Effect of Uniform Horizontal Magnetic Field on Thermal Convection in a Rotating Fluid Saturating a Porous Medium

Transcription

1 Journl of Computer nd Mthemtcl Scences, Vol.8, Novemer 07 An Interntonl Reserch Journl, ISSN rnt ISSN 9-8 Onlne Effect of Unform Horzontl Mgnetc Feld on Therml Convecton n Rottng Flud Sturtng orous Medum Rovn Kumr nd Vjy Meht Deprtment of Mthemtcs nd Sttstcs, J Nrn Vys Unversty, Jodhpur 00 Rj. INDIA. eml: rovnprdhn86@gml.com; vjymehtnrs@gml.com Receved on: Novemer, Accepted: Novemer, 07 ABSTRACT In ths pper we study therml convecton n rottng flud sturtng porous medum n unform horzontl mgnetc feld nd otned dsperson relton. Usng norml mode nlyss, from ths dsperson relton we oserved tht the medum permelty k hs stlzng effect, n the sence of rotton the medum permelty hs destlzng effect. In the sence of mgnetc feld, the medum permelty hs stlzng effect, the rotton hs stlzng effect, whtever the mgnetc feld s ppled nd mgnetc feld hs stlzng effect nd necessry condton for osclltory mode s otned wth the condton E r m Er nd o k. Keywords: Therml convecton, orous medum, Mgnetc feld, Rotton.. INTRODUCTION The prolem of convecton n horzontl lyer of flud heted from elow referred to s therml nstlty prolem, under vryng ssumptons of hydrodynmcs nd hydromgnetcs, hs een dscussed n detl y Chndrsekhr 96. The effect of Hll currents on the therml nstlty of horzontl lyer of conductng flud hs een studed y Gupt 967. Lpwood 98 hs nvestgted the stlty of convectve flow n hydrodynmcs n porous medum usng Rylegh's procedure. Sstry nd Ro 98 notced tht when the flud, s heted from elow, the rotton of the system delys the onset of stlty.

2 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, Recently, Shrm et l. 006 hve nlyzed the effect of mgnetc feld nd rotton on the stlty of strtfed elstco-vscous flud n porous medum. The effect of Hll currents on the therml nstlty of electrclly conductng flud n the presence of unform vertcl mgnetc feld hs een studed y Gupt 967. Shrm nd Kumr 997 hve studed thermosolutl convecton n Rvln Erckson flud n hydromgnetcs sturtng porous medum. Bht nd Stener 97 hve studed the prolem of therml nstlty of Mxwelln vsco-elstc flud n the presence of rotton nd found tht rotton hs destlzng nfluence n contrst to the stlzng effect on vscous Newtonn flud. Just s n hydrodynmcs, when conductng flud permetes porous mterl n the presence of mgnetc feld the ctul pth of n ndvdul prtcle of flud cn not e followed nlytclly. The gross effect, s the flud slowly percoltes through the pores of the rock, must e represented y mcroscopc lw pplyng to msses of flud whch s the usul Drcy's lw. The usul vscous term n the equtons of flud moton wll e replced y the resstnce term q, where, s the vscosty of the flud, k the permelty of the k medum whch hs the dmenson of length squred, nd q the velocty of the flud, clculted from Drcy's lw. In ll the ove studes, the Boussnesq pproxmton hs een used whch mens tht densty vrtons re dsregrded n ll the terms n the equtons of moton except the one n the externl force. The equtons governng the system ecome qute complcted when the fluds re compressle. To smplfy them, Boussnesq tred to justfy the pproxmton for compressle fluds when the densty vrtons rse prncplly from therml effects. For sttonry convecton the medum permelty hs stlzng effect under the condton mkx ok nd the mgnetc feld hs stlzng effect under or the condton o k In the present pper, I studed the effect of unform horzontl mgnetc feld on therml convecton n rottng flud sturtng porous medum. To the est of my knowledge f t unnvestgted so fr.. MATHEMATICAL FORMULATION In ths prolem, we consder n nfnte, horzontl, electrclly non-conductng ncompressle flud lyer of thckness d. Ths lyer s heted from elow such tht the lower oundry s held t constnt temperture T T o nd the upper oundry s held t fxed dt temperture T T so tht T0 T, therefore unform temperture grdent s dz 577

3 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, mntned. The physcl structure of the prolem s one of nfnte extent n x nd y drectons ounded y the plnes z 0 nd z d. The flud lyer s ssumed to e flowng through n sotropc nd homogenous porous medum of porcty nd the medum permelty k, whch s cted upon unform rotton 0, 0, o nd grvty feld g 0, 0, g. A unform mgnetc feld H H o, 0, 0 s ppled long x-xs. The mgnetc Rynold numer s ssumed to very smll so tht the nduced mgnetc feld cn e neglected n comprson to the ppled feld. We lso ssumed tht oth the oundres re free nd no externl couples nd the het sources re present. Fg. Geometry of the rolem The equton governng the moton of rottng fluds sturtng porous medum. Followng Boussnesq pproxmton re s follow: The equton of contnuty for ncompressle flud s q 0 The equton of momentum, followng Drcy lw s gven y o q q. q t ˆ o g e q q e z H H k The equton of energy s gven y T C C C q. T o v s s o v T T t nd the equton of stte s o[ T To] 578

