Airman e al. [ 5]. he crren sd is relean o high emerare elecromagneic rheological flos in energ generaors and magneo-rheological maerials fabricaion ssems (here hermal radiaion hea ransfer is also significan). I is knon ha he arise in man dierse areas of echnolog inclding combsion in gas rbines conecie flos se here he bonding srfaces absorb hea b solar radiaion design of efficien hea exchangers ec. hese flos reqire a more sohisicaed aroach o radiaie hea ransfer in he ssem hich can sbsaniall inflence erformance and modif characerisics of manfacred rodcs. In addiion o his sch regimes are srongl inflenced b hermal bondar condiions. Earlier Rais and Perdikis [6] resened he resence of radiaion in iscoelasic flo. An imoran analsis in his regard as resened b Abo-Eldahab and Ghonaim [7] ho addressed hermal radiaion effecs in hea ransfer of a microolar flid hrogh a oros medim. Rahman and Slan [8] imlemened an efficien ieraie

2 Shamshddin MD.: Hea and Mass ransfer on he nsead MHD Flo of hemicall Reacing Microolar Flid ih Radiaion and Jole Heaing finie difference mehod o sd he hermal radiaion ineracion of he bondar laer flo of microolar flid as a heaed erical oros lae embedded in a oros medim ih ariable scion as ell as hea flx a he lae. Ibrahim e al. [9] ealaed he iscos dissiaion and radiaion effecs on mixed conecion flo of microolar flid. Sdheer Bab e al. [] described he effecs of mass ransfer on nsead magneo-conecion flo of microolar flid along a erical moing oros lae hrogh oros medim ih iscos dissiaion. Frhermore deailed reies of oher inesigaions ino radiaie hea ransfer in microolar flos are roided b Redd [] Olaon e al. [] and Mamaha e al. []. In he aboe inesigaions he combined effecs of Jole dissiaion and chemical reacion in hdromagneic microolar ransor hae been exclded. In mos simlaions of magneic hea ransfer he Jole dissiaion erm is conenionall negleced on he remise ha nder normal condiions he Ecker nmber is small based on an order of magnide analsis also in man indsrial rocesses e.g. maerials fabricaion of oders chemical reacion ma exer an inflenial role. hese chemical reacions ake lace hich ma be desrcie or consrcie in nare and can inflence significanl hea and mass diffsion henomena. Generall bondar laer flo models ilize firs order chemical reacion effecs and assme he reacion o be desrcie. Rahman [] sdied he effecs of iscos dissiaion and Jole heaing in conecie flos of a microolar flid obsering ha hea ransfer raes are decreased ih increasing Jole heaing effec. Haqe e al. [5] examined he sead magneic naral conecion hea ransfer in microolar flid ih Jole heaing and iscos dissiaion. In nmeros rocess engineering ssems chemical reacions ake lace hich ma be desrcie or consrcie in nare and can inflence significanl hea and mass diffsion henomena. Generall bondar laer flo models ilize firs order chemical reacion effecs and assme he reacion o be desrcie. Seeral inesigaions hae considered reacie hea and mass ransfer in exernal bondar laer flos for microolar and oher flids. Sheri and Shamshddin [6] hae addressed he roblem of coled hea and mass ransfer in magneohdrodnamic microolar flo ih boh iscos dissiaion and chemical reacion effecs. Sheri and Shamshddin [7] hae frher resened finie elemen nmerical solions for diffsohermal and chemical reacion effecs on ransien free conecion microolar flo. Recenl Aano and Mahna [8] sdied he combined effec of chemical reacion and radiaion on microolar flid flo oer erical lae ih ariable emerare. An ineresing analsis of exohermic and endohermic kind of chemical reacion oer microolar flid as inesigaed b Koriko e al. [9]. Frher sdies of reacie microolar flos inclde Pal and alkdar [] Pal and Bisas [] Modaher e al. []. In he resen aricle moiaed b simlaing non- eonian hermal maerials rocessing of oders e exended he analical ork of Sdheer Bab e al. [] b aking ino accon of hermal radiaion Jole dissiaion and firs order chemical reacion effecs and deriing finie elemen nmerical solions for generalized microolar radiaie-conecion flo from a erical srface in a oros