INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: ), VOL. 5, NO. 2,

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1 INTERNTIONL JOURNL OF MTHEMTICS ND SCIENTIFIC COMPUTING (ISSN: -5) VOL. 5 NO. 5 Unsead MHD Flow hrogh Poros Medim Pas an Implsivel Sared Inclined Oscillaing Plae wih Variable Temperare and Mass Diffsion in he Presence of Hall Crren U. S. Rajp and Garav Kmar BSTRCT- Unsead MHD flow hrogh poros medim pas an implsivel sared inclined oscillaing plae wih variable emperare and mass diffsion in he presence of Hall crren is sdied here. The flid considered is gra absorbing-emiing radiaion b a non-scaering medim. The Governing eqaions involved in he presen analsis are solved b he Laplaceransform echniqe. The veloci profile is discssed wih he help of graphs drawn for differen parameers like Grashof nmber mass Grashof Nmber Prandl nmber Hall crren parameer permeabili parameer phase angle he magneic field parameer and Schmid nmber and he nmerical vales of skin-fricion have been ablaed. Index Terms MHD flow oscillaing inclined plae poros medim Variable Temperare Mass Diffsion and Hall crren. MSC Codes 76D5 76S5 76W5. T I. INTRODUCTION he sd of MHD flow hrogh poros medim associaed wih hea and mass ransfer plas imporan roles in differen areas of science and echnolog like chemical engineering mechanical engineering biological science perolem engineering biomechanics irrigaion engineering and aerospace echnolog. Sch problems freqenl occr in pero-chemical indsr chemical vapor deposiion on srfaces cooling of nclear reacors hea exchanger design fores fire dnamics and geophsics. The inflence of magneic field on viscos incompressible and elecricall condcing flid is of grea imporance in man applicaions sch as magneic maerial processing glass manfacring conrol processes and prificaion of crde oil. The response of laminar skin fricion and hea ransfer o flcaions in he sream veloci was sdied b Lighhill[. The sd of MHD flow pas an implsivel sared verical plae wih variable emperare and mass diffsion was sdied b Rajp and Kmar[5. MHD flow Dr. U S Rajp is an ssisan Professor in he Deparmen of Mahemaics & sronom Lcknow Universi Lcknow-67 Uar Pradesh India. Garav Kmar is a research scholar in he Deparmen of Mahemaics & sronom Lcknow Universi Lcknow-67 Uar Pradesh India. (Corresponding ahor. rajpgaravlko@gmail.com). beween wo parallel plaes wih hea ransfer was invesigaed b ia e al[. Rapis and Kafosias[ have sdied flow of a viscos flid hrogh a poros medim bonded b a verical srface. Daa and Jana[5 have sdied oscillaor magneohdrodnamic flow pas a fla plae will Hall effecs. Sondalgekar[6 has invesigaed free convecion effecs on he oscillaor flow an infinie verical poros plae wih consan scion. Hea ransfer in flow hrogh a poros medim bonded b infinie verical plae nder he acion of magneic field was sdied b Rapis and Kafosias[. Longidinal vorices in naral convecion flow on inclined plaes were sdied b Sparrow and Hsar[9.The researchers have sdied he effec of Hall crren in varios flow models. ia[9 has considered he effec of variable properies on he nsead Harmann flow wih hea ransfer considering he Hall effec. ia and hmed[8 have sdied he Hall effec on nsead MHD coee flow and hea ransfer of a Bingham flid wih scion and injecion. Deka[6 has considered Hall effecs on MHD flow pas an acceleraed plae. Mhcmaraswam and Janakiraman[7 have analed mass ransfer effec on isohermal verical oscillaing plae in he presence of chemical reacion. Hall effecs on free and forced convecive flow in a roaing channel was sdied b Rao e al[4. Pop [7 has invesigaed he effec of Hall crren on hdromagneic flow near an acceleraed plae. Pop and Waanabe[ have frher sdied Hall effecs on magneohdrodnamic bondar laer flow over a coninos moving fla plae. The effec of Hall crren on he magneo hdrodnamic bondar laer flow pas a semi-infinie plae was sdied b Kaagiri[8. Combined effecs of radiaion and Hall crren on MHD flow pas an exponeniall acceleraed verical plae in he presence of roaion were sdied b Thamihsdar and Pandrangan[. Maripala and Naikoi[ have analed Hall effecs on nsead MHD free convecion flow over a sreching shee wih variable viscosi and viscos dissipaion. Slochana[ has invesigaed Hall effecs on nsead MHD hree dimensional flow hrogh a poros medim in a roaing parallel plaes channel wih effec of inclined magneic field. Singh[4 has considered hea and mass ransfer in MHD bondar laer flow pas an inclined plae wih viscos dissipaion in poros medim. We are considering nsead MHD flow hrogh poros medim pas an implsivel sared inclined oscillaing plae wih variable

