Stability of Stratified Rotating Viscoelastic Rivlin Ericksen Fluid in The Presence of Variable Magnetic Field

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1 vailabl oli at.plaiasacliba.co vacs i ppli Scic Rsac,, (5):5-58 ISSN: COEN (US): SRFC Stabilit of Statifi Rotati iscolastic Rivli Eics Flui i T Psc of aiabl Matic Fil Rajs Kua Gupta a Mai Si patt of Matatics (Eii), ovl Pofssioal Uivsit,Paaa-, Iia. patt of Matatics, Govt Post Gauat Coll, Sa(Rou), iacal Pas uivsit Sila-77,Iia BSTRCT T Rali- Talo istabilit of statifi otati viscolastic (Rivili- Eics) fluis i t psc of vaiabl atic fil is cosi. Nuicall a apicall sults so tat t psc of atic fil stabilis a ctai av- ub ba, as t sst is ustabl fo all av-ubs i t absc of t atic fil, otatio a fo o- viscolastic flui. T av- ub a, fo ic t pottial ustabl sst ts stabilii ffct, icass it t icas i t atic fil a cass it t icas i iatic viscolasticit Kos: Rali-Talo Istabilit Statifi Rotati Rivili- Eics iscolastic Flui, aiabl Matic Fil. MSC (): 765, 765, 76, 76S5 INTROUCTION T stabilit iv fo t caact of t quilibiu of a icopssibl av flui of vaiabl sit (i.. of a toous flui) as ivstiat b Rali (88). ostat tat t sst is stabl o ustabl accoi as t sit cass v o icass v. pital ostatio of t vlopt of t Rali Talo istabilit as pfo b Talo (95). Ri (96) stui t ffct of sufac tsio a viscosit o t stabilit of to suppos fluis. T Rali Talo istabilit of a Ntoia flui as b stui b sval autos accpti vai assuptios of oaics a oatics a Caasa (98) i is clbat ooap as iv a tail accout of ts ivstiatios. Gall, t atic fil as a stabilii ffct o t istabilit, but t a a f cptios also. Fo apl, Kt 966) as stui t ffct of a oiotal atic fil ic vais i t vtical ictio o t stabilit of paalll flos a as so tat t sst is ustabl u ctai coitios, il i t absc of atic fil t sst is o to b stabl. I stlla atosps a itios, t atic fil a b (a quit oft is) vaiabl a a altot alt t atu of t istabilit. Coiolis foc also plas a ipotat ol o t stabilit of t sst. I all t abov stuis t flui as b assu to b Ntoia. Wit t oi ipotac of o Ntoia fluis i o tcolo a iustis, t ivstiatios of suc fluis a siabl. T a a viscolastic fluis ic caot b caacti it b Mall s costitutiv latios o b Olo s costitutiv latios. O suc class of t viscolastic fluis is t Rivli Eics flui. Rivli a Eics (955) av popos a totical ol fo suc viscolastic flui. Tis a 5