medim. he errbaion aroximaion form solions resened b Sdheer Bab e al. [] roide a benchmark for he resen finie elemen comaional solions. he effecs of arios emerging hermo-hsical arameers on he eloci micro-roaion (anglar) eloci emerare and concenraion rofiles as ell as on local skin fricion coefficien and all cole sress is isalized and ablaed. he crren roblem o he bes knoledge of he ahors has no been commnicaed hsfar in he echnical lierare.. Mahemaical Model Free conecie flo of an elecricall condcing incomressible micro-olar flid as a erical lane in he resence of hermal radiaion Jole dissiaion and chemical reacion is considered. he lane considered is ermeable and is moing ih consan eloci in a oros medim. he hsical configraion is illsraed in Fig.. he Eringen [] microolar model is emloed since i sccessfll cares microsrcral characerisics of comlex magneic olmers i.e. i robsl simlaes roaor moions graion of flid micro-elemens. Microolar flids can sor cole sresses shear sresses bod coles and also exhibi microroaional effecs and ineria. Darc s la is assmed and lo Renolds nmber flo (iscos-dominaed regime). A magneic field of niform srengh B is alied in a direcion arallel o he axis hich is erendiclar o he flo direcion. I is assmed ha he indced magneic field is negligible in comarison o he alied magneic field []. Magneic Renolds nmber is er small. he magneohdrodnamic (MHD) bod force erm is deried from an order of magnide analsis of he fll aier-sokes eqaion. I is also assmed ha alied or olarized olage is negleced so ha no energ is added or exraced from he flid b elecrical means. he flid is considered o be a gra absorbing-emiing b non-scaering medim and he Rosseland aroximaion is sed o describe he radiaie hea flx. he radiaie hea flx in he x direcion is considered negligible in comarison ih ha of direcion. Boh all emerare and concenraion ar ih he disance along he lae and he are alas greaer han heir niform ambien ales exising far from he lae srface. he magneic microolar flid conains a secies hich is reacie and obes firs order chemical reacion. o simlif he formlaion of he bondar condiions e assmed he size of holes in he oros lae is significanl larger han he characerisic microscoic lengh scale of he microolar flid. I is assmed ha he lae is infinie in exen and hence all hsical qaniies deend onl on and. Bossinesq aroximaion is aken in momenm eqaion. he balances of mass linear momenm anglar momenm energ and concenraion secies in he aresian frame are rien as follos: he conini eqaion:

3 Inernaional Jornal of heoreical and Alied Mahemaics 7; (): - Figre. Geomer and coordinae ssem of he roblem. () ( ) ( ) ( ) r B K g g r () γ () q r B α () ( ) K D m (5) Where and are he eloci comonens in x and axis reseciel. is he micro-roaion comonen is he kinemaic iscosi r is he kinemaic microroaion iscosi is he consan flid densi is he elecrical condcii of he microolar flid g is he acceleraion de o grai ( ) and ( ) denoe he hermal and concenraion boanc effecs reseciel B is he srengh of he ranserse magneic field is he secific hea a consan ressre K is he ermeabili of he oros medim m D is he moleclar diffsii of secies and c K is he dimensional chemical reacion rae consan. Imlici in he resen analsis is consan ermeable lae eloci in he direcion of he flid flo. he aroriae iniial and bondar condiions for eloci anglar eloci (microroaion) emerare and concenraions fields are { ( ) ( ) > as a n e n e P all for ε ε (6)

4 Shamshddin MD.: Hea and Mass ransfer on he nsead MHD Flo of hemicall Reacing Microolar Flid ih Radiaion and Jole Heaing Where is lae eloci i is clear from he eqaion of conini ha scion eloci normal o he lae is eiher a consan or a fncion of ime. Hence i is assmed ha he scion eloci akes he form: (7) Where is a scale of scion eloci and >. he negaie sign indicaes ha he scion eloci is oards he lae. he radiaie hea flx erm is gien b q r k (8) Here and k are he Sefan-Bolzmann consan and mean absorion coefficien reseciel. he assmed Rosseland model has been shon o be generall alid for oicall-hick flid media as considered in iscos flids []. Imlemening eqn. (8) resls in a highl nonlinear energ eqaion in and i is difficl o obain a solion. Hoeer researchers hae resoled his roblem b assming small emerare differences ihin he flid flo [5 6]. In his siaion Rosseland s model can be linearized abo ambien emerare assming ha he difference in he emerare ihin he flo is sch ha can be exressed as a linear combinaion of he emerare. sing alor s series exansion abo he exansion of can be rien as follos neglecing higher order erms: ( ) ( )... 