Rotation and Radiation Effects on MHD Flow through Porous Medium Past a Vertical Plate with Heat and Mass Transfer

Transcription

1 NLELE UNIVERSITĂŢII EFTIMIE MURGU REŞIŢ NUL XXIV NR. 7 ISSN Rotation and Radiation Effets on MHD Flow thogh Poos Medim Past a Vetial Plate with Heat and Mass Tansfe Uday Singh Rajpt Gaav Kma Effets of otation and adiation on nsteady MHD flow past a vetial plate with vaiable wall tempeate and mass diffsion in the pesene of Hall ent is stdied hee. Ealie we stdied hemial eation effet on nsteady MHD flow past an exponentially aeleated inlined plate with vaiable tempeate and mass diffsion in the pesene of Hall ent. We had obtained the eslts whih wee in ageement with the desied flow phenomenon. To stdy fthe we ae hanging the model by onsideing adiation effet on flid and hanging the geomety of the model. Hee in this pape we ae taking the plate positioned vetially pwad and otating with veloity Ω. Fthe medim of the flow is taken as poos. The plate tempeate and the onentation level nea the plate inease linealy with time. The govening system of patial diffeential eqations is tansfomed to dimensionless eqations sing dimensionless vaiables. The dimensionless eqations nde onsideation have been solved by Laplae tansfom tehniqe. The model ontains eqations of motion diffsion eqation and eqation of enegy. To analyze the soltion of the model desiable sets of the vales of the paametes have been onsideed. The govening eqations involved in the flow model ae solved by the Laplae-tansfom tehniqe. The eslts obtained have been analyzed with the help of gaphs dawn fo diffeent paametes. The nmeial vales obtained fo the dag at bonday and Nsselt nmbe have been tablated. We fond that the vales obtained fo veloity onentation and tempeate ae in onene with the atal flow of the flid. Keywods: MHD flow otation adiation Hall ent. 338

2 . Intodtion The MHD flow past a flat plate thogh poos medim is one of the lassial poblems in the flid dynamis. ppliations of the stdy aise in magneti field ontolled mateials poessing systems planetay and sola plasma flid dynamis systems and otating MHD indtion mahine enegy geneatos et. Radiation effet on ombined onvetion ove a vetial flat plate embedded in a poos medim of vaiable poosity was analyzed by Pal and Mondal [4]. Pasad et al [] have woked on adiation and mass tansfe effets on two dimensional flow past an implsively stated infinite vetial plate. Radiative MHD flow ove a non isothemal stething sheet in a poos medim was onsideed by Vyas and Sivastava [5]. Chahan and Rastogi [3] have investigated Hall ent and heat tansfe effets on MHD flow in a hannel patially filled with a poos medim in a otating system. Ghhait et al. [6] have examined ombined effets of Hall ent and adiation on MHD fee onvetive flow in a vetial hannel with an osillatoy wall tempeate. Pasada et al. [] have stdied Hall effet on fee and foed onvetive flow in a otating hannel. Hall effet on nsteady MHD fee onvetion flow ove a stething sheet with vaiable visosity and visos dissipation was analyzed by Sinivas and Naikoti [7]. Rajpt and Kanajia [8] have woked on ombined effets of Hall ent and hemial eation on nsteady MHD flow past an implsively stated vetial plate with onstant wall tempeate and mass diffsion. Chemial eation effet on nsteady MHD flow past an exponentially aeleated inlined plate with vaiable tempeate and mass diffsion in the pesene of Hall ent was stdied by s [9]. The pesent stdy is aied ot to examine the effets of otation and adiation on nsteady MHD flow thogh poos medim past a vetial plate with vaiable tempeate and mass diffsion in the pesene of Hall ent. The poblem is solved by the Laplae tansfom tehniqe. seleted set of gaphial eslts illstating the effets of vaios paametes involved in the poblem ae pesented and disssed. The nmeial vales of skin-fition and Nsselt nmbe have been tablated.. Mathematis nalysis. The geometial model of the poblem is shown in fige- 339

