Viscous Dissipation Effects on Radiative MHD Boundary Layer Flow of Nano fluid Past a Wedge through Porous Medium with Chemical Reaction

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1 IOSR Journal of Mathmatics (IOSR-JM) -ISSN: , p-issn: X. Volum 1, Issu 5 Vr. IV (Sp. - Oct.016), PP iosrjournals.org Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nano fluid Past a Wdg through Porous Mdium ith Chmical Raction Syam Sundar Majty 1,, K. Gangadhar 3 1 (Dpartmnt of Nanotchnology, Acharya Nagarjuna Univrsity, Andhra Pradsh, India) (Rsarch Scholar, Dpartmnt of Mathmatics, Acharya Nagarjuna Univrsity, Andhra Pradsh, India) 3 ( Dpartmnt of Mathmatics, Acharya Nagarjuna Univrsity, Ongol, Andhra Pradsh, India) Abstract: Magntohydrodynamics (MHD) boundary layr flo past a dg through porous mdium ith th influnc of thrmal radiation, hat sourc, viscous dissipation and chmical raction has bn analyzd in th prsnt study. Th govrning nonlinar boundary layr quations for momntum, thrmal nrgy and concntration ar transformd to a systm of nonlinar ordinary coupld diffrntial quations by using suitabl similarity transformation. Th transmutd modl is shon to b controlld by a numbr of thrmophysical paramtrs, viz. th magntic paramtr, hat gnration paramtr, Prmability paramtr, Porosity paramtr, Mass and Thrmal convctiv paramtrs, Prandtl numbr, Lis numbr, Bronian motion paramtr, thrmophorsis paramtr, chmical raction paramtr and prssur gradint paramtr. Numrical lucidations ar obtaind ith th lgndary Nactshim-Sigrt shooting tchniqu togthr ith Rung Kutta six ordr itration schms. Kyords: Porous mdium; MHD; dg; radiation; viscous dissipation; chmical raction I. Introduction It is no ll accptd fact that th study of MHD boundary layr flo ith thrmal radiation and chmical raction in hat and mass transfr has bn ncountrd in various nginring, tchnological and manufacturing applications, such as principls of calculation of flo in piping systm and oil and gas piplins, th flo in various pumps, comprssors and turbo-machins, problms of hat and mass transfr as found in hat xchangrs, boilrs and condnsrs, applications in th fild of agricultural nginring, chmical nginring for filtration and purification procsss and applications in txtil chmical nginring, clothing and thrmo physics. At high tmpraturs, thrmal radiation can significantly affct th hat transfr and th tmpratur distribution in th boundary layr flo of participating fluid. Thrmal radiation and chmical raction ffcts may play an important rol in controlling hat transfr in industry hr th quality of th final product dpnds on hat controlling factors to som xtnt. Manjula t al. [1] invstigatd a boundary layr analysis to study th combind ffcts of thrmal radiation and chmical raction on MHD flo; hat and mass transfr ovr a strtching surfac. Sulochana t al. [3] discussd th influnc of radiation and chmical raction ffcts on MHD thrmosolutal nanofluid flo ovr a vrtical plat in porous mdium. Ramachandra t al. [15] rportd on th influnc of radiation and mass transfr ffcts on to-dimnsional flo past an impulsivly startd isothrmal vrtical plat. Hat and mass transfr ffcts on MHD oscillatory flo in a channl filld ith porous mdium in prsnc of radiation and chmical raction as drivd by Ibrahim t al. [7]. Viscous dissipation ffcts, hich ar usually charactrizd by th Eckrt numbr, ar important in gophysical flos and also in crtain industrial applications such as oil products transportation through ducts. Th study of hat transfr in hydrodynamic boundary layr flo ovr a porous strtching sht bcoms mor intrsting hn intrnal hat sourc or absorption occurs. Hat sourc is important in th contxt of xothrmic or ndothrmic chmical ractions Srinivasasai t al. [0] analyzd th study of chmical raction and radiation ffcts on MHD flo ovr an xponntially strtching sht ith viscous dissipation and hat sourc. Th ffcts of hat sourc