2 ot class of pols is us i t aufactu of pats of spac cafts, aoplas, ts, blt covs, ops cusios, sats, foas, ii quipts tc. Rctl, pols a also us i aicultu, couicatio appliacs a i bioical appliacs. T stabilit of patiall ioi suppos plasas i t psc of vaiabl oiotal atic fil as b stui b Saa a Tau (98). Saa a Kuai (99) av stui t stabilit of statifi flui i poous iu i t psc of susp paticls a vaiabl atic fil. Kua () also av stui t Rali- Talo istabilit of a Ntoia viscous flui ovli a Rivili-Eics viscolastic flui cotaii susp paticls i a poous iu.lso Kua a al (5) av stui t stabilit of to suppos viscous-viscolastic fluis. Kua a Si [] av stui t stabilit of suppos viscous-viscolastic fluis tou poous iu; fou tat fo t stabl cofiuatio t sst is fou to b stabl o ustabl. ov, t sst is fou to b ustabl fo t ustabl cofiuatio. T sst is fou to b ustabl fo t pottiall ustabl cas, fo il viscous fluis, i t psc of a uifo otatio. T bavio of ot ats it spct to iatic viscosit a iatic viscolasticit paats a ai uicall a it is fou tat bot iatic viscosit a iatic viscilasticit av stabilii ffct. Kpi i i t ipotac of o Ntoia fluis i o tcolo, iustis, cical ii a oi to t ipotac of otatio a vaiabl atic fil i astopsics tc., a otivat to stu t stabilit of statifi otati viscolastic Rivli Eics flui i t psc of vaiabl oiotal atic fil i t pst pap.. Foulatio of t pobl a ptubatio quatios Cosi a ifiit oiotal la of ticss bou b t plas a. T caact of t quilibiu of tis statioa stat is ti b supposi tat t sst is slitl istub a t,, a uifo vtical otatio folloi its fut volutio. T flui is act o b avit foc (,, ) Ω (,,Ω) a a vaiabl oiotal atic fil (,, ). t ρ,,, p a v (,,) ot, spctivl, t sit, t viscosit, t viscolasticit, t pssu a t vlocit of flui. T t quatios pssi cosvatio of otu, ass, icopssibilit a Mall s quatios fo t viscolastic Rivli Eics flui a v ρ ( v ) v p v ( v Ω) t ρ ρ ( ), () v, () ρ ( v ) ρ, (), () ( ) v ( v ), (5), t atic pabilit is, assu to b costat. Equatio () psts t fact tat t sit of a paticl ais uca as follo it it its otio. δρ a t, δp, v ( u, v, ),, ot, spctivl, t ptubatios i sit ρ, pssu p ( ), vlocit v (,,) a vaiabl oiotal atic fil ( ( ),, ). T t quatios () (5) aft ptubatios, lcti o lia ts, i t liai fo il u ρ δp u ρ vω, (6) v ρ δp ρ δp v ρ uω, (7) δρ, (8) 5

3 u v ρ, (9) δρ, () u v, (), (), (). () ali t istubacs ito oal os, s solutios os pc o iv b f ( i i t ),, a ti t is p, (5) f is t so fuctio of ol a, spctivl, a t av-ubs i t a ictios, is t sultat av-ub a is t ot at of t istubac ic is i al a copl costat. Equatios (6) () usi pssio (5), substituti t valus of, a i sulti quatios, t i ρ u i δp ( )( ) u ρ vω, (6) ρ v i ( δp )( ) v i i ζ ρ uω, (7) ρ δp ( )( ) i u i u ( ) ζ i v i u, is t copot of voticit. ρ, (8) Multipli quatios (6) a (7) b ( )( Ω ) i a i, spctivl, a t ai, t ζ, (9) Substituti t valu of ζ i quatio (7), t Ω i ρ v i δp v i ( ) ρ uω Multipli quatios (6) a () b i a i, spctivl, a t ai,), obtai ( ) ( ). () ρ δp ρ ( )( ) ( )( ) Ω ρ. () 55

4 56 Eliiatiu, v a p δ fo quatios (6) (), usi quatio (7), t ρ ρ ρ Ω ρ.(). T cas of potiall vai statificatios I o to obtai t solutio of t stabilit pobl of a la of Rivli Eics flui, suppos tat t sit ρ, viscosit, viscolasticit a atic fil va potiall alo t vtical ictio, tfo quatio () tasfos to [ ] [ ] Ω. () Cosii t cas of to f bouais, ust av at a. () T appopiat solutio of quatio () satisfi t abov boua coitio is si : is a it a is a costat. (5) Substituti t valu of fo quatio (5) i quatio (), obtai ispsio latio [ ] [ ] v Ω ;. (6) Equatio (6) is biquaatic i a is t ispsio latio ovi t ffcts of uifo otatio, vaiabl oiotal atic fil, viscosit a viscolasticit o t stabilit of statifi Rivli Eics flui. RESUTS N ISCUSSION (a) Cas of stabl statificatios (i.. < ). Equatio (5) os ot ait a positiv al oot o copl oot it positiv al pat usi Rout uit citio; tfo, t sst is alas stabl fo istubacs of all av-ub. (b) Cas of ustabl statificatios (i.. > ): If >, > a <, t costat t i t quatio (6) is ativ a tfo as at last o oot it positiv al pat usi Rout uit citio, so t sst is ustabl fo all av-ubs satisfi t iqualit sc θ <, (7) θ is t al bt a i.. cosθ.

