MHD Oscillatory Flow in a Porous Plate

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1 Global Jounal of Mathematcal Scences: Theoy and Pactcal. ISSN 97-3 Volume, Numbe 3 (), pp Intenatonal Reseach Publcaton House MHD Oscllatoy Flow n a Poous Plate Monka Kala and H.R. Gupta Depatment of Mathematcs Hayana College of Technology & Management, Kathal, Hayana, Inda E-mal: monka.kala@gmal.com Abstact The study epoted heen deals wth the MHD heat and mass tansfe flow of a mxed convecton, ncompessble, electcally conductng, and vscous flud past an nfnte vetcal poous plate wth tme dependent sucton velocty. A unfom magnetc feld s appled n the decton nomal to the plate. Solutons fo the velocty feld and tempeatue dstbutons ae obtaned usng multpaamete petubaton technque. Appoxmate solutons fo velocty, tempeatue, skn fcton and ate of heat tansfe have been obtaned. Keywods: Unsteady flow, MHD, Heat Tansfe, Skn-fcton, Fee steam. Intoducton Unsteady mxed Convecton MHD flows ae of geat mpotance n aeonautcs, mssle aeodynamcs etc. Due to these pospects, many authos have studed fee convecton and mass tansfe flow of a vscous flud though poous medum. Lghthll (95) studed the effects of fee oscllatons on the flow of a vscous ncompessble flud past an nfnte plate. Nanda and Shama (963) analyzed fo fee convecton bounday layes along a sem-nfnte vetcal plate. Sheekanth et al. have nvestgated the effects of pemeablty vaaton on fee conventon flow past a vetcal poous wall n a poous medum when the pemeablty vaes n tme. Sngh et.al. have dscussed hydomegnetc fee convectve and mass tansfe flow of a vscous statfed flud consdeng vaaton n pemeablty wth decton. Moe ecently, Achaya et al have studed fee conventon and mass tansfe n steady fee convecton and mass tansfe n steady flow though poous medum wth constant sucton n the pesence of magnetc feld. Messha (966) studed two dmensonal ncompessble flud flow poblems along an nfnte flat plate wth no heat tansfe between the flud and the plate when the sucton velocty nomal to the plate as well

2 3 Monka Kala and H.R. Gupta as the extenal flow vaes peodcally wth tme. Futhe Rapts and Pedks (985) studed the unsteady two dmensonal fee convectve flows though hghly poous medum. Recently, Ahmed and Ahmed () analyzed the effect of two dmensonal MHD oscllatoy flows along a unfomly movng nfnte vetcal poous plate bounded by poous medum. Ahmed (7) wee nvestgated the effects of unsteady fee convectve MHD flow though a poous medum bounded by an nfnte vetcal poous plate. Soundalgeka (973 a, b) consdeed the fee convecton effects on the oscllatoy flow past an nfnte vetcal poous plate wth constant sucton. Vghnesam and Soundalgeka (998) studed the combned fee and foced convecton flow of wate fom a vetcal plate wth vaable tempeatue. Sahoo et.al. (3) have analyzed the effects of MHD unsteady fee convecton flow past an nfnte vetcal plate wth constant sucton and heat snk. The study epoted heen analyses the effect of tme dependent sucton and peodc heat tansfe on unsteady mxed convecton MHD flow past a vetcal poous flat plate. The vaous govenng equatons of the poblem unde consdeaton ae solved by usng petubaton technque. Fomulaton of the poblem Consde the two dmensonal flow of an ncompessble, electcally conductng, unsteady and mxed convecton flud. The plate heen s assumed to be nfnte n length, vetcal n decton and flat. The pemeablty and sucton velocty s taken to be tme dependent. In the Catesan coodnate system, let x-axs be along the plate n the decton of the flow and z -axs nomal to t. A unfom magnetc feld s ntoduced nomal to the decton of flow. We assume that the magnetc Reynolds numbe s vey small so that the ncluded magnetc feld s neglected. The nfluence of the densty vaatons s also neglgble. Wth the above assumptons, the govenng equatons of contnuty, momentum and enegy and heat tansfe ae gven by, w w w + Α e z wt ( ε ) u u u du σ B u + w gβ( T T ) + ν + t z z dt ρ Whee w > () () T T T ν u w s ( T T ) κ t z z cp z (3) Also the fee steam velocty oscllates wth tme s assumed to be of the fom: ( ) ( wt t +εe ) Intoducng the followng dmensonless quanttes:

3 MHD Oscllatoy Flow n a Poous Plate 33 wz u U z u U t tw,,, / ν, ν U U ( w ) w ων ν S T T ω, S, ν u/ ρ, T, w w T T νgβ T T ν M σb ν / ρw, G,P, κ κ / κ ( ρc p), Ec c T T p w ( w ) Wth the help of dmensonless quanttes, the equatons () and (3) educe to u wt wt ( ) u + εα e GT u w + + ε e Mu t z z P T wt ( ) T T e P P u + εα + ST + Ec t z z z () (5) And the non-dmensonal fee steam s ( t) wt +εe (6) The elevant bounday condtons n non-dmensonal fom ae wt u, T + εe at y u, T at y (7) Method of Soluton In ode to solve the equatons () and (5) unde the bounday condton (7), we assume wt u( z, t) u( z) + εe u( z) wt (8) T( z, t) T ( z) εe T ( z) + Whee u and T ae espectvely the mean velocty and mean tempeatue. Substtutng (8) nto the equatons () and (5), equatng the hamonc and non hamonc tems and neglectngε, we get u + u Mu G T (9) T T ST Ec u + P + P / P ( ) () ω u + u ( M + ω /) u GT Au () T + P T + P ( S ω ) T / P Ecu u () Whee the pmes denote dffeental wth espect to z.

4 3 Monka Kala and H.R. Gupta The equatons (9) to () ae stll coupled fo the vaables u, u, T and T. To solve them, t s to be noted that E c << fo all ncompessble flud and assumed that: ( ) F y F ( ) ( ) ( ) z + E z + o E (3) c c u Whee F stands fo, u, T and T E On usng (3) nto equatons (9) to () and equatng the lke powes of c, the followng equatons ae obtaned: u + u Μ u G Τ () u + u Μ u G Τ (5) ω u + u ( Μ+ ω /) u GΤ Α u (6) u + u ( Μ+ ω /) u GΤ Α u (7) Τ +ΡΤ + Ρ Τ (8) Τ ( ) +ΡΤ + Ρ Τ Ρ u (9) Τ +Ρ ( ) Τ +Ρ S ω T () ( ω ) Τ +Ρ Τ +Ρ S T / P u u () Subject to the bounday condtons: u, u u u, () T, T, T, T atz u, u, u, u, (3) T T T T atz In vew of the bounday condtons () and (3) the solutons of the dffeental equatons () to () ae: T e Az Whee, ()

5 MHD Oscllatoy Flow n a Poous Plate 35 Α P + P P S P+ P P S, A ( ) u e + R e R e (5) Bz B z A z Whee, + + M M B, B + +, R G Α ΑM ( ) T De R e R e R e + R e Β Β Α z Bz Bz Α z z 3 5 ( + ) ( ) B A z B A z 6 Re 7 + Re (6) Whee D R + R3 + R R5 R6 + R7 u D e R e + R e + R e + R e Bz Α z Β z Β z Αz 8 9 ( Β Β ) ( Β +Α ) ( Β Α ) z z z 3 R e R e + R e (7) Whee D R8 R9 R R+ R + R3 R 3 T e Az Whee, A (8) ( ω) ( ω) P + P P S P + P P S 3, A u e + D e Ae + A Ae + Ae Ae (9) Bz 3 Bz Az 3 Bz Bz Az Whee [ ] D + A A + A A + A B M + ω +, M + B ω + +, 3 A5 G A3 A3 M ω + ( B + B ) z ( ) ( B A ) z ( ) ( ) Az 3 3 B B z 3 B B z B A z T D e A e A e A e A e A e

6 36 Monka Kala and H.R. Gupta ( ) ( B + B ) z ( + ) ( B + A ) z ( ) B3 B z B A3 z B3 A z A e A e + A e + A e A e ( ) ( + ) A+ B z A A3 z Bz Bz Az 3 + A e A e + A e A e A e (3) Whee, D [ A A + A + A + A A + A A A + A A + A A + A + A ] ( Β +Β ) z ( Β Β ) Βz Α3z 3 z Α 5 +Α6 Α7 u D e e e e ( Β Α ) ( Β Β ) ( Β +Β ) z ( Β +Α ) e z 9e z z 3e 3e +Α Α +Α Α ( Β Α ) ( Α +Β ) z ( Α + ) 3 z A3 z Βz 3e 33e 3e 35e +Α Α +Α +Α Α z Β z Β z Α z ( ) Α e Α e +Α e +Α e +Α e Β Β ( ) z ( Β Α) Β +Α z z Α e + Α e (3) Whee, D A A + A A + A A A A + A + A A A + A A A A + A A ] and the othe constants ae not pesented hee fo the sake of bevty. Sepaatng eal and magnay pats of the velocty and tempeatue expesson (8) and takng only the eal pats, the velocty and tempeatue felds n tems of the fluctuatng pats gven by: u u ( z) + ε ( M cosω t- M Snω t) (3) ( ) T T z + ε ( T cosω t- T Snω t) (33) Hence expessons fo tansent velocty and tempeatue fo ωt Π ae: Π Π u z, u( z) ε M, T z, T( z) εt ω ω (3) Skn Fcton and Rate of Heat Tansfe Skn fcton coeffcent at the plate can be calculated n non-dmensonal fom s: u ωt τw u ( ) + εe u ( ) (35) z z