on MHD convctiv flo past a strtching sht in a micropolar fluid in prsnc of radiation as studid by Mabood t al. [11]. Suntha t al. [4] rportd radiation and mass transfr ffcts on MHD fr convctiv dissipativ fluid in th prsnc of hat sourc/sink. Chamkha [3] invstigatd thrmal radiation and buoyancy ffcts on hydromagntic flo ovr an acclrating prmabl surfac ith hat sourc or sink. In anothr articl, Shamsuddin t al. [18] dscribd a numrical study on MHD boundary layr flo ith viscous dissipation and chmical raction ovr a strtching plat in porous mdium. Rcntly, Hadibandhu and Mohapatra [6] studid Dufour and radiation ffcts on MHD boundary layr flo past a dg through a porous mdium ith hat sourc and chmical raction. Similarly, Rout t al. [16] invstigatd th influnc of DOI: / iosrjournals.org 71 Pag

2 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. chmical raction on MHD hat and mass transfr fluid flo ovr a moving vrtical plat in prsnc of hat sourc ith convctiv boundary conditions. On th othr hand, th rsarch of nanofluids has gaind hug attntion in rcnt yars. Choi [4] as th first to introduc th ord nanofluid that rprsnts th fluid in hich nano scal particls hos diamtr is lss than 100 mm. rcntly; svral numrical studis on th modling of natural convction hat transfr in nanofluids ar publishd. Eshtu and Shankar [5] prsntd a study on hat and mass transfr through a porous mdia of MHD flo of nanofluids ith thrmal radiation, viscous dissipation and chmical raction ffcts. Srinivasacharya t al. [1] analyzs th stady laminar MHD flo, hat and mass transfr charactristics in nanofluid ovr a dg in th prsnc of a variabl magntic fild. Shakhaoath t al. [17] rportd on th MHD radiativ boundary layr flo of a nanofluid past a strtching sht. Chamkha [] studid radiation ffcts on mixd convction ovr a dg mbddd in a porous mdium filld ith nanofluid. Similarly, Wahiduzzaman t al. [5] invstigatd viscous dissipation and radiation ffcts on MHD boundary layr flo of a nanofluid past a strtching sht. Sugunamma t al. [] prdictd th ffcts of radiation and magntic fild on th flo and hat transfr of a nanofluid in a rotating fram. Ahmad and Khan [1] mployd a quasi linarization tchniqu in studying unstady hat and mass transfr MHD nanofluid ovr a strtching sht ith hat sourc. Our main objctiv is to xtnd th analysis of Shakhaoath t al [19]. This study finds th ffct of viscous dissipation and hat sourc on MHD boundary layr flo past a dg through porous mdium ith thrmal radiation and chmical raction. Th govrning quations ar solvd numrically using th Nacthshim- Sigrt shooting itration tchniqu and Rung-Kutta six ordr itration schm and dtrmin th vlocity, tmpratur and concntration profils. II. Mathmatical Modl W considr th to dimnsional MHD laminar boundary layr flo past through a porous mdium of an imprmabl strtching dg in prsnc of thrmal radiation, hat gnration and chmical raction and moving ith th vlocity u() x in a nanofluid, and th fr stram vlocity is u( x ), hr x is th coordinat masurd along th surfac of th dg (Khan and Pop [8]. Hr u( x) 0 corrsponds to a strtching dg surfac vlocity and u( x) 0 corrsponds to a contracting dg surfac vlocity, rspctivly. Instantanously at tim t>0, tmpratur of th plat and spcis concntration ar raisd to T C rspctivly, hich ar thraftr maintaind constant, hr T, C ar T and C T C tmpratur and spcis concntration at th all and, aay from th plat rspctivly. A strong magntic fild B 0, B,0 DOI: / iosrjournals.org 7 Pag ar tmpratur and spcis concntration far 0 is applid in th y-dirction. Undr th abov assumptions and usual boundary layr approximation, th MHD mixd convctiv nanofluid flo govrnd by folloing quations (Nild and Kuzntsov [14], Kuzntsov and Nild [9] u v 0, x y (1) u v du u B0 u v u g T T T gc C C u u x y dx y k () 3 T T T Q 16 1T T u T C DT T u v T T D B x y y Cp 3 kcp y Cp y y y T y (3) C C C DT T u v DB k rc C (4) x y y T y Hr in quation () th 3rd trm on th right hand sid is th convction du to thrmal xpansion and gravitational acclration, th 4th trm on th right hand sid is th convction du to mass xpansion and gravitational acclration, th 5th trm gnratd by th magntic fild strngth bcaus a strong magntic fild B 0, B,0 is applid in th y-dirction and 6th trm rprsnts th porous mdium. Again in quation (3) 0 th nd trm on th right hand sid is th ffct of hat gnration on tmpratur flo, 3rd trm on th right