6 (9) eglecing higher order erms beond he firs degree in ( ) e hae () Differeniaing eqaion (8).r. and sing () e obain: 6 k r q () o siml relacing in Eq. (8) ih Eq. () can be exressed as follos: 6 B k κ α () In order o rie he goerning eqaions and bondar condiions in dimensionless form he folloing non-dimensional qaniies are inrodced: ( ) ( ) ( ) Λ Λ Ec k R g Gm g Gr c K Kc m D Sc Pr K K B M o n n 6 κ κ α γ λ γ ϕ θ () In ie of eqaion () he basic field eqaions ()- (5) can be exressed in non-dimensional form as ( ) ϕ θ B Gm Gr () λ (5) Γ M Ec θ θ θ (6) ϕ ϕ ϕ ϕ Kc Sc (7) Here Pr R Γ. K M B

5 Inernaional Jornal of heoreical and Alied Mahemaics 7; (): - Where Gr Gm M K P r Ec Sc Kc denoes he Eringen microolar orex iscosi arameer Grashof nmber Modified Grashof nmber Magneic field arameer Permeabili of he oros medim Prandl nmber hermal radiaion arameer Hea absorion > : : { θ εen θ arameer iscos dissiaion Schmid nmber and hemical reacion arameers reseciel. he corresonding iniial and bondar condiions in nondimensional form as follo ϕ for all ϕ εen a (8) he mahemaical saemen of he roblem is no comlee and embodies he solion of Eqs. ()- (7) ih modified bondar condiions (8). he ssem is ellosed.. Mehod of merical Solion he finie elemen mehod (FEM) is emloed o sole he ransformed coled bondar ale roblem defined b eqns. () - (7) nder (8). FEM is he mos olar and adaable mehod aailable o engineers. he general deails of he ariaional finie elemen mehod are described a lengh in Redd [7] Rao [8] and Bahe [9]. FEM has been alied o sd man ransor roblems of microolar flids and magneic liqids. he fndamenal ses inoled in he finie-elemen analsis of a roblem are as follos: Discreizaion of he infinie flid domain ino finie elemens: he hole domain is diided ino a finie nmber of sb domains rocesses knon as discreizaion of he domain. Each sb domain is ermed as finie elemen. he collecion of elemens is hen denoed he finie-elemen mesh. Deriaion of elemen eqaions: he deriaion of finie elemen eqaions i.e. algebraic eqaions among he nknon arameers of he finie elemen aroximaion inoles he folloing hree sages. A ical elemen is isolaed from he mesh and consrcing he ariaional formlaion of he differenial eqaion on ha ical elemen. Assme he form of he aroximae solion oer a ical finie elemen. Derie he finie elemen eqaions b sbsiing he aroximae solion ino ariaional formlaion. hese ses resl in a marix (knon as siffness marix) eqaion of he form [ K ]{ e e } { F e } hich defines he finie elemen model of he original eqaion. Assembl of elemen eqaions: he algebraic eqaions so obained are assembled b imosing he iner-elemen conini condiions (i.e. he ales of he nodal ariables a he nodes are idenical for o or more elemens). his ields a large nmber of algebraic eqaions knon as he global finie elemen model. his goerns he hole flo domain. Imosiion of bondar condiions: he iniial and final bondar condiions defined in eqaion (8) are imosed on he aboe obained assembled eqaions. Solion of assembled eqaions: he final marix eqaion obained can be soled b an efficien ieraie scheme. In one-dimensional sace linear and qadraic elemens or higher order can be aken. Here he enire flo domain is considered b diiding i ino sccessiel sized grids of order 8x8 x and x in he -axis direcion. Afer man ess a grid size ih inerals has been adoed. hs all he comaions are execed ih inerals of eqal se size.. A each node fncions are o be ealaed and afer assembl of he elemen eqaions a se of non-linear eqaions are obained hich necessiae an ieraie solion sbec o