3 34 Fige. Physial model The x axis is taken along the vetial plate and z axis is nomal to it. Ths the z axis lies in the hoizontal plane. The flid and the plate otate as a igid body with a nifom angla veloity Ω abot z-axis. The magneti field B of nifom stength is applied pependila to the flow. Sine the flid is eletially ondting whose magneti Reynolds nmbe is vey small theefoe the inded magneti field poded by the flid motion is negligible in ompaison to the applied one. Initially it has been onsideed that the plate as well as the flid is at the same tempeate T. The speies onentation in the flid is taken as C. t time t > the plate stats moving with a veloity in its own plane and tempeate of the plate is aised to T w. The onentation C w nea the plate is aised linealy with espet to time. The flow modal is as nde:. K υ m mv B C C g T T g z v t = Ω ( ( ( ( * ρ σ β β υ (. ( ( K v m v m B z v t v υ ρ σ υ = Ω (. z C D t C = (3. z q z T k t T C p = ρ (4

4 The initial and bonday onditions ae t : = v = T = T C = C fo evey z t : = v = T = T (Tw T C = C (Cw C v T T C C as > atz= (5 z. Hee is the pimay veloity v-the seonday veloity g-the aeleation de to gavity β-volmeti oeffiient of themal expansion t-time m = ω τ ( e e is the Hall ent paamete with ωe-yloton feqeny of eletons and τe- eleton ollision time T-tempeate of the flid * β - volmeti oeffiient of onentation expansion C- speies onentation in the flid ν - the kinemati visosity ρ- the density C p- the speifi heat at onstant pesse k- themal ondtivity of the flid D- the mass diffsion oeffiient K- the pemeability paamete Tw- tempeate of the plate at z= Cw- speies onentation at the plate z= B- the nifom magneti field σ- eletial ondtivity. The loal adiant fo the ase of an optially thin gay gas is expessed by q = * 4 4 4a σ ( T T (6 z * a is absoption onstant. The tempeate diffeene within the flow is 4 whee onsideed sffiiently small hene T an be expessed as the linea fntion of 4 tempeate. This is aomplished by expanding T in a Taylo seies abot T and negleting highe-ode tems T 4T T 3T. (7 Using eqations (6 and (7 eqation (4 beomes T T * 3 ρ C p = k 6a σt ( T T (8 t z The following non-dimensional qantities ae intoded to tansfom eqations ( ( (3 and (8 into dimensionless fom: 34

5 z z = = υ * 6a συ R = k (C C C = (C C w 3 T v (T T µ υ p v = θ = S = µ = ρυ P = (Tw T D k * gβυ( T T σbυ gβ υ(cw C G = M = G m = 3 ρ K = υ w 3 t K t = υ υω Ω =. (9 The symbols in dimensionless fom ae as nde: is the pimay veloity v- the seonday veloity t- time θ - the tempeate C - the onentation G- themal Gashof nmbe Gm- mass Gashof nmbe µ- the oeffiient of visosity R- Radiation paamete K- the pemeability paamete P- the Pandtl nmbe S- the Shmidt nmbe M- the magneti paamete. The model now beomes t Ωv = G z v M ( m v Ω = v. t z ( m K C t C =. S z θ θ Rθ =. t P z P The oesponding bonday onditions (5 beome: M ( mv θ Gm C. ( ( m K ( ( (3 t : = v = θ = C = fo evey z t > : = v = θ = t C = t at z = (4 v θ C as z. Dopping bas in the above eqations we get 34

6 t Ωv = G z v v M ( m v Ω = v. t z ( m K C C =. t S z θ θ Rθ =. t P z P M ( mv θ Gm C. (5 ( m K The following bonday onditions beome t : = v = θ = C = fo evey z t > : = v = θ = t C = t at z= (9 v θ C as z. Witing the eqations (5 and (6 in ombined fom (sing q= i v (6 (7 (8 q q = G θ G t z C C = t S z θ θ Rθ =. t P z P m M ( im C q( iω m K ( ( ( Finally the bonday onditions beome: t : q = θ = C = fo all z t > : q = θ = tc = t at z= (3 q θ C as z. The dimensionless govening eqations ( to ( sbjet to the bonday onditions (3 ae solved by the sal Laplae - tansfom tehniqe. The soltion obtained is as nde: 343

