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1 Intenational Reseah Jounal of Engineeing and ehnology (IRJE e-issn: Volume: 03 Issue: 0 Feb-06.ijet.net -ISSN: Effets of Vaiable Visosity and hemal ondutivity on MHD Fee onvetion Flo of Dusty Fluid along a Vetial Stething Sheet ith Heat Geneation Jadav Konh, G.. Hazaika Reseah Shola, Deatment of Mathematis, Dibugah Univesity, Assam, India Pofesso, Deatment of Mathematis, Dibugah Univesity, Assam, India *** Abstat - A numeial study has been aied out on a steady to-dimensional lamina MHD heat and mass tansfe fee onvetion flo of a visous inomessible dusty fluid nea an isothemal linealy stething sheet in the esene of a unifom magneti field ith heat geneation. he esent ok examines the effets of temeatue deendent visosity and themal ondutivity on veloity, temeatue and seies onentation of the fluid. Effets of visous dissiation and Joule heating ae taken into onsideation in the enegy equation. Non-linea atial diffeential equations govening the motion ae edued to a system of odinay diffeential equations using suitable similaity tansfomations. Resultant equations ae then solved numeially using fouth ode Runge-Kutta shooting method. Some imotant featues of veloity, temeatue and seies onentation of fluid and solid atiles ae obtained fo vaious hysial aametes involved in the oblem hih ae of hysial and engineeing inteest ae analyzed, disussed and esented though gahs. Finally, numeial values of skin-fition oeffiient, Nusselt numbe and Sheood numbe ae esented in tabula fom, hih esetively give the all shea stess, ate of heat tansfe and ate of mass tansfe. Key Wods: Dusty fluid, vaiable visosity and themal ondutivity, visous dissiation, Joule heating..inroduion Investigations of to-dimensional bounday laye flo of fee onvetion heat and mass tansfe ove a vetial stething sufae ae imotant due to its aliations in industies, and many manufatuing oesses suh as aeodynami extusion of lasti sheets, ooling of metalli sheets in a ooling bath, hih ould be in the fom of an eletolyte, and olyme sheet extuded ontinuously fom a die ae fe atial aliations of moving sufaes. A losed fom exonential solution fo lana visous flo of linea stething sheet has been esented by ane []. Initially, Gebhat [] obseved the oblem taking into aount the visous dissiation. Effets of MHD and visous dissiation on heat tansfe analysis ee studied by many authos suh as Mahmoud [3], Vajavelu and Hadjinialaou [4], Samad et al. [5] and Anjali Devi [6]. otell[7] analyzed MHD flo of a oe-la fluid ove a stething sheet. hen [8] studied mixed onvetion of oe la fluid ast a stething sufae ith themal adiation and magneti field. otell [9] obseved the influene of visous dissiation and ok done by defomation on MHD flo and heat tansfe of visoelasti fluid ove a stething sheet. sai et al. [0] have disussed unsteady flo ove a stething sufae ith non-unifom heat soue. o study the to-hase flos, in hih solid sheial atiles ae distibuted in a fluid ae of inteest due to ide ange of eal old aliations. he study of heat tansfe in the bounday laye indued by stething sufae ith a given temeatue distibution in a onduting dusty fluid is imotant in seveal manufatuing oess in industies like extusion of lasti sheets, glass fibe and ae odution, metal sinning and the ooling of metalli late in a ooling bath. In vie of these aliations, Saffman [] aied out ioneeing ok on the stability of lamina flo of a dusty gas hih desibes the fluid-atile system and deived the equations of motion of a gas aying dust atiles. Mable[] studied dynamis of a gas ontaining small solid atiles. Baal[3] obseved lane aallel flo of a onduting dusty gas. hakabati [4] investigated the bounday laye flo of a dusty