2 INTERNTIONL JOURNL OF MTHEMTICS ND SCIENTIFIC COMPUTING (ISSN: -5) VOL. 5 NO. 5 emperare and mass diffsion in he presence of Hall crren. The resls are shown wih he help of graphs and able. II. MTHEMTICL NLYSIS In his paper we have considered MHD flow beween wo parallel elecricall non condcing plaes inclined a an angle from verical. x axis is aken along he plae and normal o i. ransverse magneic field B of niform srengh is applied on he flow. The viscos dissipaion and indced magneic field has been negleced de o is small effec. Iniiall i has been considered ha he plae as well as he flid is a he same emperare T and he concenraion level C everwhere in he flid is same in saionar condiion. ime > he plae sars oscillaing in is own plane wih freqenc and emperare of he plae is raised o T and he concenraion level near he plae is raised w linearl wih respec o ime. De o he Hall effec here will be wo componens of he momenm eqaion he flow modal is as nder: gcos ( T T ) () * B ( mv) g Cos ( C C ) ( m ) K v v B ( m v) v ( m ) K () C C D () T T C p k (4) wih he corresponding iniial and bondar condiions: : v T T C C for all : Cos v ( ) T T T w T ( ) C C C w C a v T T C C as Here is he Primar veloci v - he secondar veloci g- he acceleraion de o gravi - volmeric coefficien of hermal expansion - ime m ) is he Hall parameer ( e e wih e - ccloron freqenc of elecrons and e - elecron collision ime K- he permeabili parameer T- emperare * of he flid - volmeric coefficien of concenraion (5) expansion C- species concenraion in he flid - he kinemaic viscosi - he densi C - he specific hea a consan pressre k- hermal condcivi of he flid D- he mass diffsion coefficien T - emperare of he plae a = w C - species concenraion a he plae = B w - he niform magneic field - elecrical condcivi. The following non-dimensional qaniies are inrodced o ransform eqaions () () () and (4) ino dimensionless form: υ S c ωυ ω υ P r D M v μc p gβ υ (T w T ) G r k σb υ * gβ (C w C ) G m ρ (C C ) μ ρυ C (C w C ) v θ p (T T ) (T w T ) K υ υ where is he dimensionless Primar veloci v - he secondar veloci - dimensionless ime - he dimensionless emperare C - he dimensionless concenraion K - he dimensionless permeabili parameer G - hermal Grashof nmber G - mass Grashof nmber r - he coefficien of viscosi r m K. (6) P - he Prandl nmber he Schmid nmber M- he magneic parameer. Ths he model becomes G G C r Cos m Cos M ( mv ) ( m ) K v M ( m v) v ( m ) K C C S c P r wih he following bondar condiions: : v C for all (7) (8) (9) () Sc -