7 z z S S z S 4t C = t ( ef[ ] e S t t π t - e θ = 4 Rz { ef[ R P at S Rt zp ]( Pt Rt zr e Rz ef[ [az π Rt zp ]( Pt Rt zr} az G az az az q = e 33 [e ( P ( ( te a R z3e a 4( a R R Gm az az 4 ( P ] [ e ( a3 e ( S at a 4a P G 3 5 ( S ] [ 6 6 π z( at Rt P 4 7 ( π z π ( a R 3 9 P G 6 8 π z( a R] R a π ( S ] m S e z S 4t t 5 π ( az S The expessions fo the onstants involved in the above eqations ae given in the appendix. 3. Skin fition The dimensionless skin fition at the plate z= is obtained by dq = τ x iτ y dz z = The nmeial vales of σ x and σ y ae given in table fo diffeent paametes. 4. Nsselt Nmbe The dimensionless Nsselt nmbe is given by θ Rt R P e Rt R P N = = ef[ ]( Rt t ef[ ]( t z tp 4 R tp 4 R z= Rt P π tp The nmeial vales of N ae given in table- fo diffeent paametes. 344

8 5. Reslts and Disssion In this pape we have stdied the effets of otation adiation and pemeability of poos medim on the flow. The behavio of othe paametes like magneti field Hall ent and themal boyany is almost simila to the ealie model stdied by s [9]. The analytial eslts ae shown gaphially in figes to. The nmeial vales of skin-fition and Nsselt nmbe ae pesented in table- and table- espetively. Fom figes 3 and 6 it is obseved that pimay and seonday veloities inease with pemeability paamete (K. This eslt is de to the fat that ineases in the vale of (K eslts in eding the dag foe and hene ineasing the flid veloity. Effet of otation on flid flow behavio is shown by figes 4 and 7. It is obseved that an inease in otation paamete Ω pimay veloity deeases thoghot the bonday laye egion wheeas seonday veloity ineases ontinosly nea the sfae of the plate. This implies that otation tends to aeleate seonday veloity wheeas it etads the pimay veloity in the bonday laye egion. Figes and 5 indiates the effet of adiation paamete R on both omponents of the veloity; and it is obseved that it etads the flow. s the adiation paamete ineases the tempeate of the system deeases; as a eslt flid flow beomes slow. Fthe it is obseved that the tempeate deeases when Pandtl nmbe and adiation paamete ae ineased (figes 8 and 9. Howeve fom fige it is obseved that the tempeate ineases with time. This is de to the heat whih is tanspoted to the system ontinosly. Skin fition is given in table. The vales of τ x and τ y deeases with the ineases in adiation and pemeability paamete. If otation paamete is ineased then τ x deeases bt τ y ineases. Nsselt nmbe is given in table. The vale of N deeases with inease in Pandtl nmbe adiation paamete and time. Fige. Veloity fo diffeent vales of R 345

9 Fige 3. Veloity fo diffeent vales of K Fige 4. Veloity fo diffeent vales of Ω Fige 5. Veloity v fo diffeent vales of R 346

10 Fige 6. Veloity v fo diffeent vales of K Fige 7. Veloity v fo diffeent vales of Ω Fige 8. Tempeate θ fo diffeent vales of P 347

11 Fige 9. Tempeate θ fo diffeent vales of R Fige. Tempeate θ fo diffeent vales of t Table. Skin fition fo diffeent Paametes. M m P S Gm G R Ω K t τ x τ y 348