gas. Guta and Guta [5] have disussed flo of dusty gas in a hannel ith abitay time vaying essue. Datta and Misha[6] have disussed the flo of dusty fluid in bounday laye ove a semi-infinite flat late. Vajavelu and Nayfeh[7] analyzed the MHD flo of a dusty fluid ove a stething sheet ith the effets of fluid-atile inteation, atile loading and sution on the flo haateistis and omaed thei analytial solution ith numeial ones. Unsteady hydomagneti bounday laye flo and heat tansfe of dusty fluid ove a stething sheet ith vaiable all temeatue (VW and vaiable heat flux (VHF have been studied by Gieesha et al. [8,9]. In these aes the effets of magneti field on an unsteady bounday laye flo and heat tansfe of a dusty fluid ove an unsteady stething sufae in the esene of non- unifom heat soue/sink ae disussed. 06, IRJE Imat Fato value: 4.45 ISO 900:008 etified Jounal Page 09

2 Intenational Reseah Jounal of Engineeing and ehnology (IRJE e-issn: Volume: 03 Issue: 0 Feb-06.ijet.net -ISSN: (u,v veloity omonents of the fluid, (u,v veloity omonents of dust hase, density of the fluid, density of the atile hase, oeffiient of dynami visosity, oeffiient of dynami visosity of ambient fluid, kinemati visosity of the fluid in the fee steam, N numbe density of the atile hase, K Stokes esistane(dag o-effiient, m mass of the dust atile, g aeleation due to gavity, β* volumeti oeffiient of themal exansion, temeatue of the fluid inside the bounday laye, temeatue of the fluid at fee steam, Nomenlatue temeatue of the dust atiles inside the bounday laye, temeatue of the dust atiles in the fee-steam, and seies onentation of dust atiles at the sufae of the late and suffiiently fa aay fom the flat sufae esetively, density atio, eletial ondutivity, stething ate, l haateisti length, m seifi heat of fluid at onstant essue, seifi heat of dust atiles at onstant essue, τ τ v λ themal equilibium time and is the time equied by the dust loud to adjust its temeatue to the fluid, elaxation time of the dust atile i.e., the equied by a dust atile to adjust its veloity elative to the fluid, themal ondutivity of the fluid, themal ondutivity of the ambient fluid, seies onentation of the fluid, onentation of the dust hase ithin the bounday laye, Dm oeffiient of mass diffusivity of the fluid, D oeffiient of mass diffusivity of dust hase, m visosity vaiation aamete, themal ondutivity vaiation aamete, l* mass onentation, τ elaxation time of atile hase, β fluid atile inteation aamete, G Gashof numbe, M magneti field aamete, Re Reynolds numbe, P Pandtl numbe, E Eket numbe, S Shmidt numbe fo fluid hase, Shmidt numbe fo dust hase, S f Nu Sh skin-fition oeffiient, Nusselt numbe, Sheood numbe. Evgeny and Segei [0] disussed the stability of the lamina bounday laye flo of dusty gas on a flat late. Futhe, XIE Ming-liang et al. [] extended the ok of [6] and studied the hydodynami stability of a atile-laden flo in going flat late bounday laye. Palani and Ganesan [] studied heat tansfe effets on dusty gas flo ast a semi-infinite inlined late. Aganat [3] studied dusty bounday laye flo and heat tansfe, ith the effet of essue gadient. Ezzat et al. [4] analyzed the sae aoah to the hydo-magneti flo of a dusty fluid though a oous medium. Kannan and Venkataaman [5] examined the fee onvetion in an infinite oous dusty medium indued by ulsating oint heat soue. Reently, Gieesha et al. [6] studied the bounday laye flo and heat tansfe of a dusty fluid flo ove a stething sheet ith non-unifom heat soue/sink. In all the above mentioned aes on dusty fluid, the themohysial tansot oeties of the ambient fluid ee assumed to be onstant. Hoeve, it is ell knon fom the ok of Heig and Wiken[7], Lai and Kulaki[8], Setayesh and