4 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, Where q, p,, o, s,, e, k, T, t, T, To, Cv, Cs nd êz denote respectvely flter velocty, pressure, flud densty, reference densty, densty of sold mtrx, flud vscosty, mgnetc permelty, medum permelty, temperture, tme, therml conductvty, reference temperture, specfc het t constnt volume, specfc het of sold mtrx nd unt vector long z-drecton. The Mxwell s equton ecome H q H m H 5 t nd H 0 6 where m s the mgnetc vscosty.. BASIC STATE OF THE ROBLEM The sc stte of the prolem s tken s q q 0, 0, 0, 0, 0, o, z, z nd H H Ho, 0, 0 usng ths sc stte, equton to 6 yeld dp g 0 dz 7 T z T o 8 o z 9. ERTURBATION EQUATIONS Let q u, v, w, ', ', nd h h x, h y, h z represent the erturton n q,,, T nd H respectvely usng ove perturtons, the new vrles re q q q ', ', T T, H H h Usng these new vrle nd usng 7, 8 nd 9 equton -6 fter lnerzton yeld. q ' 0 0 o q ' ' ' o g q q ' e h H t k C C C q '. T o v o s s v p T dt 579

5 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, ' o h H. q ' m h t. h 0 5 Mkng the equton 0- nto non-dmensonl lnerlzed form y usng the followng non-dmensonl vrle nd droppng the strs. We hve * * * k,,, ' T x x d y y d z z d q q *, t o d t *, d * k, ' T * d d *, h H h *, o where k T T s the therml dffusvty, we otn o d o C v. q 0 6 q ˆ o Rez q [ q eˆ z ] h eˆ x t K 7 E r W t 8 h q r r h t x m 9 nd. h 0 0 g d Where R o H d s the Rylegh numer, Q e o s the Chndrsekhr numer k T k T K K, r C d k the rndtl numer, s s E, m s the mgnetc T ocv om rndtl numer nd eˆ x, eˆ y, eˆ z re the unt vector long x, y nd z-drecton respectvely nd W q. eˆ z s the z-component. 5. BOUNDARY CONDITIONS Here, we consder the cse when oth oundres re free s well s eng perfect conductor of het, whle the djonng medum s perfectly conductng, then we hve W W 0 t z 0 t z z 580

6 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, DISERSION RELATIONS Applyng curl twce, to equton 7 nd tkng z-component, we hve o W R W D z Q hz t K z Where, x y z x y z q z s the z-component of vortcty vector hz h. eˆz, W q. eˆz, D z Apply curl once to equton 7 nd tkng z-component, we hve, z o m DW Q z z t K x Where mz h z s the z-component of current densty Applyng curl once to equton 9 nd tkng z-component, we hve m or z r r mz t x m Tkng z-component of the equton 9, we get hz W r r hz t x m 5 7. NORMAL MODE ANALYSIS The norml mode nlyss cn e defned s follows: [ W, z,, mz, hz ] [ W z, X z, z, M z, B z]exp.[ kx ky t] Where s the stlty prmeter whch s n generl complex constnt nd k k x y s the wve numer. Anlyzng ove norml mode, we hve D W R o DX k Q D x B k 6 r r D M kx X m 7 r D r B kxw m 8 58

7 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, X 0 DW kxqm 9 K [ E r D ] W 0 The oundry condton how ecomes W D W 0 DX M, 0, DB 0 t z 0 nd Also DM 0, B 0 on perfectly conductng oundry lso D 0, D M 0, D B 0, D X 0 nd z 0 nd z Elmntng X,, M nd B etween 6 to 0 we hve from 9 nd 7 we get r X k D o x r DW K m W [ E D ] r From 6, 8,, we get D W R E D r W K 0 D r k D o D x r W K m k Q D r D x r W m From the equton we oserve tht D n W 0, n s postve nteger t z 0, Therefore the proper soluton W chrcterzng the lowest mode s W Wo sn z Where W o s constnt Susttutng for W n equton we hve [ E ] E r k r r xq K m r k Q R r r x r K m m r o k Q [ E ] r r x r r 5 K m m 58

8 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, STATIONARY CONVECTION For sttonry mrgnl stte we put 0 n 5 we get r k Q r k x x Q K m K m r r o r R kxq m K m m When Q 0 n the sence of mgnetc feld we hve R o K 7 K When o 0 In the sence of rotton we hve R 8 K From 6 we get k Q m x o R 9 K r mk xq K r To nvestgte the effects of medum permelty K, rotton o, mgnetc feld Q. We dr dr dr exmne the ehvor of,, nlytclly. dk d o dq From 9, we otn dr dk o K 0 k m xq K r dr 0 dk When m k x o K or Thus, the medum permelty hs stlzng effect when m k x o K or 58 6