3 INTERNTIONL JOURNL OF MTHEMTICS ND SCIENTIFIC COMPUTING (ISSN: -5) VOL. 5 NO. 5 : Cos v C a = () v C as. Dropping bars in he above eqaions we ge G r Cos α θ G m Cos α C M( mv) ( m ) K v v M ( m v) v ( m ) K C C S c P r wih he following bondar condiions: () () (4) (5) : v C for all : Cos v C a = (6) v C as. Wriing he eqaions () and () in Combined form: q q G r Cos α θ G m Cos α C qa (7) C C (8) S c (9) P r wih he following bondar condiions: : q C for all : q Cos C a = () q C as. M ( im) Here q= + i v a. m K The dimensionless governing eqaions (7) o (9) sbjec o he bondar condiions () are solved b he sal Laplace - ransform echniqe. The solion obained is as nder: C S ( c S c S c ) erfc[ e 4 Sc P ( r P r ) erfc[ 8 P r a e ae S c G m { a 6 P r } a S c P r e 4 Pr iw Cos a q e 5 [ G { r 4 9 ae 4a a e P r P r } G r P r { a P r a 4 P r P r 4 5 S c } S c }. 4 S c 6 The expressions for he consans involved in he above eqaions are given in he appendix. G m { 9 S c 4 7 S c III. SKIN FRICTION The dimensionless skin fricion a he plae = dq x i d Separaing real and imaginar par in dq he d dimensionless skin fricion componen d and x d dv can be comped. d IV. RRESULT ND DISCUSSIONS The veloci profile for differen parameers like hermal Grashof nmber Gr magneic field parameer M Hall parameer m Prandl nmber Pr and ime is shown in figres. o.9. I is observed from figres. and. ha he primar and secondar velociies of flid decrease when he angle of inclinaion ( ) is increased. I is observed from figre. and. when he mass Grashof nmber is increased hen he velociies are increased. From figres. and. i is dedced ha when hermal Grashof nmber Gr is increased hen he velociies are increased. I is observed ha velociies increase when permeabili parameer is increased (figres. and.). If Hall crren parameer m is increases hen he

4 INTERNTIONL JOURNL OF MTHEMTICS ND SCIENTIFIC COMPUTING (ISSN: -5) VOL. 5 NO. 5 primar veloci increases and secondar veloci decreases (figres.4 and.4). I is observed from figres.5 and.5 ha he effec of increasing vales of he parameer M resls in decreasing and increasing v. i is dedced ha when phase angle is increased hen he velociies ge decreased (figres.6 and.6). Frher i is observed ha velociies decrease when Prandl nmber is increased (figres.7 and.7). When he Schmid nmber is increased hen he velociies ge decreased (figres.8 and.8). Frher from figres.9 and.9 i is observed ha velociies increase wih ime. Skin fricion is given in able. The vale of x increases wih he increase in hermal Grashof nmber mass Grashof Nmber Permeabili parameer Hall crrens parameer and ime and i decreases wih he angle of inclinaion of plae he magneic field phase angle Prandl nmber and Schmid nmber. The vale of increases wih he increase in hermal Grashof nmber he magneic field Permeabili parameer mass Grashof Nmber and ime and i decreases wih he angle of inclinaion of plae Hall crren parameer phase angle Prandl nmber and Schmid nmber. Figre.: v/s Figre.: v/s Figre.: v/s Figre.4: v/s Figre.: v/s Figre.5: v/s

5 INTERNTIONL JOURNL OF MTHEMTICS ND SCIENTIFIC COMPUTING (ISSN: -5) VOL. 5 NO. 5 4 Figre.6: v/s Figre.: v v/s Figre.7: v/s Figre.: v v/s Figre.8: v/s Figre.: v v/s Figre.9: v/s Figre.: v v/s

6 INTERNTIONL JOURNL OF MTHEMTICS ND SCIENTIFIC COMPUTING (ISSN: -5) VOL. 5 NO. 5 5 Figre.4: v v/s Figre.8: v v/s Figre.5: v v/s Figre.9: v v/s V. CONCLUSION Figre.6: v v/s The conclsions of he sd are as follows: Primar Veloci increases wih he increase in hermal Grashof nmber mass Grashof Nmber permeabili parameer Hall crren parameer and ime. Primar Veloci decreases wih he angle of inclinaion of he plae he magneic field phase angle Prandl nmber and Schmid nmber. Secondar Veloci increases wih he increase in hermal Grashof nmber mass Grashof Nmber permeabili parameer he magneic field and ime. Secondar Veloci decreases wih he angle of inclinaion of plae Hall crren parameer Prandl nmber and Schmid nmber. x increases wih he increase in Gr Gm m and and i decreases wih angle of inclinaion of plae M Pr and Sc. increases wih he increase in Gr Gm M and and i decreases wih angle of inclinaion of he plae m Pr and Sc. Figre.7: v v/s