7 MHD Oscllatoy Flow n a Poous Plate 37 Splttng the equaton (35) nto eal and magnay pats and takng eal pat only τ w τ + ε N cos( ωt+ α) (36) Ν Ν Ν +Ν,tan α, Ν Re ( Ν ) M, Ν Im( Ν ) M. Ν Heat tansfe coeffcent can be calculated n non-dmensonal fom... T qw T + e T z z wt ( ) ( ) (37) Splttng equaton (37) nto eal and magnay pats and takng eal pats only qw q + Q cos( ωt+ β ), Q Q Q + Q,tanβ, Q Re ( ), Im ( ) ( ) ( ) Q Q T Q Q T q T A + EcD + Ec ( ) ( ) ( ) BR + BR + AR + B B R B+ A R B A R (38) Expesson fo Ν, Ν, Q and Q ae not pesented hee fo the sake of bevty Dscusson of the Result We have solved the poblem of MHD oscllatoy flow n a poous plate. Now t s vey dffcult to study the effects of all the paametes nvolved n the poblem. Theefoe a few of them, whch ae compaatvely mpotant, have been selected and the nfluence on the flow has been dscussed. Solutons fo the velocty feld and tempeatue dstbuton ae obtaned usng the petubaton technque. Appoxmate solutons fo the velocty, tempeatue, skn fcton and ate of heat tansfe have been obtaned. Nomenclatue ( uw, ) Velocty components along x and z decton espectvely, w u g Β t Mean sucton velocty, Dmensonless velocty component, Acceleaton due to gavty, Magnetc feld, Tme,

8 38 Monka Kala and H.R. Gupta U T w T T T P G S M τ q E c Fee steam velocty, Tempeatue at the plate, Fee steam tempeatue. Flud tempeatue, Dmensonless tempeatue, Pandtl numbe, Gashof numbe, Snk stength, Hatman numbe, Mean skn-fcton, Mean heat tansfe and Ecket numbe, Geek symbols β Coeffcent of volume expanson, ε Ampltude paamete, κ Themal conductvty, ρ Densty, μ Coeffcent of vscosty, υ Knematcs vscosty, ω Fequency paamete, Subscpts w Evaluated at wall condtons Evaluated at fee steam condtons Refeences [] Lghthll, M.J., The esponse of lamna Skn fcton and heat tansfe to fluctuatons n the steam velocty, Poc. Roy. Soc. A, -6 (95) [] Nanda, R.S. and V.P.Shama, Fee Convecton bounday layes along a semnfnte vetcal plate, J. Flud Mech. 5, 9-8(963).

9 MHD Oscllatoy Flow n a Poous Plate 39 [3] S.Sheekanth, S. Venkataamana and S. Ramakshna, Acta Cenca Indca, M (996) [] N.P. Sngh, Ajay Kuma sngh, M.K.Yadav and Atul Kuma Sngh, J, Enegy Heat Mass Tansfe, (999) -5 [5] M. Achaya. G.C. Dash and L.P. Sngh, Ind. J. Pue app. Math, 3 () -8. [6] Messha, S.A.S., Lamna bounday layes n oscllatoy flow along an nfnte flat plate wth vaable sucton, Poc. Camb. Phl. Soc., 6, (966) [7] Rapts, A.A. and C.P. Pedks, Oscllatoy flow though a poous medum by the pesence of fee convectve flow, Int. J. Engng. Sc., 3, 5-55 (985). [8] Ahmad, S. and N. Ahmad, Two-dmensonal MHD oscllatoy flow along a unfomly movng nfnte vetcal poous plate bounded by poous medum, IJPAM, 35, () [9] Ahmed, S., Effects of unsteady fee convectve MHD flow though a poous medum bounded by an nfnte vetcal poous plate, Bull. Cal. Math. Soc., 99, 5-5 (7). [] Soundagelka, V.M. (973), Fee convecton effects on the oscllatoy flow past an nfnte vetcal poous plate wth constant sucton., Poc. Royal Soc. Landon A 333: [] Soundagelka, V.M. (973), Fee convecton effects on the oscllatoy flow past an nfnte vetcal poous plate wth constant sucton., Poc. Royal Soc. London A 333: [] Vghnesam, N. V. and V.M. Soundagelka, Combned fee and foce. convecton flow of wate at c fom a vetcal plate wth vaable tempeatue, Ind.J. Of Engneeng and Mateal Scence, 5, -6 (998). [3] Sahoo, P.K. N. Datta and S. Bswal, MHD unsteady fee convecton flow past an nfnte vetcal plate wth constant sucton and heat Snk, IJPAM, 3, 5-55 (3).

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