he secified bondar condiions. he ieraie rocess is erminaed hen he folloing condiion is me: n n 6 ξ ξ i here ξ θ and ϕ are eloci along x axis microroaion emerare and concenraion reseciel. n denoe he ieraie se. o see he effecs of se size (h) he finie elemen code is rn ih se sizes as h. and er good agreemen is obained for differen rofiles. he skin fricion coefficien cole sress coefficien ssel nmber and Sherood nmber are imoran arameers for his e of bondar laer flo and freqenl sed in maerials rocessing simlaions and design. he skin-fricion a he lae in non-dimensional form is gien b: f (9) he cole sress coefficien a he lae in nondimensional form is gien b: () ssel nmber is comed in non-dimensional form as: θ / Re () x Sherood nmber is ealaed as in non-dimensional form b:

6 5 Shamshddin MD.: Hea and Mass ransfer on he nsead MHD Flo of hemicall Reacing Microolar Flid ih Radiaion and Jole Heaing ϕ Sh / Re () x Here o x Re x is he local Renolds nmber based on he lae scion eloci. able. ariaions of in Re and Sh Re for differen GrGm and M. f x x Gr Gm M f Re Sh Re x x able. ariaions of in Re and Sh Re for differen Pr REcand Sc. f x x Pr R Ec Sc f Re Sh Re x x able. Effecs of Kc on Re andshre for ih n. ε M Gr Gm Pr 7. R Ec.. f x x arameer ales f Re Sh Re x x Kc able and able docmen he fricion facor srface hea ransfer and mass ransfer rae deendenc on he emerging hermo-hsical arameers. able deics he effec of Gr Gm M on f / Re Sh/ Re x x reseciel. I is obsered ha he skin fricion decreases as Gmincreases hile i increases as Gr increases. As M decreases skin fricion increases. he same rend is obsered in case of all cole sress. Frher i is obsered ha he ssel nmber decreases as Gr Gmincreases b as M increases ssel nmber decreases. Sherood nmber has no significan effec on GrGm M. able deics he effec of Pr R Ec Sc on f / Re Sh/ Re x x reseciel. he skin fricion coefficien decreases as Pr EcR increases hile i decreases as Sc decreases. he same rend is comed in he case of all cole sress (all micro-roaion gradien). Frher i is obsered ha he ssel nmber decreases as Rdecreases. ssel nmber increases as Sc decreases hile i increases as Pr Ec increases. Sherood nmber decreases as Scdecreases o angible modificaion is comed in Sherood nmber (all mass ransfer rae) ih a change in Pr Ec R. merical ales of he coefficiens roorional o he skin fricion f cole sress coefficien ssel nmber and Sherood nmber Sh are gien in able for he general model ih all arameers inoked. I is eiden from able homogeneos chemical reacion arameer Kc increase he skin fricion coefficien f and all cole sress coefficien boh decrease. Also i is aaren ha as chemical reacionkc increases a significan increase is comed in Sherood nmber Sh.. Resls and Inerreaion he nonlinear bondar ale roblem soled in he reios secion is dicaed b an exensie nmber of hermal and hdrodnamic arameers. In order o gain a clear insigh ino he hsical roblem nmerical calclaions for disribion of he eloci microroaion (anglar) eloci emerare and concenraion for differen ales of hese arameers is condced ih grahical illsraions (Figres -7). For he rose of or comaions e adoed he folloing defal arameers: ε. n.. 5. Gr

7 Inernaional Jornal of heoreical and Alied Mahemaics 7; (): - 6 Gm M K Pr. 7 Sc. 6 R. Ec. Kc. and all grahs herefore corresond o hese ales nless secificall indicaed oherise on he aroriae grah. he ermeabili in all he figres loed is se a.5 hich corresonds o a highl oros regime characerisic of cerain maerials rocessing ssems. he ale of Pr is aken o be.7 hich corresonds o air a and amosheric ressre and he ale of Sc is.6 (aer-aor). merical ales of he coefficiens roorional o he skin fricion f cole sress roorional o he skin fricion f cole sress coefficien ssel nmber and Sherood nmber Sh are gien in able able and able for he general model ih all arameers inoked. Figres - illsrae he resonse of microroaion eloci arameer on eloci and micro-roaion. I is seen ha as increases he eloci gradien near he oros lae decreases and hen aroaches o he free sream eloci. Also i is noeorh ha