Sahai[9], Attia[30] that these hysial oeties may hange ith temeatue, eseially the fluid visosity and the themal ondutivity. Fo lubiating fluids, the oeties of the fluid ae no longe assumed to be onstant. An inease in temeatue leads to inease in the tansot henomena by eduing the hysial oeties aoss the themal bounday laye. heefoe to edit the flo and heat tansfe haateistis moe auately, it is neessay to take into aount the temeatue deendent fluid oeties. In vie of these aliations, the oblem studied hee the effets of temeatue deendent visosity and themal ondutivity on MHD fee onvetion flo of dusty fluid along a vetial stething sheet ith heat geneation. Hee the themohysial tansot oeties namely, the visosity and the themal ondutivity ae assumed to be funtion of temeatue. In addition to this, e have also inluded the ontibution of visous dissiation and Joule heating in the enegy equation. 06, IRJE Imat Fato value: 4.45 ISO 900:008 etified Jounal Page 030

3 Intenational Reseah Jounal of Engineeing and ehnology (IRJE e-issn: Volume: 03 Issue: 0 Feb-06.ijet.net -ISSN: MAHEMAIAL FORMULAION onside a steady to-dimensional heat and mass tansfe flo of an eletially onduting visous inomessible dusty fluid along an isothemal stething vetial sheet ith heat geneation/absotion. he stething sheet oinides ith the lane y = 0 hee the flo is onfined to y > 0. A unifom magneti field of stength B 0 is alied along the y- axis. his is the only magneti field in the oblem as the indued magneti field is negleted by assuming a vey small magneti Reynolds numbe [3, 3]. he visous dissiation and Joule heating tems ae taken into onsideation in the enegy equation. o equal and oosite foes ae intodued along the x-axis so that the sheet is stethed keeing the oigin fixed as shon in Fig.. N u J ( u u Q0 ( v y u v Dm x y y y Fo the dust hase: ( u ( v 0 x y u u K u v ( u u x y m (3 (4 (5 (6 v (x u=d x O x B 0 B 0 g u/u v/v y v v K u v ( v v x y m u v ( x y u v Dm x y y y m (7 (8 (9 u=d x he aoiate bounday onditions ae given by: Fig.: Flo onfiguation and oodinate system In the esent study dust atiles ae assumed to be sheial and unifom in size and shae and ae unifomly distibuted thoughout the fluid. Also it is assumed that fluid has onstant hysial oeties exet visosity and themal ondutivity. Using the above assumtions and based on Saffman[] model of the steady motion of an inomessible fluid and folloing Vajavelu and Nayfeh[7] and Eldable et al.[33], govening equations of the flo ae: Fo the fluid hase: u v 0 x y ( u u u KN * B0 u v ( u u g ( u x y y y ( N u v ( x y y y At y 0 : u U x, v 0, x, A l x B l, x E l As y : (0 u 0, u 0, v v,,,,, hee >0 is the stething ate, ω is the density atio, A, B and E ae ositive onstants and l is a haateisti length, and ae the seies onentation of the dust atiles at the sufae of the late and suffiiently fa aay fom the flat sufae esetively. o onvet the govening equations into a set of similaity equations, e intodue the folloing tansfomations as, 06, IRJE Imat Fato value: 4.45 ISO 900:008 etified Jounal Page 03

4 Intenational Reseah Jounal of Engineeing and ehnology (IRJE e-issn: Volume: 03 Issue: 0 Feb-06.ijet.net -ISSN: u xf (, v f (, y, u xf (, v G(, H( ( ( hee hee, (, ( x A (, l x B (, l, x E (. l, ( is the elative density and the ime ( denotes deivative ith eset to η. he visosity of the fluid is assumed to be an invese linea funtion of temeatue, and it an be exessed as folloing Lai and Kulaki [4]: (, o, hee and Also themal ondutivity of the fluid is vaying ith temeatue. Folloing houdhuy and Hazaika [0], e assumed themal ondutivity of the fluid as: (3, o, hee and. Hee, α,,,, and ae onstants and thei values deend on the efeene state and themal oeties of the fluid i.e., ( kinemati visosity and λ (themal ondutivity. o dimensionless efeene temeatues and an be defined as ( ( and (4 and alled the visosity vaiation aamete and the themal ondutivity vaiation aamete esetively. Substituting Eqs. (-(4 in the Eqs. (-(0, e get f f ff f l H F f ( Mf G 0 (5 GF F F f * ( ( 0 (6 GG ( f G 0 (7 ( 0 (8 G H H f G GFH N P( f f P( ( N v (9 P E( F f PQ PEf PEMf 0 G F ( 0 (0 m S( f f 0 ( ( ( 0 S F G ( ( hee the dimensionless aametes ae defined as follos: 06, IRJE Imat Fato value: 4.45 ISO 900:008 etified Jounal Page 03

