Heat Generation and Radiation Effects on MHD Casson Fluid Flow Over a Stretching Surface Through Porous Medium

Transcription

1 vailable online Euoean Jounal o dvanes in Engineeing and Tehnology, 017, 4(11): Reseah tile ISSN: X Heat Geneation and Radiation Eets on MHD Casson Fluid Flow Ove a Stething Suae Though Poous Medium Noua S lsedais Deatment, o Mathematial Siene, Piness Nouah bint bdulahman Univesity, College o Sienes, Riyadh, KS nsalsudais@nu.edu.sa BSTRCT In this ae, the heat geneation and adiation eets on magnetohydodynami stagnation-oint low o non- Newtonian Casson luid low ove a stething suae though oous medium have been obtained and studied numeially. The govening ontinuity and momentum equations ae onveted into a system o non-linea odinay dieential equations by means o similaity tansomation. The esulting system o ouled non-linea odinay dieential equations is solved numeially. Numeial esults wee esented o veloity and temeatue oiles o dieent aametes o the oblem. lso the eets o the etinent aametes on the skin ition, the ate o heat and mass tanse ae obtained and disussed numeially. Keywods: MHD, Heat Geneation, Radiation, Poous medium, Casson luid INTRODUCTION Thee is ontinuous ineasing inteest o eent eseahes in low oblems o non-newtonian luids due to thei high aliations in industy and engineeing. It is well known now that the stethed low oblems o non- Newtonian luids ou in odution o lasti, ae and ood mateials. Heat tanse involvement has geat ole in these oesses. Vaious eent eseahes ae engaged in exloing the heat tanse haateistis in the low o non-newtonian luids ove a stething suae. Mahaata and Guta [1] eonsideed the stagnation oint low oblem towads a stething sheet taking dieent stething and staining veloities and they obseved two dieent kinds o bounday laye nea the sheet deending on the atio o the stething and staining onstants. Mahaata and Guta [] esented heat tanse in stagnation oint low towads a stething sheet. Naza. min, et al [3] showed the unsteady bounday laye low in the egion o the stagnation oint on a stething sheet. Heat and mass tanse analysis o bounday laye stagnation-oint low towads a heated oous stething sheet with heat absotiongeneation and sution blowing by Layek et al [4]. ttia [5] initiated the study o on the eetiveness o oosity on stagnation oint low towads a stething suae with heat geneation; Nadeem et al [6] studied the HM solutions o bounday laye low in the egion o the stagnation oint towads a stething sheet. Dual solutions in bounday laye and unsteady stagnation-oint low and mass tanse with hemial eation ast a stethingshinking sheet by [7, 8]. Salem and Fathy [9] who analyzed the eets o vaiable oeties on MHD heat and mass tanse low nea a stagnation oint towads a stething sheet in a oous medium with themal adiation. The detailed disussion on the stagnation-oint low ove stethingshinking sheet an be ound in the woks o [10-14]. In the esent wok, we extend and genealize the wok o [15] to inlude the numeial study o MHD Stagnation- Point Flow o Casson Fluid and Heat Tanse ove a Stething Sheet with Themal Radiation. The aim o the esented ae is to study the eets o heat geneation and adiation on magnetohydodynami stagnation-oint low o non-newtonian Casson luid low ove a stething suae though oous medium ae used to similaity tansomation into a system o nonlinea odinay dieential equations whih ae solved numeially. Numeial esults wee esented o veloity and temeatue oiles o dieent aametes o the oblem. lso the eets o the etinent aametes on the skin ition, the ate o heat and mass tanse ae also disussed. 850

2 Noua S lsedais Euo. J. dv. Engg. Teh., 017, 4(11): MTHEMTICL NLYSIS Conside the steady two-dimensional inomessible low o a non-newtonian Casson luid ove a stething suae though oous medium unde the inluene o a tansvesely alied magneti ield B in the dietion with y 0 the indued magneti ield being negleted, with the low being onined in. It is also assumed that the heologial equation o state o an isotoi and inomessible low o a Casson luid an be witten as [16-17]. whee ij y eij, y eij, is lasti dynami visosity o the non-newtonian luid, o the omonent o deomation ate with itsel, namely, omonent o the deomation ate, and y e ij e ij is a itial value o π based on non-newtonian model. The govening bounday laye equations ae: u v 0 x y,, y (1) is the yield stess o luid, π is the odut e ij is the ( i, j) u u d U s 1 u B0 u v U s 1 u U, s u U s x y d x y (3) T T k T 1 q Q0 u v ( T T ), x y C y y (4) With the aoiate bounday onditions: y 0 : u Uw x, v 0, T Tw, (5) y : u Us a x T T. Whee and a ae the ositive onstants that eesent the haateisti stething intensity and the ee steam stength., and C ae the density, dynami visosity and the seii heat at onstant essue, esetively. is the emeability o the oous medium, temeatue assumed to be onstant, being the staining onstant, y T w is the onstant temeatue at the sheet and T () is the ee steam U s a x is the staining veloity o the stagnation-oint low with a (>0) is the kinemati luid visosity, is the luid density, is the non-newtonian o Casson aamete, is the eletial ondutivity o the luid, U w x is stething veloity o the sheet with (>0) being the stething onstant, k ae the eetive themal ondutivity, Q 0 is the volumeti heat geneationabsotion ate, q ae the themal-diusion ate, onentation susetibility, luid mean temeatue and the adiative heat lux, esetively. Using the Rosseland aoximation (Rashed [18]), the adiative heat lux q ould be exessed by whee the q 4 3k 4 T y eesents the Stean-Boltzman onstant and k is the Rosseland mean absotion oeiient. 4 ssuming that the temeatue dieene within the low is suiiently small suh that T ould be aoahed as the linea untion o temeatue T T T 3T (7) 4 (6) 851