7 INTERNTIONL JOURNL OF MTHEMTICS ND SCIENTIFIC COMPUTING (ISSN: -5) VOL. 5 NO. 5 6 e e a 6 e a 6 e PPENDEX a 7 a 7 ap r P r 8 ap r P r a P r a P r P r 5 9 as c S c 6 9 P r as c S c a S c S c a S c S c 9 a e ( a) P r e 4 P ( ) P r r S c a apr a as Pr Pr S c e Sc c 4 e a 5 6 apr a 7 Pr 8 e 9 asc S c aiw aiw e e e e e e aiwiw e e aiwiw aiw aiw e aiwiw a iw aiwiw REFERENCES a iw. a iw a iw [ M. Srinivas and N. Kishan Hall effecs on nsead MHD free convecion flow over a sreching shee wih variable viscosi and viscos dissipaion World pplied Sciences Jornal vol. no. 6 pp [ M. Thamihsdar and J. Pandrangan Combined effecs of radiaion and Hall crren on MHD flow pas an exponeniall acceleraed verical plae in he presence of roaion Inernaional Jornal of Innovaive Research in Comper and Commnicaion Engineering vol. no. pp [ P. Slochana Hall effecs on nsead MHD hree dimensional flow hrogh a poros medim in a roaing parallel plaes channel wih effec of inclined magneic field merican Jornal of Compaional Mahemaics vol. 4 pp [4 P. K. Singh Hea and mass ransfer in MHD bondar laer flow pas an inclined plae wih viscos dissipaion in poros medim Inernaional Jornal of Scienific & Engineering Research vol. no. 6 pp. -. [5 U. S. Rajp and K. Srendra MHD Flow Pas an Implsivel Sared Verical Plae wih Variable Temperare and Mass Diffsion pplied Mahemaical Sciences vol. 5 no. pp [6 R. K. Deka Hall effecs on MHD flow pas an acceleraed plae Theore. ppl. Mech. vol. 5 no. 4 pp [7 R. Mhcmaraswam and B. Janakiraman Mass ransfer effecs on isohermal verical oscillaing plae in he presence of chemical reacion Inernaional Jornal of pplied Mahemaics and Mechanics vol. 4 no. pp [8. H. li and. M. Eissa Saed The Hall effec on nsead MHD Coee flow wih hea ransfer of a Bingham flid wih scion and injecion pplied Mahemaical Modelling vol. 8 pp [9 H.. ia The effec of variable properies on he nsead Harmann flow wih hea ransfer considering he Hall effec ppl. Mah. Model. vol. 7 no. 7 pp [ H.. ia and N.. Kob MHD flow beween wo parallel plae wih hea ransfer ca Mechanica vol. 7 pp [ I. Pop and T. Waanabe Hall effecs on magneohdrodnamic bondar laer flow over a coninos moving fla plae ca Mech.vol. 8 pp [. Rapis and C. Perdikis Flow of a viscos flid hrogh a poros medim bonded b a verical srface In. J. Eng. Sci. vol. pp [. Rapis and N. Kafosias Hea ransfer in flow hrogh a poros medim bonded b infinie verical plae nder he acion of magneic field. Inernaional Jornal of Energ Research vol. 6 pp [4 D.R.V. Prasada Rao and D.V. Krishna Hall effecs on free and forced convecive flow in a roaing channel. ca Mechanica vol. 4 no. pp [5 N. Daa and R. N. Jana Oscillaor magneo hdrodnamic flow pas a fla plae will Hall effecs J. Phs. Soc. Japan vol. 4 pp [6 V. M. Sondalgekar Free convecion effecs on he oscillaor flow an infinie verical poros plae wih consan scion I Proc. R. Soc. vol. pp

8 INTERNTIONL JOURNL OF MTHEMTICS ND SCIENTIFIC COMPUTING (ISSN: -5) VOL. 5 NO. 5 7 [7 I. Pop. The effec of Hall crrens on hdromagneic flow near an acceleraed plae J. Mah. Phs. Sci. vol. 5 pp [8 M. Kaagiri The effec of Hall crren on he magneo hdrodnamic bondar laer flow pas a semi-infinie plae Jornal of he Phsical Socie of Japan vol. 7 no. 4 pp [9 E. M. Sparrow and R. B. Hsar Longidinal vorices in naral convecion flow on inclined plaes J. Flid. Mech. vol. 7 pp [ M. J. Lighhill The response of laminar skin fricion and hea ransfer o flcaions in he sream veloci Proc. R. Soc. vol. 4 pp Table : Skin fricion for differen parameers M m Pr Sc Gm Gr K x