5 Intenational Reseah Jounal of Engineeing and ehnology (IRJE e-issn: Volume: 03 Issue: 0 Feb-06.ijet.net -ISSN: l*= mn/ρ mass onentation, τ= m/k elaxation time of the atile hase, β=/τ fluid atile inteation aamete, ρ =ρ /ρ elative density, * g ( G Gashof numbe, x S S M P Q D Dm Shmidt numbe fo fluid hase, m Shmidt numbe fo dust hase, B 0 magneti field aamete, Pandtl numbe, Q0 heat geneating aamete, U E ( Eket numbe. he bounday onditions defined in Eqn.(0 ill beomes: f 0, f,,, at 0 f 0, F=0, G=-f, H=, 0, 0, 0, (3 0 as. Skin-fition, Nusselt numbe and Sheood numbe Skin-fition oeffiient ( f, Nusselt numbe(nu and Sheood numbe(sh ae the aametes of hysial and engineeing inteest fo the esent oblem, hih hysially indiate the all shea stess, ate of heat tansfe and ate of mass tansfe esetively. Skin-fition oeffiient is defined as f u0, hee u y y0 is the sheaing stess. Afte using the non-dimensional vaiables e get the skinfition o-effiient as / f Re f (0. he Nusselt numbe hih is defined as Nu xq (, hee q the heat tansfe fom the sheet. Using the non-dimensional vaiables e get y y0 is Nu / Re (0. he Sheood numbe hih is defined as xm Sh, Dm ( hee and m Dm y y0 is the mass flux at the sufae Dm is the diffusion onstant at fee steam. Using the non-dimensional vaiables e get / Sh Re S (0. 3. RESULS AND DISUSSION Equations (5-( ith the bounday onditions (3 ee solved numeially using fouth ode Runge-Kutta method using shooting tehnique. Numeial values ae omuted by develoing suitable odes in MALAB fo the method. Numeial omutations of these solutions have been aied out to study the effets of vaious hysial aametes suh as visosity vaiation aamete(, themal ondutivity vaiation aamete(, magneti aamete(m, Pandtl numbe(p and numbe density aamete of dust atile(n on veloity, temeatue and seies onentation ofiles of both the fluid and dust atiles. A eesentative set of numeial esults ae shon gahially fo veloity, temeatue and seies onentation ofiles fom Fig.-Fig.. In the folloing disussion some aametes ae given the folloing fixed values: Re=00, G=, M=, P=, Q=5, S=S =0., l * =0., β=, N=, ρ=, =, τ =, τ v=, ω=0., E=0.05, = m=0., =5 and =3, unless otheise stated. he effet of visosity vaiation aamete on veloity ofiles of fluid hase f ( and dust hase F ( veses is deited in Fig.. It is obseved that the veloity ofiles deease ith an ineasing values of fluid visosity aamete fo both the hases. his is beause of the fat that ith an inease in the value of fluid visosity vaiation aamete deeases the veloity bounday laye thikness. Fig. 3 illustates the effet of themal ondutivity vaiation aamete on veloity ofiles. It is seen that veloity ineases ith the ineasing values of. It is due to the eason that temeatue deeases ith the ineasing 06, IRJE Imat Fato value: 4.45 ISO 900:008 etified Jounal Page 033