3 Noua S lsedais Euo. J. dv. Engg. Teh., 017, 4(11): The govening atial dieential equations ()-(5) admit similaity solutions o obtaining the dimensionless steam untion and temeatue. The elative aametes ae intodued as () u x () ( ), v ( ), T T ( ) T T w, y. The equation o ontinuity is satisied automatially. te intoduing the similaity tansomation, the equations o (3) and (4) an be tansomed into a set o ollowing oms in tems with and, exessed as () 1 1 ( M S)( ) 0, 1 () (8) 4 3 R (9) The bounday onditions edue to: Whee (0) 0, P (0) 1, (0) 1, ( ), ( ) 0. P Q 0, M B is the magneti ield aamete, S is the veloity atio aamete, aamete and Q 0 Q 0 is the oous medium aamete, (10) a 3 P k is the Pandtl numbe, R 1 4 T k k R is the adiation is the heat geneation aamete. SKIN-FRICTION COEFFICIENT ND NUSSELT NUMBER The aametes o engineeing inteest o the esent oblem ae the loal skin-ition oeiient and loal Nusselt numbe whih indiate the hysial wall shea stess and ate o heat tanse, esetively. Whih ae deined as: C and N x q ( T T ) w uw w w u, (11) Whee ) is the shea stess o skin ition along the stething sheet and q ) is the heat lux om the sheet ( w and those ae deined as w 1 Hene, loal skin-ition oeiient whee R e C y u y y0 and C and the Nusselt numbe x is the loal Reynolds numbe. N u ( w T qw. (1) y as ollows: 1 1 (0) and 1 (0) e N u RESULTS ND DISCUSSION y0 R. (13) The system o non-linea odinay dieential Eqs. (8)- (9) togethe with the bounday onditions (10) ae solved numeially by using the oth ode o Runge-kutta integation aomanied with the shooting method. In ode to get a hysial insight, the veloity and temeatue oiles have been disussed. Figs. 1 (a and b) show the eet o Casson aamete on the veloity and the temeatue oiles. We ound that the veloity oile ineases at. 0, and deeases at 0. 1 with an inease o Casson aamete. We obseve om this igue that the bounday laye thikness ineases as β deeases at 0. 1, likewise, this igue deits that o ineasing values o the Casson aamete, it edues the luid veloity distibution inside the bounday laye away om the sheet but the evese is tue along the sheet. Physially, with an inease in the non-newtonian Casson aamete, the luid yield stess is deeasing auses a odution o esistane oe whih make the luid veloity deeases. lso, we ound that the temeatue oile ineases at 0. 1, but deeases at. 0 with the inease o Casson aamete. On the othe hand, the eet o Casson aamete on temeatue ield in the ase o 0. 1is muh moe onouned than that o the ase o

4 Noua S lsedais Euo. J. dv. Engg. Teh., 017, 4(11): The dimensionless veloity oile o seleted values o magneti aamete M ( M 1.0,.0,3.0) is lotted in Fig () (a). It is aaent that the veloity deeases along the suae with an inease in the magneti aamete at 0.1. The tansvese magneti ield ooses the motion o the luid and the ate o tansot is onsideably.0, we ound that the veloity ineases with an inease in the magneti aamete, and this is be- edued, this is beause with the inease in M, Loentz oe ineases and it odues moe esistane to the low. t ause the stething veloity is geate than the staining veloity. lso, it is ound that the temeatue distibution along the bounday laye, themal bounday thikness and the temeatue o the stething suae inease with an inease in the same aamete at 0. 1,as we an see om Fig()(b).So the temeatue inside the themal bounday laye ineases due to exess o heating. Theeoe, the magneti ield an be used to ontol the low haateistis. lso, it is aaent that thee is no eet o M on at.0. Figs. (3)(a and b) show the eets o oous medium aamete S on the veloity and the temeatue oiles, we ound that the veloity oile deeases at 0.1, but the temeatue oile inease at 0.1 with the ineases o oous medium aamete. lso, it is aaent that thee is no eet o S on ' and at. 0. (a) (b) Fig. 1 Eet o Casson aamete on (a) the veloity oile and (b) the temeatue oile (a) (b) Fig. Eet o magneti ield aamete on (a) the veloity oile and (b) the temeatue oile 853

