Combined effects of Hall current and rotation on free convection MHD flow in a porous channel

Transcription

1 Idia Joural of Pur & Applid Physics Vol. 47, Sptbr 009, pp Cobid ffcts of Hall currt ad rotatio o fr covctio MHD flow i a porous chal K D Sigh & Raksh Kuar Dpartt of Mathatics (ICDEOL, H P Uivrsy, Shila Dpartt of Mathatics, Govt Collg for Girls (RKMV, Logwood Shila 7 00 E-ail: kdsighshila@gail.co Rcivd 4 March 009; rvisd 7 Ju 009; accptd 3 July 009 A thortical aalysis of a oscillatory fr covctiv MHD flow of a viscous icoprssibl ad lctrically coductig fluid i a vrtical porous chal i th prsc of Hall currt has b carrid out. Th two isulatig porous plats ar subjctd to a costat ijctio ad suctio. A uifor agtic fild is applid i th dirctio oral to th plats. Th tir syst rotats about th axis oral to th plats wh uifor agular vlocy Ω. For sall ad larg rotatios, th dpdc of th stady ad ustady rsultat vlocis ad thir phas diffrcs o various paratrs hav b discussd i dtail. Th rsults show that both stady ad ustady rsultat vlocis icras rapidly fro ro ar th statioary plat ad th approach to uy i th for of dapd oscillatios. For larg valus of rotatios ad ijctio/suctio at th plats, a phas lag is also obsrvd for both stady ad ustady phas agls. Kywords: Oscillatory fr covctiv flow, Rotatig, Hall currt, MHD flow Itroductio Th hydroagtic covctio wh hat trasfr i a rotatig diu has b studid du to s iportac i th dsig of agtohydrodyaics (MHD grators ad acclrators i gophysics, th udr groud watr rgy storag syst, uclar powr ractors, soil scics, astrophysics ad MHD boudary layr cotrol of rtry vhicls. I rct yars, th probl of fr covctio has attractd th atttio of a larg ubr of rsarchrs du to s divrs applicatios. O accout of thir varid iportac, such a flow has b studid by Myr. Attia ad Kotb studid th MHD flow btw two paralll porous plats. I th rct yars, a ubr of studis hav b do o th fluid phoo o arth ivolvig rotatio to a gratr or lssr xtt. Sigh 3 studid th ustady fr covctiv flow through rotatig porous diu boudd by a ifi vrtical porous plat. Th xact solutio of oscillatory Eka boudary layr flow through a porous diu boudd by two horiotal flat plats i rotatig syst has b studid by Sigh t al 4. Th fluid flows i rotatig chals hav b studid by ay rsarchrs 5-8. Wh th strgth of th agtic fild is strog, o caot glct th ffcts of Hall currts. It is of cosidrabl iportac ad itrst to study how th rsults of th hydrodyaical probls gt odifid by th ffcts of Hall currts. Th Hall currts giv ris to a cross flow akig th flow thr-disioal. Fr covctiv flows i th prsc of Hall currts hav b studid by ay rsarchrs 9-4. I th prst study, a oscillatory fr covctiv MHD flow i a rotatig vrtical porous chal has b studid wh th tir syst rotats about a axis prpdicular to th plas of th plats ad is also th axis alog which a strog agtic fild of uifor strgth is applid. Forulatio of obl Cosidr a oscillatory fr covctiv flow of a viscous icoprssibl ad lctrically coductig fluid btw two isulatig ifi vrtical porous plats distac d apart. A costat ijctio vlocy, w 0, is applid at th statioary plat 0 ad th sa costat suctio vlocy, w 0, is applid at th plat d, which is oscillatig i s ow pla wh a vlocy U(t about a o-ro costat a vlocy U 0. Th origi is assud to b at th plat 0 ad th chal is oritd vrtically upward alog th x-axis. Th chal rotats as a rigid body wh agular vlocy Ω about th -axis prpdicular to th plas of th plats. A strog trasvrs agtic fild of uifor strgth H 0 is also applid alog th axis of rotatio. Th valu of this uifor agtic fild is assud to b ualtrd

