Effect of Uniform Magnetic Field on Rayleigh Bènard Marangoni Convection in a Relatively Hotter or Cooler Layer of Liquid

Transcription

1 Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Volume Issue 5 ay 5 PP 4-3 ISSN (Prit) & ISSN (Olie) Effet of Uiform ageti Field o ayleigh Bèard aragoi Covetio i a elatively Hotter or Cooler Layer of Liquid A.K. Gupta Himahal Pradesh Uiversity Cetre for Eveig Studies Shimla Idia guptakdr@gmail.om S.K. Dhima Govt. Post Graduate College Dharamshala Idia smaths8@gmail.om. Abstrat: The effet of uiform vertial mageti field o the oset of steady ayleigh-bèard-aragoi ovetio i a relatively hotter or ooler layer of eletrially odutig liquid is studied theoretially by meas of modified liear stability aalysis. The top surfae of the layer is o-deformable free where surfae tesio gradiets arise o aout of variatio of temperature ad the bottom surfae is rigid ad thermally odutig. The eigevalue equatio is obtaied by usig the Fourier series method. Numerial results are obtaied ad preseted. The results of this aalysis idiate that the ritial eigevalues i the presee of mageti field are greater i a relatively hotter layer of liquid tha a ooler oe uder idetial oditios otherwise. A detail desriptio of margial stability urves showig the ifluee of the mageti field is preseted ad disussed. The asymptoti behavior of the mageti field for large values of Chadrasekhar umber is also obtaied. Keywords: Buoyay Covetio Liear stability Surfae tesio Chadrasekhar umber.. INTODUCTION The mehaism of the oset of surfae tesio idued ovetio i a thi horizotal fluid layer heated from below with free upper surfae was reported experimetally by Blok [5] ad explaied theoretially by Pearso [4]. They established that the pattered hexagoal ells observed by Béard [3 4] ad explaied by ayleigh [5] i terms of buoyay were i fat due to temperature depedet surfae tesio. Covetio drive by surfae tesio gradiets is ow ommoly kow as Béard-aragoi ovetio i otrast to the buoyay drive ayleigh- Béard ovetio. uatitative disagreemet betwee experimet ad theory has idiated that gravity was preset i Béard's experimets as well as i other experimets ivolvig ovetio i a fluid layer with free surfae i a laboratory o the earth therefore Nield [] osidered the ombied effets of both the surfae tesio ad buoyay o the oset of ovetio i a liquid layer heated from below with free upper surfae alled ayleigh-béard-aragoi ovetio ad foud that the two effets ausig istability are tightly oupled. For a detail study of ovetio oe may be referred to the work of Chadrasekhar [6] Normad et al. [3] Koshmieder [9] ad Shatz ad Neitzel [6]. Sie the proess of otrollig ovetio i a fluid has beome importat i material proessig ad beause of its appliatios extedig from produig large rystals of uiform properties to maufaturig ew materials with uique properties. eetly Gupta ad Shadil [7] studied the surfae tesio drive ovetio i a relatively hotter or ooler layer of liquid ad Gupta et al. [8] osidered the ombied effets of both surfae tesio ad buoyay o the oset of ovetio i a relatively hotter or ooler layer of liquid. They established that irrespetive of the drivig fore ausig istability amely surfae tesio or buoyay or both the hotter layer with its heat diffusivity apparetly ireased as a osequee of atual derease i its speifi heat at ostat volume must exhibit ovetio at a higher temperature differee hee more stable tha a ooler layer uder idetial oditios otherwise. The stabilizig ature of the mageti field o buoyay drive or/ad surfae tesio drive ovetio i a liquid layer heated from below has bee well established otably by Chadrasekhar [6] Nield [] aekawa ad Taasawa [] ad by Wilso [8] ad referee therei. Nevertheless the effet of the mageti AC Page 4