5 If >, ( ustabl statificatios) > a >. Equatio (6) os ot ait of a positiv al oot o copl oot it positiv al pat, tfo, t sst is stabl. T sst is clal ustabl i t absc of atic fil, otatio a fo o viscolastic flui. Tfo t atic fil, stabilis pottiall ustabl statificatios fo sall av-lt ptubatios sc θ ( ) >. (8) lso, it is cla tat t av-ub a, fo ic t pottiall ustabl sst ts stabili, icass it t icas i atic fil a cass it t icas i iatic viscolasticit. ll lo av-lt ptubatios satisfi quatio (8) ai ustabl a a ot stabili b atic fil. T baviou of ot ats it spct to iatic viscosit iatic viscolasticit a squa of t lfvé vlocit satisfi quatio (6) as b ai uicall usi Nto Rapso to tou t softa Matca. I apicall, fiu () sos t vaiatio of ot at (positiv al valu of ) it spct to t av-ub satisfi quatio (5) fo fi pissibl valus of,, 6 c,, Ω otatio/iut, 98c/s, cos 5, 55 fo t valus of, a spctivl. T vaious paat valus satisf t iqualit (7), ic povis t av-ub a fo ic t sst is ustabl. Ts valus a t pissibl valus fo t spctiv paats a a i oo at it t cospoi valus us b Caasa [96] il scibi vaious oaic a oatic stabilit pobls. T ap sos tat fo fi avubs, t ot at icass fo. 6 it t icas i iatic viscolasticit, ic iicats t stabilii ffct of viscolasticit as t ot at cass fo.6 <. 6, ipli tb t stabilii ffct of iatic viscosit o t sst. Fiu () sos t vaiatio of ot at (positiv al valu of ) it spct to t av-ub fo fi pissibl valus of,, 6 c, Ω Rotatio/iut,, 98c/s, cos 5, 55 fo t valus of, a 6 spctivl. T ap sos tat fo fi av-ubs, t ot at icass fo. 6 it t icas i iatic viscosit ic iicats t stabilii ifluc of iatic viscosit, as t ot at cass fo.6 <. 6, ipli tb t stabilii ffct of t squa of t lfv vlocit of t sst. / / / Fiu : aiatio of (positiv al pat of ) it av-ub fo fi pissibl valus of,, 6 c,, Ω v/i, 98c/s cos 5, 55 fo t valus of,,. v/i, 57

6 Fiu: aiatio of of,, ( positiv al pat of ) it av-ub fo fi pissibl valus 6 c, Ω v/i, 98c/s, cos 5, 55, fo t valus of,, ( positiv al pat of ) it av-ub fo fi pissibl valus of Fiu : aiatio of 6 Ω v/i, 98c/s,, c, 5, 55., fo to valus of cos 5,, REFERENCES []Caasa S, Ps. Flui, 966, 9, pp86-89 [] Kua P, Z.a. Mat., Ps.,,5, 9-9 [] Kua P,al R, Tal Scic, 5, 9(),87-95 [5] Kua P, Si G, M: It. J,,ol.5, No., pp -9. [6] Rali, o,poc. oo Mat. Soc., 88,, [7] Rivli R.S,Eics, J.. J. Rat. Mc. al. 955,, pp-5 [8] Ri W., Poc. Cab. Pil. Soc., 96, 57, 5-5. [9] Saa R. C., Tau K. P., Nuova Cito, 98, 7B, 8-6. [] Saa R. C., Saa., Ccoslova Joual of Psics, 99, (5), [] Talo G.I., Poc. Ro. Soc.(o.), 95,, [] S. Sat, R.Saavaa, S.ataaaa, R. ai R, vacs i ppli Scic Rsac,, (): 6-6. [] K. Ca, R. Kua, S. Saa, vacs i ppli Scic Rsac,, (): [] ilp S. Caua, ias Kua, vacs i ppli Scic Rsac,, ():75-9. [5] Staaalasi, G.. Raaa R, B.. C. N. Pasa, vacs i ppli Scic Rsac,, (6):6-69. [6] S. Sa, S. Naa Kiso, N S Siivas a R. ai R, vacs i ppli Scic Rsac,, (6):5-. [7] isaat B. ati, N. M. Buju, N.N. Katai, vacs i ppli Scic Rsac,, ():- 58