2 68 INDIAN J PURE & APPL PHYS, VOL 47, SEPTEMBER 009 by akig th cssary assuptios that guarat th glctio of iducd lctric ad agtic filds. Usig th rlatio. H 0 for th agtic fild H ( H x, H y, H, H H0 is obtaid vrywhr i th fluid (H 0 is a costat. If J ( J x, J y, J is th currt dsy, fro th rlatio. J 0, o hav J costat. Sic th chal is o-coductig, J 0 vrywhr. Th gralid Oh s law, i th absc of th lctric fild (Mayr, is of th for: ω τ J + ( J H σ µ V H + p H 0 whr V, σ, µ, ω, τ,, ad p ar rspctivly th vlocy, th lctrical coductivy, th agtic prabily, th cyclotro frqucy, th lctro collisio ti, th lctric charg, th ubr dsy of th lctro ad th lctro prssur. Udr th usual assuptios that th lctro prssur (for a wakly ioid gas, th throlctric prssur ad io slip ar gligibl, o hav fro th Oh s law J + ω τ J σµ H v x y 0 J ω τ J σµ H u y x 0 fro which o obtais σµ H0 J x ( u + v σµ H0 J y ( v u whr ωτ is th Hall paratr. Sic th plats ar ifi i xtt, all th physical quatis xcpt th prssur dpd oly o ad t. Th physical cofiguratio of th probl is show i Fig.. Dotig th vlocy copots u, v, w i th x, y, dirctios rspctivly ad tpratur by T, th flow i th rotatig syst i th prsc of Hall currt is govrd by th followig quatios: w 0, which itgrats to w w0, ( u u p u + w 0 + υ + Ω v t ρ x σµ H 0 ( v u + + gβ T T ρ ( + ( d ( Fig. Physical cofiguratio of th probl v v p v + w 0 + υ Ω u t ρ y σµ H 0 ( u + v, (3 ρ ( + T T k T + w t ρc 0 P, (4 whr υ is th kiatic viscosy, t is th ti, ρ is th dsy ad p is th odifid prssur, σ is th lctrical coductivy, T is th tpratur, C P is th spcific hat at costat prssur, k is th thral coductivy, g is th acclratio du to gravy, β is th cofficit of volu xpasio ad is th Hall paratr. Th boudary codios for th probl ar u v 0, T T0 + ( T0 Td cos ω t at 0, u ( t U ( t U0( cos ω t, v 0, T Td at d, (5 whr ω is th frqucy of oscillatio ad is a vry sall posiv costat. Now itroduc th followig o-disioal quatis ito Eqs (-(4: u v η, t ω t, u, v, d U U paratr, 0 0 Ω d Ω (rotatio υ ω d ω (frqucy paratr, υ w0d λ (ijctio/suctio paratr, M H 0d σ υ µ

3 SINGH & KUMAR: FREE CONVECTION MHD FLOW IN POROUS CHANNEL 69 υ gβ( T0 Td (Harta ubr, Gr (Grashoff U w 0 0 ubr, T T d µ CP θ ad T0 Td k (adtl ubr. Aftr cobiig Eqs ( ad (3 ad by takig q u+iv, Eqs (-(4 rduc to: q q q du ω + λ + ω iω( q U t η η dt ( ( M + i q U + Gr λ θ ω + λ t η η θ θ θ (6 (7 Th boudary codios (5 ca also b wrt i coplx otatios as: q 0, θ + ( + at η 0 q U ( t + ( + at η (8 I ordr to solv th syst of Eqs (6 ad (7 subjct to th boudary codios (8, is assud q( η, t q0 ( η + { q ( η + q ( η }, (9 θ( η, t θ0( η + { θ( η + θ( η }. (0 Substutig Eqs (9 ad (0 ito Eqs (6 ad (7 ad coparig th haroic ad o-haroic trs, o gts q λ q Sq S Grλ θ ( q λ q ( S + iω q ( S + iω Grλ θ ( q λ q ( S iω q ( S iω Grλ θ (3 θ λ θ 0 (4 0 0 θ λ θ iωθ 0 (5 θ λ θ + iω θ 0 (6 M ( + i whr S + iω ad dashs dot th drivativs w.r.t. η. Th corrspodig trasford boudary codios rduc to q0 q q 0, θ 0 θ θ at η 0 (7 q q q, θ θ θ 0 at η 0 0 Th solutios of Eqs (-(6 udr th boudary codios (7 ar q ( η + A ( A ( η η η λ η 0 η η ( λ + A + A ( { ( ( } q ( η + B ( B ( (8 4η 4η η η η 3η 4η B + B 3 4 ( { ( ( ( } (9 q ( η + C ( C ( 6η 6η 4η 6η 3η 5η 6η C + C 5 6 ( { ( ( ( } (0 λ η λ θ0( η λ ( θ ( η + η + η θ ( η 3 + 4η 4 + 3η 3 4 whr λ + λ + ω λ λ + ω 3 λ + λ ω 4 λ λ ω 3 4 λ + λ + λ λ + 4S 4S 4( λ + λ + S + iω 4( λ λ + S + iω ( (3