6 f', F f', F f ', F f', F f', F Intenational Reseah Jounal of Engineeing and ehnology (IRJE e-issn: Volume: 03 Issue: 0 Feb-06.ijet.net -ISSN: = 3, 5, 7 f' F Fig. 4 shos the vaiation of veloity ofile fo both the hases ith vaious values of M. Fom this lot it is seen that the effet of ineasing values of M is to deease the veloity distibution of both the fluid and dust hases. his is due to the fat that the esene of a magneti field nomal to the flo in an eletially onduting fluid odues a Loentz foe, hih ats against the flo. 0. P= 0.,, f' F Fig. : Veloity ofile fo diffeent Ө =3, 5, 7 f ' F Fig. 5: Veloity ofile fo diffeent P Vaiation of veloity ofile fo diffeent values of Pandtl numbe(p and numbe density of the atile hase(n ae deited in Fig.5 and Fig. 6, esetively. We obseved that the veloity ofile deeases ith ineasing values of both the aametes Fig. 3: Veloity ofile fo diffeent Ө N=,, 3 f' F values of and as a esult visosity deeases and so veloity ineases. M=,,.5, f' F Fig. 4: Veloity ofile fo diffeent M Fig. 6: Veloity ofile fo diffeent N he fluid and dust atiles temeatue ofiles ae esented in Fig. 7-Fig. 0. Fig. 7 eesents that an inease in the visosity vaiation aamete( ineases the temeatue distibution. his is due to the eason that an inease in the value of, the themal bounday laye thikness ineases, hih esults an inease in the temeatue. 06, IRJE Imat Fato value: 4.45 ISO 900:008 etified Jounal Page 034

7 Intenational Reseah Jounal of Engineeing and ehnology (IRJE e-issn: Volume: 03 Issue: 0 Feb-06.ijet.net -ISSN: = 3, 7, Fig. 0 dislays the effet Pandtl numbe(p on temeatue ofile. he inease in the value of P deeases the temeatue of fluid and dust hase. Physially, by this e mean that fo highe Pandtl numbe fluid has a elatively high themal ondutivity hih edues temeatue., P=,,, Fig. 7: emeatue ofile fo diffeent Ө Fom the lots in Fig. 8, one an obseved that fo ineasing values of themal ondutivity vaiation aamete( enhanes the temeatue distibution, hih is tue fo both the fluid hase as ell as dust hase. Fig. 9 eveals the effet of magneti aamete M on temeatue ofile. It is notied that temeatue ofile ineases fo ineasing values of magneti aamete. his is beause of the Loentz foe., =3, 7,, Fig. 8: emeatue ofile fo diffeent Ө M=,,.5,, Fig. 0: emeatue ofile fo diffeent P he fluid and dust atiles seies onentation ofiles esented in Fig. and Fig. fo vaious values of and, esetively. Fom these lots it is obseved that seies onentation ofiles deeases fo ineasing values of, hile it ineases ith an ineasing values of, = 3, 7,., Fig. 9: emeatue ofile fo diffeent M Fig. : onentation ofile fo diffeent Ө he alulated values of skin-fition oeffiient( f, Nusselt numbe(nu and Sheood numbe(sh fo vaious values of flo govening aametes ae esented in able. Fom the table it is obseved that skin-fition oeffiient ineases ith the ineasing values of visosity vaiation aamete, 06, IRJE Imat Fato value: 4.45 ISO 900:008 etified Jounal Page 035

8 Intenational Reseah Jounal of Engineeing and ehnology (IRJE e-issn: Volume: 03 Issue: 0 Feb-06.ijet.net -ISSN: , =3, 8, 3 themal ondutivity vaiation aamete and magneti aamete. Visosity vaiation aamete leads to a deease in the values of Nusselt numbe and Sheood numbe. It is also notied that the value of Nusselt numbe deeases ith inease in the value of magneti aametes, but it ineases fo the ising value of Eket numbe. We obseved that Sheood numbe deeases ith inease in the value of themal ondutivity vaiation aamete, magneti aamete, Eket numbe and Shmidt numbe Fig. : onentation ofile fo diffeent Ө able: Effets of,, M, E and S on loal skin-fition oeffiient ( f, loal Nusselt numbe ( Nu and loal Sheood numbe( Sh M E S f Nu Sh , IRJE Imat Fato value: 4.45 ISO 900:008 etified Jounal Page 036