eloci disribion across he bondar laer is loer for eonian flid ( ) as comared ih sronger microolar flid (.) for he same condiions and flid roeries. Microolari (i.e. increasing orex iscosi of micro-elemens) herefore consisenl indces deceleraion in he flo adacen o he lae. All rofiles are arabolic and eak a some disance from he all decaing smoohl o anish in he free sream. In addiion he magnide of microroaion a he all is decreased as increases. Hoeer he disribion of microroaion across bondar laer does no sho consisen ariaions ih increase of. he effec of hermal radiaion-condcion arameer (R ) on emerare is resened in Figre. his arameer is defined as R 6 / k k and feares in he agmened hermal diffsion erm in eqn. (6) i.e. / ( / ) Γ θ. I defines he relaie conribion of hermal radiaion hea ransfer o hermal condcion hea ransfer. When R < hermal condcion dominaes. When R boh hermal condcion and hermal radiaion conribions are eqal. For R > hermal radiaion dominaes oer hermal condcion. In he resen simlaions e confine aenion o he las of hese hree cases i.e. R > herein hermal radiaie flx is sbsanial. Increasing radiaion-condcion arameer is fond o decrease emerares in he bondar laer. hermal bondar laer hickness is herefore redced ih greaer ales of R. Figre 5 reresen he inflence of chemical reacion arameer ( Kc ) on concenraion rofiles. he reacion arameer is based on a firs-order irreersible chemical reacion hich akes lace boh in he blk of he flid (homogeneos) as ell as a he lae hich is assmed o be caalic o chemical reacion. We consider he desrcie e of homogenos chemical reacion. I is noiced ha concenraion disribions decrease hen he chemical reacion increases. Phsicall for a desrcie case chemical reacion akes lace and rogressiel desros he original secies. his in rn sresses moleclar diffsion of he remaining secies hich leads o a fall in concenraion magnides and a decrease in concenraion bondar laer hickness. Figre 6 illsraes he inflence of Prandl nmber (Pr) on emerare rofiles. Prandl nmber reresens he relaie rae of momenm diffsion o energ diffsion. WihPr > he momenm diffsion rae also exceeds he hermal diffsion rae in he flid. I is obsered ha here is a srong deression in emerare ih greaer Prandl nmber (Pr) greaer Prandl nmber corresonds o a loer hermal condcii. his leads o a redcion in hermal energ coneced hrogh he flid from he lae (Gr > i.e. lae cooling) and deresses he hermal bondar laer hickness. Figre 7 resen he effecs of he iscos dissiaion arameer i.e. he Ecker nmber Econ emerare field. Ecker nmber signifies he qani of mechanical energ conered ia inernal fricion o hermal energ i.e. hea dissiaion. Increasing Ec ales ill herefore case an increase in hermal energ conribing o he flo and ill hea he regime. Posiie Ecker nmber imlies cooling of he all and herefore a ransfer of hea o he microolar flid also conecion is enhanced. emerares are markedl increased ih greaer Ecker nmber. For all nonzero ales of Ec he emerare oershoo near he all is disinc; his oershoo migraes marginall frher ino he bondar laer ih an increase in Ec. er smooh decas in emerare rofiles are obsered for all ales of Ecker nmber and he conergence of rofiles in he free sream indicaes ha an adeqael large infini bondar condiion has been imosed in he finie elemen model. he rofiles of he eloci and microroaion in he bondar laer for arios ales of he lae moing eloci are shon in Figres 8-9 in he direcion of he flid flo. I is noiced ha he eak ale of eloci across he bondar laer increases near he oros lae as he lae eloci increases. he resls also sho ha he magnide of microroaion on oros lae decreases as increases. he linear flo is herefore acceleraed ih greaer lae eloci hereas he micro-roaion (anglar flo) of micro-elemens is inhibied i.e. deceleraed. Figres - reresens he inflence of Grashof nmber Gr and modified Grashof nmber Gm on eloci and microroaion rofiles. he hermal Grashof nmber Gr qanifies he relaie magnide of he boanc force and he oosing fricional (iscos) forces acing on he microolar flid. Phsicall he osiie negaie and zero ( i.e. Gr> Gr< and Gr ) ales of he Grashof nmber corresond o cooling heaing of he bondar srface and absence of free conecion crrens reseciel. he secies (solal) Grashof nmber i.e. Gm embodies he relaie conribion of secies boanc force o iscos hdrodnamic force. I is obsered ha he eloci increases as Gr or Gm increases. Frhermore he eak ale of eloci increases raidl near he all of he oros lae. Hoeer he conerse behaior is comed in he case of micro-roaion rofiles. hermal and secies boanc