3 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. hand sid xprssd th radiativ hat transfr flo, th scond trm on th right sid is th viscous-dissipation trm and th last trm indicats th Bronian motion du to nanofluid hat and mass transfr flo. Also in quation (4) th nd trm on th right hand sid is th thrmophorsis diffusion trm du to nanofluid flo and th 3rd trm is th rat of chmical raction on th nt mass flos. And th boundary condition for th modl is; u u ( x) u ( x), v = 0,,, at y = 0 T T C C u u ( ), x T T, C C, as y hr u and v ar x (along th sht) and y (normal to th sht) componnts of th vlocitis rspctivly, g is th acclration du to gravity, T is th fluid tmpratur, T is th ambint tmpratur as y tnds to infinity, T is th tmpratur at th dg surfac, C is th nano particl concntration, C is th ambint nano particl concntration as y tnds to infinity, k is th prmability of porous mdium, nanoparticl concntration at dg, D B is th Bronian diffusion cofficint, diffusivity, is th cofficint of thrmal xpansion, T (5) C is th C p is th spcific hat capacity, D is th cofficint of mass diffusivity, DT is th thrmophorsis diffusion cofficint, is th thrmal C is th cofficint of mass xpansion, 1 is th Stfan-Boltzmann constant, k is th man absorption cofficint, is th lctric conductivity of th fluid, is th dnsity of th fluid, is th ratio btn th ffctiv hat capacity of th nano particl matrial and hat capacity of th fluid, is th constant moving paramtr and kr is th raction rat paramtr. In ordr to conqurs a similarity solution to qs. (1) to (4) ith th boundary conditions (5) th folloing similarity transformations, dimnsionlss variabls ar adoptd in th rst of th analysis; 1 mu y uxv T T, f C C xv, 1 m T T C C and u, v y x (6) For th similarity solution of quation (1) to (4) ith considring th valu (from th proprtis of m m dg, s Lin and Lin [10]) u ( x) ax, u ( x) cx, 0m 1 ar positiv hr a, c and m constant. Thrfor, th constant moving paramtr is dfind as c, hras 0 rlats to a a strtching dg, 0 rlats to a contracting dg, and 0 corrsponds to a fixd dg, rspctivly. From th abov transformations th non-dimnsional, nonlinar, coupld ordinary diffrntial quations ar obtaind as; f f f f M K f (7) ( ) ( ) ( ) 1 ( ) T ( ) M( ) ( ) ( ) 0 1 R ( ) f ( ) ( ) Nb ( ) ( ) Nt ( ) Q ( ) Ecf ( ) 0 Pr Nt ( ) ( ) L f ( ) ( ) R L ( ) 0 (9) N b Th transformd boundary conditions ar as follos; f 0, f, 1, 1 at 0 f 1, 0, 0 as (10) hr th notation prims dnot diffrntiation ith rspct to and th paramtrs ar dfind as: (8) DOI: / iosrjournals.org 73 Pag

4 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. Magntic fild paramtr, numbr, N b Ec u CpT T Cp DB C C p Cp f B0 M mu 1 hat gnration paramtr, D Qx 0 Q C m 1 u p, Eckrt, Lis numbr, L, Bronian motion paramtr,, Thrmophorsis paramtr, paramtr, m m 1 G m R 16 T C k v p 1, Chmical raction paramtr,, Grashof numbr, T Gr m1 v 3 gcc C x m1 v Gm M, Prmability paramtr R numbr, P r, Hat gnration. g T T x, Thrmal convctiv paramtr, B N t k r u 1 m 3 Cp DT T T p T Cp, Radiation f., Prssur gradint paramtr,, Modifid Grashof numbr, G T R x K, Local Rynolds numbr, kp (1 m ) u r, Mass convctiv paramtr, xu R v, Prandtl For th typ of flo undr considration, th main physical quantitis of intrst ar th skin-friction cofficint f (0), th local Nusslt numbr (0), hich rprsnt th all shar strss and th hat transfr rat at th surfac and local Shrood numbr (0), hich