9 Intenational Reseah Jounal of Engineeing and ehnology (IRJE e-issn: Volume: 03 Issue: 0 Feb-06.ijet.net -ISSN: ONLUSIONS Fom the esent investigation the folloing onlusions ee made: When the effets of vaiable visosity and themal ondutivity ae taken into aount, the flo haateistis ae signifiantly hanged omaed to the onstant oety ase. he veloity ofile deeases ith the ineasing values of visosity vaiation aamete and Pandtl numbe hile the oosite effet is obseved in the ase of themal ondutivity vaiation aamete. he effet of magneti aamete leads to deease in the veloity ofile. On the othe hand, it enhanes the temeatue fo both the fluid and dust hases. An ineasing visosity vaiation aamete auses inease in the temeatue. But seies onentation edues ith it fo both the fluid and dust hases. Seies onentation field enhanes ith the inease of themal ondutivity vaiation aamete. Skin-fition oeffiient ineases due to magneti field. he all shea stess ineases ith the inease of the visosity vaiation aamete and themal ondutivity vaiation aamete. he ate of heat tansfe to the fluid deeases ith the inease in and M heeas it ineases fo ineasing value of and E. An inease in the value of visosity vaiation aamete leads to deease in the values of Sheood numbe. he oeffiient of skin-fition and Sheood numbe deease ith the inease in and M. Rate of mass tansfe deeases ith the inease in the themal ondutivity vaiation aamete. AKNOWLEDGEMEN Authos ae gateful to the eviees fo thei valuable suggestions hih heled us to imove the quality of this eseah ae. Authos ae also thankful to the UG, Ne Delhi, India fo oviding finanial suot to ay out this ok. REFERENES [] L. J. ane, Flo ast a stething late, Z Ange Math Phys (ZAMP, Vol., , 970. [] B. Gebhat, Effets of visous dissiation in natual onvetion, J. Fluid Meh, Vol. 4,. 5-3, 96. [3] M. A. A. Mahmoud, Vaiable visosity effets on hydomagneti bounday laye flo along a ontinuously moving vetial late in the esene of adiation, Al. Math. Si., Vol., , 007. [4] K. Vajavelu and A. Hadjinialaou, Heat tansfe in a visous fluid ove a stething sheet ith visous dissiation and intenal heat geneation, Int. omm. Heat Mass ans., Vol. 0, , 993. [5] M. A. Samad and M. Mohebujjaman, MHD heat and mass tansfe fee onvetion flo along a vetial stething sheet in esene of magneti field ith heat geneation, Res. J. Al. Si. Eng. ehnol, Vol. (3, , 009. [6] S. P. Anjali Devi and B. Ganga, Dissiation effets on MHD nonlinea flo and heat tansfe ast a oous sufae ith esibed heat flux Jounal of Alied Fluid Mehanis, Vol. 3 (,. -6, 00. [7] R. otell, A note on magnetohydodynami flo of a oe-la fluid ove a stething sheet, Al. Math. om. Vol.68, , 005. [8]. H. hen, MHD mixed onvetion of a oe la fluid ast a stething sufae in the esene of themal adiation and intenal heat geneation/absotion, Int. J. Nonlinea Mehanis, Vol. 44, , 008. [9] R. otell, Effets of visous dissiation and ok done by defomation on the MHD flo and heat tansfe of a visoelasti fluid ove a stething sheet, Phys. Lett. A., Vol. 357, , 006. [0] R. sai, K. H. Huang and J. S. Haung, Flo and Heat ansfe ove an Unsteady Stething Sufae ith Nonunifom Heat Soue, Intenational ommuniations in Heat and Mass ansfe, Vol. 35(0, , 008. [] P. G. Saffman, On the stability of lamina flo of a dusty gas, Jounal of Fluid Mehanis, Vol. 3,. 0-8, 96. [] F. E. Mable, Dynamis of a gas ontaining small solid atiles, Poeedings of Fifth AGARD olloquium ombustion and Poulsion (96, Pegamon Pess, Oxfod,.75-3, 963. [3] M.. Baal, Plane aallel flo of a onduting dusty gas, J. Phy. So. Jaan, 33: 73, 968. [4] K. M. hakabati, Note on bounday laye in a dusty Gas, AIAA Jounal, Vol. ( , 974. [5] P. K. Guta and S.. Guta, Flo of a dusty gas though a hannel ith abitay time vaying essue gadient, Jounal of Al. Math. and Phys., Vol. 7, 9, 976 [6] N. Datta and S. K. Misha, Bounday laye flo of a dusty fluid ove a semi-infinite flat Plate, AtaMehania, Vol. 4(,. 7 83, , IRJE Imat Fato value: 4.45 ISO 900:008 etified Jounal Page 037

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