8 7 Shamshddin MD.: Hea and Mass ransfer on he nsead MHD Flo of hemicall Reacing Microolar Flid ih Radiaion and Jole Heaing herefore modif linear and anglar eloci fields in a differen fashion ih differen imlicaions for bondar laer hicknesses. Figres -5 sho he aern of he eloci and microroaion for differen ales of magneic field arameer M. I is obsered ha he amlide of he eloci as ell as he bondar laer hickness decreases hen M is increased. Phsicall in magneohdrodnamic maerials rocessing he alied magneic field exers a rearding effec on he free conecie flo ranserse o he direcion of imosiion of he magneic field. Wih increasing he ales ofm his e of resising force slos don he flid i.e. ih sronger magneic field srengh he flo is deceleraed and his is confirmed ih he decreasing eloci disribion across he bondar laer. In case of figre 5 an increase in magneic arameer is obsered o significanl accelerae he anglar eloci i.e. enhance he magnide of micro-roaion alhogh he effec is more localized a he lae srface and rogressiel gros frher from he lae. In boh figres and 5 asmoicall smooh solions are obained indicaing ha a sfficienl large infini bondar condiion is rescribed in he free sream. Linear momenm bondar laer hickness is herefore increased ih greaer magneic arameer hereas anglar momenm bondar laer hickness is redced. Figres 6-7 isalize he effec of he oros medim ermeabili arameer (K) on boh eloci and microroaion fields. his arameer characerizes he hdralic ransmissii of he oros medim. Wih increasing ermeabili he regime he qani of solid fibers rogressiel decreases. he Darcian blk imedance o flo is herefore also decreased. his resls in acceleraion in he eloci as obsered in Fig. 6. his behaior is ssained across he bondar laer i.e. for all ales of ranserse coordinae. I is also aaren ha micro-roaion i.e. anglar eloci is enhanced ih greaer ermeabili arameer alhogh he effec is more rominen near he lae srface and is eakened ih frher disance ino he bondar laer. Since he ermeabili arameer does no arise in he anglar momenm conseraion (bondar laer) eqn. (5) he acceleraing effec on micro-roaion is ssained ia he boos in linear momenm exerienced hrogh he coling erms hich link boh linear and anglar momenm fields. he increase in ermeabili imlies greaer oid sace in he oros medim. his allos an enhancemen in graor moions as he micro-elemens are afforded greaer sace in hich o sin. he concenraion rofiles for differen ales of Schmid nmber Sc are illsraed in Figres 8. he Schmid nmber embodies he raio of he momenm o he mass diffsii i.e. Sc / D. he Schmid nmber herefore qanifies he relaie effecieness of momenm and mass ransor b diffsion in he hdrodnamic (eloci) and concenraion (secies) bondar laers. ForSc > momenm diffsion rae exceeds he secies diffsion rae. he oosie alies forsc <. For Sc boh momenm and concenraion (secies) bondar laers ill hae he same hickness and diffsii raes ill be eqal. Hoeer i is aaren ha secies (concenraion) rofiles gradall increase ih higher Schmid nmber. Smaller ales of Sc are eqialen o increasing he chemical moleclar diffsii and ice ersa for larger ales of Sc. Figre. eloci rofiles for ario ales of. Figre. Microroaion rofiles for arios ales of. Figre. emerare rofiles for arios ales of R.

9 Inernaional Jornal of heoreical and Alied Mahemaics 7; (): - 8 Figre 5. oncenraion rofiles for arios ales of Kc. Figre 8. eloci rofiles for arios ales of. Figre 6. emerare rofiles for arios ales of Pr. Figre 9. Microroaion rofiles for arios ale. Figre 7. emerare rofiles for arios ales of Ec. Figre. eloci rofiles for arios ales of Gr.