5 Noua S lsedais Euo. J. dv. Engg. Teh., 017, 4(11): (a) (b) Fig. 3 Eet o oous medium aamete on (a) the veloity oile and (b) the temeatue oile (a) (b) Fig. 4 Eet o veloity atio aamete on (a) the veloity oile and (b) the temeatue oile The veloity and temeatue oiles o vaious values o veloity atio aamete ae lotted in Fig. (4)(a and b). Deending on the veloity atio aamete, two dieent kinds o bounday layes ae obtained. In the ist kind, the veloity o luid inside the bounday laye deeases om the suae towads the edge o the laye (o and in the seond kind the luid veloity ineases om the suae towads the edge (o >1). Those haates an be seen om veloity oiles in Fig. 4(a). lso, it is imotant to note that i 1 ( a ), that is, the stething veloity and the staining veloity ae equal, and then thee is no bounday laye o Casson luid low nea the sheet. <1) It is obseved in Fig. (5) that ineasing themal adiation aamete R (1.0,0.5,0. ) ineases temea- R tue oiles, this is due to the at that as moe heat is geneated within the luid, the luid temeatue ineases leading to a sha inlination o the temeatue gadient between the suae and the luid. In Fig. (6), it is lea that the temeatue oile deease with an ineasing o Pandtl numbe, at 0.1 and. 0. lso, by oma

6 Noua S lsedais Euo. J. dv. Engg. Teh., 017, 4(11): ison these esults with the esults o Kishnendu [15] we ound that they ae simila at 0.1 and. 0, whih ae exellent and ageement. Fig.(7) shows that, the temeatue oile ineases with an ineasing in heat geneation aamete at 0.1 and. 0.This is beause when heat is absobed the buoyany oes and themal diusivity o the luid inease, whih aeleate the low ate and theeby ise to an inease in the temeatue oile o the low. Fig. 5 Eet o adiation aamete on the temeatue oile Fig. 6. Eet o Pandtl numbe aamete on the temeatue oile Fig. 7 Eet o heat geneation aamete on the temeatue oile Table 1 shows the data o heat and mass tanse ate. Fom the table, we ould detemine that the value Skinition oeiient ineases and Nusselt numbe deeases at 0. 1, but the value Skin-ition oeiient deeases and Nusselt numbe ineases at. 0 with an ineasing in Casson aamete. The value Skin-ition oeiient and Nusselt numbe deease at 0. 1, while the value Skin-ition oeiient and Nusselt numbe inease at. 0 with the inease o eah o magneti ield and oous medium aametes. With inease o Pandtl numbe aamete, we ound that Nusselt numbe deeases at 0. 1, while Nusselt numbe ineases at.0. nd Nusselt numbe deeases with an ineasing in heat geneation aamete at 0.1and

7 Noua S lsedais Euo. J. dv. Engg. Teh., 017, 4(11): Table -1 Numeial o the values the loal Skin-ition oeiient ( ) and the loal Nusselt numbe ( ) with M S R P, M, S, R, P Q 1 C andq N u C N u CONCLUSION The MHD bounday laye low o Casson luid ove a stething sheet in the esene o tansvese magneti ield has been investigated. The main indings o the esent study an be summaized as ollows: Momentum bounday laye thikness ineases at. 0, and deeases at 0. 1 with the inease in values o Casson aamete, magneti ield aamete and the oous medium aamete. The themal bounday laye thikness at 0. 1 ineases with the inease in values o Casson aamete, magneti ield aamete, the oous medium aamete and the heat geneation aamete, and deeases with the inease in values o the veloity atio aamete. The Skin-ition at suae ineases at 0. 1and deeases at. 0 with an ineasing in Casson aamete, and vie vesa with the inease o eah o magneti ield and oous medium aametes. The Nusselt numbe deeases at 0. 1, and ineases at. 0 with the inease o Casson aamete, magneti ield aamete and the oous medium aamete. REFERENCES [1] TR Mahaata and S Guta, Magnetohydodynami Stagnation-Point Flow Towads a Stething Sheet, ta Mehania, 001,15, (1-4), [] TR Mahaata and S Guta, Heat Tanse in Stagnation Point Flow Towads a Stething Sheet, Heat and Mass Tanse, 00, 38 (6), [3] R Naza, N min, D Fili and I Po, Unsteady Bounday Laye Flow in the Region o the Stagnation Point On a Stething Sheet, Intenational Jounal o Engineeing Siene, 004, 4 (11-1), [4] GC Layek, S Mukhoadhyay and S Samad, Heat and Mass Tanse nalysis o Bounday Laye Stagnation- Point Flow Towads a Heated Poous Stething Sheet with Heat bsotiongeneation and SutionBlowing, Intenational Communiation on Heat and Mass Tanse, 007, 34 (3), [5] H ttia, On the Eetiveness o Poosity on Stagnation Point Flow Towads a Stething Suae with Heat Geneation, Comutational Mateials Siene, 007, 38, [6] S Nadeem, Hussain and M Khan, HM Solutions o Bounday Laye Flow in the Region o the Stagnation Point Towads a Stething Sheet, Communiation on Nonlinea Siene and Numeial Simulation, 010,15 (3),

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