4 60 INDIAN J PURE & APPL PHYS, VOL 47, SEPTEMBER ( λ + λ + S iω λ λ + 4( S iω 6 λ λ Gr A λ S( λ Gr A λ ( λ ( S λ Gr B ( λ ( S + iω λ Gr B ( λ ( S + iω 3 λ Gr C 3 4 ( 4 λ4 ( S iω 4 λ Gr C 3 4 ( 3 λ3 ( S iω 3 Rsults ad Discussio For th rsultat vlocis ad th shar strsss of th stady ad ustady flow: u ( η + iv ( η q ( η ( ad Fig. Rsultat vlocy R 0 for sall ad larg rotatios du to u 0 ad v 0 ( η + ( η ( η + ( η (5 u iv q q Th Eq. (8 corrspods to th stady part which givs u 0 as th priary ad v 0 as th scodary vlocy copots. Th aplud ad th phas diffrc du to ths priary ad scodary vlocis for th stady flow ar giv by R u + v, α ta ( v / u ( Th rsultat vlocy R 0 for th stady part is show i Fig. for sall ad larg valus of rotatios of th vrtical porous chal. Th two valus of th adtl ubr as 0.7 ad 7.0, ar chos to rprst air ad watr rspctivly. It is clar fro Fig. that R 0 icrass wh th Grashoff ubr ad suctio vlocy. It is itrstig to ot that th icras of th rotatio of th chal ad Harta ubr lad to a icras of R 0 ar th statioary plat, but to a dcras ar th oscillatig plat. Howvr, th ffcts of th Hall paratr ad th adtl ubr ar rvrsd, i.. th aplud R 0 Fig. 3 Phas agl α 0 for sall ad larg rotatios du to u 0 ad v 0 dcrass ar th statioary plat ad icrass thraftr. Th phas diffrc α 0 for th stady flow is show graphically i Fig. 3 for sall ad larg rotatios. Fig. 3 shows that th phas agl α 0 dcrass wh th icras of Grashoff ubr, Harta ubr, th suctio/ijctio paratr, th rotatio of th chal ad adtl ubr. But th icras of Hall paratr lads to a icras of α 0. Th aplud ad th phas diffrc of shar strsss at th statioary plat (η0 for th stady flow ca b obtaid as: τ τ + τ, β ta ( τ / τ (7 0r 0x 0y 0 0y 0x

5 SINGH & KUMAR: FREE CONVECTION MHD FLOW IN POROUS CHANNEL 6 whr q0 η η 0 τ + iτ + A A ( λ 0x 0y ( { A ( A ( λ + + } (8 Hr τ 0x ad τ 0y ar, rspctivly, th shar strsss at th statioary plat du to th priary ad scodary vlocy copots. Th urical valus of th aplud τ 0r of th stady shar strsss ad th phas diffrc of th shar strsss at th statioary plat (η0 for th stady flow ar prstd i Tabl. Th τ 0r icrass ad β 0 dcrass wh th icras of Grashoff ubr, Harta ubr ad suctio/ijctio paratr. But th ffcts ar rvrsd wh th icras of Hall paratr, i.. τ 0r dcrass ad β 0 icrass wh th icras of. It is otd that both τ 0r ad β 0 icras du to a icras i th rotatio of th chal. But a icras i th adtl ubr lads to dcras both i τ 0r ad β 0. Th Eqs (9 ad (0 togthr giv th ustady part of th flow. Th ustady priary ad scodary vlocy copots u (η ad v (η, rspctivly, for th fluctuatig flow ca b obtaid as: { } { I ( I ( } si u ( η, t R al q ( η + R al q ( η cost q η q η t (9 { } { } v ( η, t R al q ( η R al q ( η si t + I q ( η + I q ( η cost (30 Th rsultat vlocy or aplud ad th phas diffrc of th ustady flow ar giv by: For th ustady part, th rsultat vlocy or aplud R is show i Fig. 4 for rotatio Ω sall ad larg. It is obsrvd fro Fig. 4 that R icrass wh th Grashoff ubr, Hall paratr ad suctio/ijctio paratr, but dcrass wh th icras of frqucy of oscillatios. It is itrstig to ot that icras i Harta ubr, rotatio of th chal ad th adtl ubr lad to a icras i R ar th statioary plat, but to a dcras ar th oscillatory plat. Th phas diffrc α for th ustady part is show i Fig. 5. It is vidt fro Fig. 5 that icras i Grashoff ubr or Harta ubr or suctio/ijctio paratr or rotatio of th chal lad to a dcras i α but α icrass wh th icras of Hall paratr, adtl ubr ad frqucy of oscillatios. Fig. 4 Rsultat vlocy R for sall ad larg rotatios du to u ad v at t (π/4. R u + v, α ta ( v / u. (3 Tabl Valus of τ 0r ad β 0 for various Gr, M,, λ, Ω ad Gr M λ Ω τ r β Fig. 5 Phas agl α for sall ad larg rotatios du to u ad v at t (π/4