12 Table. Nsselt nmbe fo diffeent paamete. P R t N Conlsion In this pape a theoetial analysis has been done to stdy the nsteady MHD flow thogh poos medim past a moving vetial plate with heat and mass tansfe in the pesene of Hall ent otation and adiation. The eslts obtained ae in ageement with the sal flow whih stenthenes the flid model. It has been fond that the veloity in the bonday laye egion ineases with the vales of pemeability paamete. Bt tend is evesed with adiation paamete; that is the veloity deeases when adiation paamete is ineased. The otation paamete etads the pimay flow wheeas it aeleates the seonday flow. It is also obseved that adiation and pemeability paametes deease the dag at the plate sfae. Nsselt nmbe deeases with inease in adiation paamete Pandtl nmbe and time. The eslts obtained will have appliations in the eseah elated to the sola physis dealing with the snspot development the stte of otating magneti stas ooling of eletoni omponents of a nlea eato bed themal stoage and heat sink in the tbine blades. ppendex M ( im t az a = iω. = = e ( 8 7 = m K υ az = e ( = ( = ( = 3 6 ( 3 7 = 9 7 ( 3 8 = 3 6 ( 3 9 = 4 8 ( 5 = 9 = bs [ z]. bs [ P ] P P P P = e = e = e at P Rt P z ap R 3 at z S as S at Rt P ( ap R bs[ z] 349

13 5 8 4 z S [ z ] P R = ef [ ] 6 e bs t = a t z = ef [ ] ap R ap R z t z t at z P P = ef [ ] 9 = ef [ ] = ef [ ] t t t as as z t z t S S = ef [ ] ef [ ] t t 7 bs[ z]. bs[ P ] tr = 3 = ef[ ] t P a a t S t S S S bs[ z] P R = ef [ ] 5 = ef [ ] 6 = e t t P ( ap R bs[ z] P S = e = e 7 z ap R P 3 = e bs[ z]. bs[ P ] tr ef [ ] 3 as z bs[ z]. bs[ P ] ( 8 t R ap 9 = ef [ ] t P ( P bs [ z]. bs [ P ] t( R ap = 3 = ef [ ] t P t P ( P az at z = 7 e = ef[ ] t 33 t Refeenes [] Pasada V.R. Kishna D.V. Debnath L. Hall effets on fee and foed onvetive flow in a otating hannel ta Mehania pp [] Pasad V.R. Bhaska R.N. Mthmaaswamy R. Radiation and mass tansfe effets on two-dimensional flow past an implsively stated infinite vetial plate Intenational Jonal of Themal Sienes 46 7 pp [3] Chahan D.S. Rastogi P. Hall ent and heat tansfe effets on MHD flow in a hannel patially filled with a poos medim in a otating system Tkish J. Eng. Env. Si. 33 (9 pp [4] Pal D. Mondal H. Radiation effets on ombined onvetion ove a vetial flat plate embedded in a poos medim of vaiable poosity Meania 9 44 pp [5] Vyas P. Sivastava N. Radiative MHD flow ove a non-isothemal stething sheet in a poos medim pplied Mathematial Sienes vol. 4 no pp. ( pp

14 [6] Ghhait S.K. Das S. Jana R.N. Combined effets of hall ent and adiation on MHD fee onvetive flow in a vetial hannel with an osillatoy wall tempeate Open Jonal of Flid Dynamis 3 3 pp. 9-. [7] Sinivas Maipala Naikoti Kishan Hall effets on nsteady MHD fee onvetion flow ove a stething sheet with vaiable visosity and visos dissipation Wold pplied Sienes Jonal 33 (6 5 pp [8] Rajpt U.S. Kanajia N. Combined effets of Hall ent and hemial eation on nsteady MHD flow past an implsively stated vetial plate with onstant wall tempeate and mass diffsion nalele Univesităţii Eftimie Mg Reşiţa nl XXIII NR [9] Rajpt U.S. Kma G. Chemial eation effet on nsteady MHD flow past an exponentially aeleated inlined plate with vaiable tempeate and mass diffsion in the pesene of hall ent nalele Univesităţii Eftimie Mg Reşiţa nl XXIII N ddesses: D. Uday Singh Rajpt Univesity of Lknow U P India. Gaav Kma Univesity of Lknow U P India. ajptgaavlko@gmail.om 35