rprsnt th all shar strss and th mass transfr rat th surfac. III. Numrical Solution Th dimnsionlss Eqs. (7) - (9) ar coupld nonlinar partial diffrntial quations, hich ar to b solvd by using boundary conditions (10). It is difficult to obtain th analytical solution of this st of quations ith prscribd boundary conditions. So, hr an attmpt has bn mad to sk approximat solutions to this st of quations. To obtain th numrical solutions of Eqs. (7) - (9) for vlocity, tmpratur and concntration fild, along ith boundary conditions (10), hav usd standard initially valu solvr th shooting mthod. For th purpos of this mthod, th Nactshim-Sigrt shooting itration tchniqu (Nachtshim and Sigrt [13] togthr ith Rung Kutta six ordr itration schm is takn and dtrmins th vlocity, tmpratur and concntration as a function of th coordinat η. Philip R. Nachtshim and Paul Sigrt prsntd a mthod for th numrical solution of th diffrntial quations of th boundary layr typ in For th purpos of prsnt problm, applid th Nacthshim-Sigrt itration tchniqu. In shooting mthod, th missing (unspcifid) initial condition at th initial point of th intrval is assumd and th diffrntial quation is intgratd numrically as an initial valu problm to th trminal point. Th accuracy of th assumd missing initial condition is thn chckd by comparing th calculatd valu of th dpndnt variabl at th trminal point ith its givn valu thr, if a diffrnc xists, anothr valu of th missing initial condition must b assumd and th procss is rpatd. This procss is continud until th agrmnt btn th calculatd and th givn condition at th trminal point is ithin th spcifid dgr of accuracy. For this typ of itrativ approach, on naturally inquirs hthr or not thr is a systmatic ay of finding ach succding (assumd) valu of th missing initial condition. Th boundary conditions quation (10) associatd ith th ordinary nonlinar diffrntial quations of th boundary layr typ is of th to-point asymptotic class. To point boundary conditions hav valus of th dpndnt variabl spcifid at to diffrnt valus of th indpndnt variabl. Spcification of an asymptotic boundary condition implis th valu of vlocity approachs to unity and th valu of tmpratur approachs to zro as th outr spcifid valu of th indpndnt variabl is approachd. Th mthod of numrically intgrating to-point asymptotic boundary valu problm of th boundary layr typ, th initial valu mthod, rquirs that th problm b rcast as an initial valu problm. Thus it is ncssary to st up as many boundary conditions at th surfac as thr ar at infinity. Th govrning diffrntial quations ar thn intgratd ith DOI: / iosrjournals.org 74 Pag

5 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. ths assumd surfac boundary conditions. If th rquird outr boundary condition is satisfid, a solution has bn achivd. Hovr, this is not gnrally th cas. Hnc a mthod must b dvisd to logically stimat th n surfac boundary conditions for th nxt trial intgration. Asymptotic boundary valu problms such as thos govrning th boundary layr quations ar furthr complicatd by th fact that th outr boundary condition is spcifid at infinity. In th trial intgration infinity is numrically approximatd by som larg valu of th indpndnt variabl. Thr is no a priori gnral mthod of stimating this valu. Slction of too small a maximum valu for th indpndnt variabl may not allo th solution to asymptotically convrg to th rquird accuracy. Slcting a larg valu may rsult in divrgnc of th trial intgration or in slo convrgnc of surfac boundary conditions rquird satisfying th asymptotic outr boundary condition. Slcting too larg a valu of th indpndnt variabl is xpnsiv in trms of computr tim. Nachtshim- Sigrt dvlopd an itration mthod, hich ovrcoms ths difficultis. From th procss of numrical computation, th skin-friction cofficint, Nusslt numbr and Shrood numbr hich ar rspctivly proportional to f (0), (0) and (0) ar also sortd out