9 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, In the sence of rotton o 0, equton reduces to dr dk K dr 0 dk Whch s lwys negtve, thus n ths cse the medum permelty hs destlzng effect n the sence of rotton. In the sence of mgnetc feld Q 0, equton 0 reduce to dr K o dk K dr 0 dk When o K Thus, n the sence of mgnetc feld the medum permelty hs stlzng effect When o K From 9, we hve dr 8 o 0 d o mk x Q K r Whch s lwys postve, thus the rotton hs stlzng effect From 9 we hve m kx Q m kxq o K K dr k r m x r dq r k m xq K r dr 0 dq When o K 58

10 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, Thus, the mgnetc feld hs stlzng effect when o K 9. OSCILLATORY CONVECTION Equton 5 cn e rewrtten E r [ ] k r m xq K m r R [ ] r [ ] k m m xq m K m E r o r [ ] m m k Q Let E,, x m r A A A, then we hve K r [ A ] [ A ] [ m ] A R [ m ] [ A ] [ ] A [ A ] [ ] m o m uttng n [ A ] m A A A m R [ m ] m A A A m [ A ] [ ] o m A A m, A m A m,, m [ A R [ ] [ o A ] Equtng rel nd mgnry prt we get Rel prt A o A R 585

11 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, Imgnry prt A o A R Elmntng R etween nd 5 we get A A o A A A A A A o A A A A 6 Equton 6 cn e rewrtten s A A A A o A A A A A A 0 7 After puttng the expressons for,,,, we otn 6 [ ] m m A m m m A A m A ma m m A A m A A A m A A A A o m o m A A 5 Am m A m A m A m AA 5 ma m A m A m A A m A AA 5 A A m A m A A A A m A A m A A A ma A m A A m A A m A A A m A 586 5

12 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, ma A m A A A A m o m A A 8 p A o m o m o m A o m A 5 A 5 A A A A A A A A A A A 5 A A m A m A m A A m A A 8 Equton 8 cn e wrtten s [for osclltory modes] 9 Where 0 m m A m m Er A A A m m m m m m A A m A A m A A m m A A A A o m o m A A Smlrly, re defned s the coeffcent n equton 8 Let r the equton 8 ecomes 0r r r 0 50 Snce r whch s lwys postve, so tht the sum of roots of equton 50 s postve, the sum of roots of equton 50 s 0 s postve. Necessry condton for the exstence of osclltory modes s gven y condton clerly 0 0, f m Er 0 nd E 0, r r, E r f m Er o k k x m kx m k 0. CONCLUSIONS For Sttonry Convecton. The crtcl Rylegh numer ncreses s the medum permelty ncreses for sttonry convecton under the condton 5. Thus the medum permelty hs stlzng effect when

13 Rovn Kumr, et l., Comp. & Mth. Sc. Vol.8, m k x o k o r In the sence of rotton, the crtcl Rylegh numer decreses s the medum permelty ncreses. In the sence of rotton the medum permelty hs destlzng effect. In the sence of mgnetc feld, the crtcl Rylegh numer ncreses the medum permelty ncreses under the condton stlzng effects when. o k o the medum permelty k. The crtcl Rylegh numer ncreses wth the ncrese of rotton, thus the rotton hs stlzng effect, whtever the mgnetc feld s ppled.. The crtcl Rylegh numer ncreses wth the ncrese of Chndrshekhr numer, thus the mgnetc feld hs stlzng effect under the condton o k For osclltory Convecton: The necessry condton for the exstence of osclltory mode re gven y condton clerly 0 0, f m Er 0 nd. E 0, r r, E r f m Er o k k x m kx m k REFERENCES. Bht,.K., Stener, J.M., Convectve nstlty n rottng vscoelstc flud lyer, Z. Angew. Mth. Mech Chndrsekhr, S. Hydrodynmc nd Hydromgnetc Stlty, Oxford Unversty ress, Chps Gupt, A.S. Rev. Roumne Mth. ures Appl., Gupt, A.S., Hll effects on therml nstlty, Rev. Roum. Mth. ures Appl Lpwood, E.R. roc. Cm. hl. Soc., Shrm, R.C., Kumr,., On the mcropolr flud heted from elow n hydromgnetcs n porous medum, Czechoslovk J. hys Shrm, V., Sunl, Gupt, U., Stlty of strtfed elstco-vscous wlters Model B flud n the presence of horzontl mgnetc feld nd rotton n porous medum, Arch. Mech Shstry, V.U.K., nd Ro, V.R., Int. J. Eng. Sc. 5, 9 98.