6 6 INDIAN J PURE & APPL PHYS, VOL 47, SEPTEMBER 009 Fig. 6 Aplud τ r of ustady shar strsss for sall ad larg rotatios at t (π/4 which givs τ τ + τ, β ta ( τ / τ. (33 r x y y x Fig. 7 Phas diffrc β of ustady shar strsss for sall ad larg rotatios at t (π/4 For th ustady part of flow, th aplud ad phas diffrc of shar strsss at th statioary plat (η0 ca b obtaid as: q q τ x + τ + η η i y η 0 η 0 + B ( B ( ( { B ( + B ( } ( 6 + C ( 6 4 C ( 6 3 ( { C ( + C ( } 6 ( (3 Th aplud τ r of th ustady shar strss is show graphically i Fig. 6 for sall ad larg rotatios. It is clar fro Fig. 6 that τ r icrass wh th icras of Grashoff ubr, Harta ubr ad th rotatio of th chal. It is also obsrvd that th icras of adtl ubr lad to a icras of τ r for sall oscillatios of frqucy, but to a dcras for largr frqucy of oscillatios. Howvr, th ffcts of th suctio/ijctio paratr at th plats ar rvrsd. It is itrstig to ot that wh th icras of Hall paratr, τ r dcrass for sall oscillatios of frqucy, icrass for odrat oscillatios ad thraftr dcrass for lagr frqucy of oscillatios. Th phas diffrc β of th ustady shar strss is show graphically i Fig. 7 for sall ad larg rotatios. Fig. 7 clarly shows that β dcrass wh th icras of Grashoff ubr, Harta ubr, th suctio/ijctio paratr at th plats ad th rotatio of th chal, but icrass wh th icras of adtl ubr or Hall paratr. 4 Coclusios Th rsults obtaid fro th prst study show that th rsultat vlocy of th stady part icrass ar th statioary plat ad dcrass ar th oscillatig plat as th rotatio of th chal ad

7 SINGH & KUMAR: FREE CONVECTION MHD FLOW IN POROUS CHANNEL 63 Harta ubr icras, howvr, th ffcts of th Hall paratr ad th adtl ubr ar rvrsd. Th rsultat vlocy of th ustady part icrass wh th Grashoff ubr, Hall paratr ad suctio/ijctio paratr, but dcrass wh th icras of frqucy of oscillatios. Th phas agls of th stady ad ustady part of shar strsss dcrass wh th icras of Grashoff ubr, Harta ubr, th suctio/ijctio paratr, th rotatio of th chal. Th aplud of shar strsss of th ustady part dcrass for sall oscillatios of frqucy, icrass for odrat oscillatios ad thraftr dcrass for lagr frqucy of oscillatios wh th icras of Hall paratr. Rfrcs Myr R C, J Arospac Sci, 5 ( Attia H A & Kotb N A, Acta Mch, 7 ( Sigh A K, oc Idia Nat Sci Acad, 5 ( Sigh K D, Gorla M G & Raj H, Idia J Pur ad App Math, 36 ( Sigh K D, J App Math ad Mch (ZAMM, 80 ( Maudr B S, ASME J Appl Mch, 58 ( Gaapathy R, ASME J Appl Mch, 6 ( Sigh K D & Mathw Alphosa, Idia J Phys, 8 ( Vkatasiva Murthy K N, Idia J Pur Appl Math, 0(9 ( Sigh N P, oc Nat Acad Sci Idia, 7(A (0043. Kuar A & Sigh, DEI, J Sci Egg Rs, ( Sigh V & Sigh N P, Id J Tho Phys, 40 ( Pop I, J Math Phys Sci, 5 ( Sigh A K, Kuar A & Sigh A K, Rflctios ds ERA, (