and thir numrical valus ar prsntd in a tabular form. IV. Rsults and Discussion In ordr to invstigat th physical rprsntation of th problm, th numrical valus of vlocity ( f ), tmpratur (θ) and concntration (φ) hav bn computd for rsultant principal paramtrs as th Magntic paramtr M, prmability paramtr K, prssur gradint paramtr β, constant moving paramtr l, Thrmal convctiv paramtr λ T, Mass convctiv paramtr λ M, local Rynolds numbr R, Prandtl numbr Pr, Viscous dissipation paramtr Ec, Hat sourc paramtr Q, Lis numbr L, Bronian motion paramtr N b, thrmophorsis paramtr N t, radiation paramtr R and chmical raction paramtr γ rspctivly. In th prsnt study, th folloing dfault paramtr valus ar adoptd for computations: M =0.5, K = 0.5, 6.0, 3.0, Q = 0.5, R = 1.0, Pr = 0.71, N b = 0.3, N t = 0.3, 1.0, L = 3.0, R = 1.0, 1.0, T M Ec = 0.1. All graphs thrfor corrspond to ths valus unlss spcifically indicatd appropriatly. In ordr to tst th accuracy of th prsnt mthod, our rsults ar compard ith thos of Whit [6], Yih [8], Yacob t al. [7], Khan and Pop [8] and Shakhaoath Khan t al. [19] (for rducd cass) and found that thr is an xcllnt agrmnt, as shon in Tabl1. Figur rvals th ffcts of thrmal convctiv paramtr λ T on th vlocity profil. It is obsrvd that incrass in th thrmal convctiv paramtr strongly acclrat th flo thrby incrasing th momntum boundary layr. Figur 3 rprsnts th vlocity profil for diffrnt valus of mass convctiv paramtr λ M. An incras in mass convctiv (spcis buoyancy) paramtr, as shon in Fig. 3, gnrally hats th momntum boundary layr and causs th vlocity fluid to incras in th nanofluid. Figur 4 dpicts th influnc of magntic fild paramtr M on vlocity profil. It is vidnt that th vlocity profil dcrass ith incras in magntic fild strngth. Th ffct of magntic fild on an lctrically conducting fluid giv ris to a rsistiv typ forc calld Lorntz forc. This forc has tndncy to slo don th motion of th fluid in th boundary layr thicknss. Figur 5 shos th ffct of th prmability paramtr K on vlocity profil ith diffrnt valus of constant moving paramtr l. It is clar that th incras in K lads to dcras in th vlocity of th nanofluid on th porous all and so nhanc th momntum boundary layr thicknss. Figur 6 xhibits th vlocity profil for diffrnt valus of prssur gradint paramtr β. An incras in th prssur gradint paramtr is obsrvd to strongly rtard th flo. Thrfor th vlocity dcrass ithin th boundary layr as prssur gradint incras. In figurs 7 and 8, vlocity and tmpratur profils ar plottd for diffrnt valus of Hat sourc paramtr Q. From th graphs it is clar that thr is an incras in both th vlocity and tmpratur profils of th fluid ith an incras in hat sourc paramtr. This is bcaus hn hat is absorbd, th buoyancy forcs and th thrmal diffusivity of th fluid incras, hich acclrat th flo rat and thrby giv ris to an incras in th vlocity and tmpratur profils. Figur 9 and 10 illustrats th ffct of radiation paramtr R on th vlocity and tmpratur profils of th flo. It is obsrvd a hik in th vlocity and tmpratur profils for highr valus of th radiation paramtr. Gnrally, an incras in th radiation paramtr rlass th hat nrgy to th flo. This causs to dvlop th momntum and thrmal boundary layr thicknss. Th influnc of viscous dissipation paramtr Ec on th vlocity, tmpratur and concntration profils is rportd in figur 11, 1 and 13 rspctivly. Th Eckrt numbr mbodis th convrsion of kintic nrgy into intrnal nrgy by ork don against th viscous fluid strss. It is obsrvd that gratr viscous dissipativ hat causs an incras in th vlocity and tmpratur profils across th boundary layr as shon in figur 11 and 1, hil rvrs trnd is obsrvd in figur 13 hrby th concntration profil gradually dcras ith incras in viscous dissipation paramtr. DOI: / iosrjournals.org 75 Pag