10 9 Shamshddin MD.: Hea and Mass ransfer on he nsead MHD Flo of hemicall Reacing Microolar Flid ih Radiaion and Jole Heaing Figre. Microroaion rofiles for arios ales ofgr. Figre. eloci rofiles for arios ales of M. Figre. eloci rofiles for arios ales of Gm. Figre 5. Microroaion rofiles for arios ales ofm. Figre. Microroaion rofiles for arios ales of Gr. Figre 6. eloci rofiles for arios ales of K.

11 Inernaional Jornal of heoreical and Alied Mahemaics 7; (): - increases. Increasing Ecker nmber Ec eleaes emerare and enhances hickness of hermal bondar laer. Increasing Schmid nmber eleaes concenraion and enhances he hickness of he secies bondar laer. Increasing homogeneos chemical reacion arameer decreases concenraion and redces concenraion bondar laer hickness. Sherood nmber (all mass ransfer rae) is enhanced ih increasing ermeabili and homogeneos chemical reacion. he finie elemen code deeloed has resoled efficienl he nonlinear microolar ransor henomena in erical lae magneohdrodnamic hea and mass ransfer. Fre sdies ill consider magneic indcion effecs and ill be reored soon. Figre 7. Microroaion rofiles for arios ales of K. Acknoledgemen he ahor graefll acknoledges he referees for heir consrcie commens hich imroed he qali of he aer. References [] A.. Eringen heor of microolar flids J. Al. Mah. Mech. ol [] A.. Eringen Micro-coninm field heories II Flen media Sringer eork (). [] G. Lkaszeicz Microolar Flids Modelling and Simlaion Birkhaser Boson Boson (999). Figre 8. oncenraion rofiles for arios ales of Sc. 5. onclsions A mahemaical model has been resened for radiaie magneic free conecion hea and mass in ransien flo of an incomressible microolar flid from an inclined lae in oros media. Hea sorce/sink and homogeneos chemical reacion effecs hae been inclded in he formlaion. he conseraion eqaions for momenm anglar momenm (micro-roaion comonen) energ and concenraion hae been non-dimensionlized ih aroriae ariables. he resling non-linear ransien coled ssem of arial differenial eqaions and se of iniial and bondar condiions has been soled nmericall sing he ariaional finie elemen mehod ih a Galerkin eighed residal scheme. omaions hae been execed in MALAB sofare and hae shon ha he flo is acceleraed and momenm bondar laer hickness decreased ih increasing ales of Gr Gm K and Ec b in case of M R Sc and Kc he flo is deceleraed and momenm bondar laer hickness increased. Anglar eloci (Microroaion) is sressed as Gr Gm and K increases conersel anglar eloci is eleaed ih M []. Ariman M. A. rk. D. Sleser Alicaions of Micro coninm flid mechanics-a reie In. J. Eng. Sci. ol [5]. Ariman M. A. rk. D. Sleser Alicaions of Micro coninm flid mechanics-a reie In. J. Eng. Sci. ol [6] A. Rais. Perdikis iscoelasic flo b he resence of radiaion. ZAMP ol [7] E. M. Abo-Eldahab A. F. Ghonaim Radiaion effec on hea ransfer of a microolar flid hrogh a oros medim A. Mah. om ol.69 o [8] M. M. Rahman Y. Slana Radiaie hea ransfer flo of microolar flid ih ariable hea flx in oros medim onlinear Analsis Modelling and onrol ol [9] F. S. Ibrahim A. M. Elai A. A. Bakr Inflence of iscos dissiion and radiaion on nsead MHD mixed conecion flo of microolar flids. Alied Mahemaics & informaion Sciences-An Inernaional Jornal ol. o [] M. Sdheer Bab J. Girish Kmar. Shankar redd Mass ransfer effecs on nsead MHD conecie flo of a microolar flid as a erical moing oros lae hrogh oros medim ih iscos dissiaion. In. J. Al. Mah. Mech. ol. 9 o