6 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. Figur 14 and 15 displays th typical tmpratur and concntration profils for diffrnt valus of th thrmophorsis paramtr N t. Th thrmophorsis forc gnratd by th tmpratur gradint crats a fast flo from th surfac. In this ay mor fluid is hatd aay from th surfac and consquntly, as N t incrass, both th tmpratur and concntration profils incrass in th boundary layr. Figurs 16 and 17 illustrat th ffcts of an incras in th Bronian motion paramtr N b on th tmpratur and concntration. It is obsrvd that as th Bronian motion paramtr incrass, th tmpratur profil incras, spcially in th rgion clos to th dg surfac. Hovr, it is rmarkd that thr is a sharp fall in th concntration profil nar th plat. Clarly, incrass in N b dcras th concntration profil nar th plat. Figur 18 rvals th influnc of diffrnt valus of Prandtl numbr Pr on th tmpratur profil. It can b sn that th tmpratur gradually dcrass ith an incras in Pr. This is bcaus, as Pr incrass, th thrmal diffusivity of th fluid rduc and causs ak pntration of hat insid th fluid and consquntly, rducs th tmpratur profil of th fluid. Figurs 19 and 0 rports th concntration profil for diffrnt valus of Lis numbr L and local Rynolds numbr R. It is clarly obsrvd that th concntration profil and its boundary layr thicknss dcras considrably as th Lis numbr incras. Also, ith th incras in local Rynolds numbr noticd similar typ of rsult as obsrvd in figur 19. Gnrally, incrasing in L and R rducs th concntration boundary layr. This rsults in th nhancmnt of hat and mass transfr. Figur 1 illustrats th concntration profil for diffrnt valus of chmical raction paramtr γ. It is vidnt that th incras in chmical raction paramtr significantly altrs th concntration boundary layr thicknss. This causs th concntration buoyancy ffct to dcras yilding a rduction in th concntration profil. Tabl displays th ffcts of non-dimnsional govrning paramtrs on th skin friction cofficint, Nusslt and Shrood numbrs rspctivly. It is obsrvd that ith dcras in thrmal convctiv paramtr λ T and mass convctiv paramtr λ M, noticd dprciation in skin friction, Nusslt numbr and Shrood numbr. But obsrvd rvrs rsults to abov ith th incras in magntic fild paramtr M and prmability paramtr K. A dcras in prssur gradint paramtr β nhancs th Shrood numbr. But quit opposit is sn as both th skin friction and Nusslt numbr dcrasd as th prssur gradint paramtr β dcrasd. In Tabl 3 it is sn that a dcras in R, N b, N t and Ec nhancs th skin friction and Shrood numbr. A rvrs trnd is sn in th cas of Nusslt numbr. It is also obsrvd from th sam Tabl that an incras in Pr and Q causs an incras in skin friction and Shrood numbr throughout th momntum and concntration boundary layr. In Tabl 4 it is obsrvd that th L, γ, R and l hav no ffct on th Shrood numbr, hras th skin friction and Nusslt numbr incrass throughout th momntum and thrmal boundary layr du to th nhancmnt of th non-dimnsional govrning paramtrs. Figur Vlocity profil for diffrnt valus of T Figur 3 Vlocity profil for diffrnt valus of M Figur 4 Vlocity profil for diffrnt valus of M Figur 5 Vlocity profil for diffrnt valus of K DOI: / iosrjournals.org 76 Pag

7 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. Figur 6 Vlocity profil for diffrnt valus of Figur 7 Vlocity profil for diffrnt valus of Q Figur 8 Tmpratur profil for diffrnt valus of Q Figur 9 Vlocity profil for diffrnt valus of R Figur 10 Tmpratur profil for diffrnt valus of R Figur 11 Vlocity profil for diffrnt valus of E c Figur 1 Tmpratur profil for diffrnt valus of E c Figur 13 Concntration profil for diffrnt valus of E c DOI: / iosrjournals.org 77 Pag

8 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. Figur 14 Tmpratur profil for diffrnt valus of N t Figur 15 Concntration profil for diffrnt valus of N t Figur 16 Tmpratur profil for diffrnt valus of N b Figur 17 Concntration profil for diffrnt valus of N b Figur 18 Tmpratur profil for diffrnt valus of Pr Figur 19 Concntration profil for diffrnt valus of L Figur 0 Concntration profil for diffrnt valus of R Figur 1 Concntration profil for diffrnt valus of DOI: / iosrjournals.org 78 Pag

9 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. Tabl 1: Comparison of th valus of skin friction cofficint f (0) T M M K R Q Ec for svral valus of m hn m Whit [4] Yih [5] Yacob t al. [6] Khan and Pop [19] Shakhaoath Khan [18] Prsnt / / / ½ Tabl : Effcts of thrmal convctiv paramtr λ T, mass convctiv paramtr λ M, magntic fild paramtr M, prssur gradint paramtr β for Pr = 0.71,R = 1.0, N b = 0.3, N t = 0.3, Ec = 0.1, Q = 0.5, L = 3.0, = 1.0, R =.0, and = 0.1. K f (0) (0) (0) M T M Tabl 3: Effcts of Prandtl numbr Pr, radiation paramtr R, Bronian motion paramtr N b, thrmophrsis paramtr N t, Eckrt numbr Ec, and hat gnration Q for λ T =6.0, λ M = 3.0, M = 0.5, K = 0.5, β = 1.0, L = Pr R N 3.0, = 1.0, R =.0, and = 0.1. Ec Q N f (0) (0) (0) b t Tabl 4: Effcts of Lis numbr L, chmical raction paramtr, Rynolds numbr 0 R, and constant moving paramtr for λ T =6.0, λ M = 3.0, M = 0.5, K = 0.5, β = 1.0, Pr = 0.71,R = 1.0, N b = 0.3, N t = 0.3, Ec = 0.1, and Q = 0.5. L R f (0) (0) (0) DOI: / iosrjournals.org 79 Pag

10 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. V. Conclusions This papr prsnts th problm radiativ MHD dissipativ hat gnrating and chmical racting nanofluid flo past a dg through a porous mdium. Th govrning nonlinar partial diffrntial quations r transformd into ordinary diffrntial quations using th similarity approach and solvd numrically using Nactshim-Sigrt shooting tchniqu togthr ith Rung Kutta six ordr itration schms. From th abov discussions conclud th folloing (i) Viscous dissipation producs hat du to drag btn th fluid particls, hich causs an incras in fluid tmpratur. (ii) Prsnc of magntic fild producs a Lorntz forc hich usually rsists th momntum fild. (iii) Prsnc of porous paramtr is bnficial to gnrat hat in th systm. (iv) High Prandtl numbr causs lo thrmal diffusivity. (v) Th skin-friction at th surfac incrass ith incrasing buoyancy paramtrs. (vi) Th hat transfr rat at th boundary surfac dcrass ith incrasing thrmal radiation, hat sourc and Eckrt numbr. (vii) Th mass transfr rat at th boundary surfac incrass ith incrasing ith Lis numbr and chmical raction paramtr. Rfrnc [1] Ahmad, R., and W.A. Khan (015). Unstady hat and mass transfr magntohydrodynamic (MHD) nanofluid flo ovr a strtching sht ith hat sourc-sink using quasi linarization tchniqu, Canadian J. of physics, 93(1), [] Chamkha, A.J., S. Abbasbandy, A. M. Rashad, and K. Vajravlu (01). Radiation ffcts on mixd convction ovr a dg mbddd in a porous mdium filld ith a nanofluid, Trans. Porous Md., 91, [3] Chamkha, A.J. (000). Thrmal radiation and buoyancy ffcts on hydromagntic flo ovr an acclrating prmabl surfac ith hat sourc or sink, Int. J. Hat Mass Transfr, 38, [4] Choi S.U.S. (1995). Enhancing thrmal conductivity of fluids ith nano particls, Dvls Appls Non Ntonian Flos, 66, [5] Eshtu, H. and B. Shankar (014). 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Radiation and mass transfr ffcts on MHD fr convctiv dissipativ fluid in th prsnc of hat sourc/sink, J. of Applid Fluid Mchanics, 4(1), [5] Wahiduzzaman, M., Md. Shakhaoath Khan, P. Bisas, Ifsana Karim, and M. S. Uddin (015). Viscous dissipation and radiation ffcts on MHD boundary layr flo of a nanofluid past a rotating strtching sht, Applid Mathmatics, 6, DOI: / iosrjournals.org 80 Pag

11 Viscous Dissipation Effcts on Radiativ MHD Boundary Layr Flo of Nanofluid Past a Wdg. [6] Whit F.M. (1991). Viscous Fluid Flo, nd dn. (McGra-Hill, N York, NY, USA. [7] Yacob N.A., A. Ishak, and I. Pop, (011). Falknr-Skan problm for a static or moving dg in nanofluids. Int. J. Thrm. Sci. 50, 1-8. [8] Yih, K.A. (1998). Uniform suction/bloing ffct on forcd convction about a dg: uniform hat flux. Acta. Mch. 18, DOI: / iosrjournals.org 81 Pag