12 Shamshddin MD.: Hea and Mass ransfer on he nsead MHD Flo of hemicall Reacing Microolar Flid ih Radiaion and Jole Heaing [] M. G. Redd Mass ransfer effecs on he insead MHD radiaie-conecie flo of a microolar flid as a erical oros lae ih ariable hea and mass flx Jornal of Engineering Phsics and hermohsics ol. 86 o [] B. I. Olaon J. I. Oahimire M. Ferdo Effec of hermal radiaion and Hall crren on hea and mass ransfer of nsead MHD flo of a iscoelasic microolar flid hrogh a oros medim. Engineering Science & echnolog-an Inernaional Jornal ol [] B. Mamaha M.. Ra S.. K. arma hermal diffsion effec on MHD mixed conecion nsead flo of microolar flid as a semi-infinie erical oros lae ih radiaion and mass ransfer. In. J. Eng. Reaserch in Africa ol [] M. M. Rahman onecie flos of microolar flids from radiaie isohermal oros srfaces ih iscos dissiaion and Jole heaing ommn. onlinear. Sci. Simla. ol [5] MD. Zial Haqe MD. Mahmd Alam M. Ferdos A. Poselnic Microolar flid behaiors on sead MHD free conecion and mass ransfer flo ih consan hea and mass flxes Jole heaing and iscos dissiaion Jornal of King Sad niersi-engineering Sciences ol [6] S. Sia Redd MD. Shamshddin Hea and mass ransfer on he MHD flo of a microolar flid in he resence of iscos dissiaion and chemical reacion Procedia Eng. ol [7] S. Sia Redd MD. Shamshddin Diffsion-hermo and chemical reacion effecs on an nsead MHD free conecion flo in a microolar flid. heoreical and Alied Mechanics ol [8] M. S. Aano J. S. Mahna hemical reacion and radiaion effecs on nsead MHD microolar flid flo oer a erical lae ih ariable emerare. Froniers in Hea and Mass ransfer ol. 7 o. 6. doi:.598/hm.7.9. [9] O. K. Koriko A. J. Omoae I. L. Animasan I. O. Babande Bondar laer analsis of exohermic and endohermic kind of chemical reacion in he flo of non- Darcian nsead microolar flid along an infinie erical srface. In. J. Eng. Reaserch in Africa ol doi:.8/.scienific.ne/jera.8-9. [] D. Pal B. alkdar Perrbaion echniqe for n-sead MHD mixed conecion eriodic flo hea and mass ransfer in microolar flid ih chemical reacion in he resence of hermal radiaion enral Eroean J. Phsics ol [] D. Pal S. Bisas Perrbaion analsis of magneohdrodnamics oscillaor flo on conecie-radiaie hea and mass ransfer of microolar flid in a oros medim ih chemical reacion Engineering Science and echnolog: An Inernaional Jornal ol [] M. Modaher A. M. Rashad A. J. hamkha An analical sd of MHD hea and mass ransfer oscillaor flo of a microolar flid oer a erical ermeable lae in oros medim rkish Jornal of Engineering and Enironmenal Sciences ol. o []. G. oling Magneohdrodnamics Wile Iner Science eork (957). [] Bég O. Anar merical mehods for mli-hsical magneohdrodnamics e deelomens in Hdrodnamics Reaserch Maximiano J. Ibragimo and A. Anisimo Eds. h.- oa Science: e York (). [5]. Adnson B. Gebhar An exerimenal and analical sd of naral conecion ih areciable hermal radiaion effecs J. Flid Mechanics ol [6] R. orell A nmerical ackling on Sakiadis flo ih hermal radiaion. hin Phsics Le: [7] J.. Redd An Inrodcion o he Finie Elemen Mehod McGra-Hill e York (985). [8] S. R. Rao he Finie Elemen Mehod in Engineering nd Ediion BP Wheaon s Ld. Exeer SA (989). [9] K. J. Bahe Finie elemen rocedres Prenice-Hall e Jerse SA (966). Biograh Shamshddin MD. is orking as an Assisan Professor of Mahemaics in he Dearmen of Basic Sciences and Hmaniies aagdei ollege of Engineering (Aonomos) Warangal India. He did his MSc. Mahemaics from Osmania niersi Hderabad in and obained M. Phil from S niersi irai in. He has ears of eaching exerience. His maor field of sd is omaional Flid Dnamics MHD non-eonian flo Hea and mass ransfer flo hrogh oros media and Bondar ale roblems. He blished more han fie naional and inernaional ornals in Mahemaics and Mahemaical Phsics.