2 A.K. Gupta & S.K. Dhima field o the oset of ayleigh-béard-aragoi ovetio i a relatively hotter or ooler layer of liquid heated from below has ot bee give ay attetio i the literature despite its importae i otrollig ad uderstadig qualitatively the mehaism of ovetive istability problems eoutered i geophysis oeaography atmospheri siees ad hemial egieerig of paits ad detergets. The aim of the preset study is to ivestigate the effet of a uiform vertial mageti filed o the oset of ayleigh-béard-aragoi ovetio i a relatively hotter or ooler layer of liquid heated from below thus extedig the work of Gupta et al. [8] to ilude the effet of mageti field. The Fourier series method is used to obtai the eigevalue equatio aalytially. The umerial results are obtaied for a wide rage of parameters relevat to the problem i the preset otext. The result of this aalysis idiate that the ritial eigevalues i the presee of mageti field are greater i a relatively hotter layer of liquid tha a ooler oe uder idetial oditios otherwise ad also establish the ihibitig effet of the mageti field. A detail desriptio of margial stability urves showig the ifluee of the mageti field is preseted. We fid that whe the mageti field beomes suffiietly large the two mehaisms ausig istability beome less tight as ompared to the ase whe there is o mageti field preset. The asymptoti behavior of the mageti field for large values of Chadrasekhar umber is also obtaied.. ATHEATICAL FOULATION OF THE POBLE The physial ofiguratio of the problem osists of a ifiite horizotal layer of visous fluid of uiform thikess d i the presee of a uiform vertial mageti field H ad whih is heated from below. The lower rigid boudary of the layer is kept at a ostat temperature T ad the upper free surfae is ope to the atmosphere at temperature T T subjet to geeral heat radiative oditio. We hoose a Cartesia o-ordiate system of axis with the x ad y-axis i the plae of the lower surfae of the fluid layer ad z-axis alog the vertially upward diretio so that the fluid layer is ofied betwee the plaes at z ad z = d as show i Fig (). The surfae tesio o the upper free surfae of fluid layer is regarded as a futio of temperature oly whih is give by the simple liear law T () T where the ostat is the uperturbed value of at the uperturbed surfae temperature T T ad / T represets the rate of hage of surfae tesio with temperature evaluated T T at temperature T ad surfae tesio beig a mootoially dereasig futio of temperature is positive. Fig. Shemati represetatio of the physial ofiguratio of the problem Uder the Boussiesq approximatio the modified liearized perturbatio equatios for a eletrially odutig liquid i the presee of a uiform mageti field i the preset otext (Baerjee et al. []) are give as v t w g H 4 z h z () Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Page 5

3 Effet of Uiform ageti Field o ayleigh Bèard aragoi Covetio i a elatively Hotter or Cooler Layer of Liquid ( T ) w t (3) t h z H w z (4) where w is the perturbatio veloity the perturbatio temperature h z is the z-ompoet of the perturbatio from the basi vertial mageti field H ad is the desity of the fluid. The adverse uiform temperature gradiet T T / d the gravitatioal aeleratio g the oeffiiet of volume expasio the kiemati visosity v the thermal diffusivity the mageti permeability ad the mageti resistivity 4 (where is the eletrial odutivity) are eah assumed ostat ad represet respetively x y z ad x y ad t represets time. Further the oeffiiet (due to variatio i speifi heat at ostat 4 volume o aout of variatios i temperature) lies i the rage to ad rage of the dimesioless parameter T overig the usual laboratory oditios is T for liquids with whih we are mostly oered. I this rage ay give value of T orrespods to the layer of liquid whih is relatively hotter ompared to that assoiated with its value less tha (iludig T ) the give oe. A detailed aout of this has bee give i the researh moograph by Baerjee ad Gupta []. Equatios () (4) must be solved subjet to appropriate boudary oditios. For the rigid ad perfetly thermally odutig (the perturbatio temperature is thus zero) lower boudary at z = the boudary oditios are w z w (5a b ) Sie the tagetial visous stress experieed by the liquid at the upper free surfae is balaed by the tratio due to variatio with temperature of surfae tesio ad the geeral heat radiatio oditio (Sparrow et al. [7]) at z = d the boudary oditios are w w k q (6a b ) z z where k is the heat odutivity ad q is the rate of hage with temperature of the time rate of heat loss per uit area from the upper surfae. We ow aalyze a arbitrary disturbae i terms of ormal modes assumig that the perturbatios w ad h are of the form z w x y z t x y z t h x y z t W z z K z exp i a x a y where a ad x y z a are ompoets of the horizotal wave umber a a x a y x y st of the disturbae ad s is the time growth rate (a omplex umber i geeral). Usig the above expressios for w ad h z i equatios () (4) ad the makig the resultig equatios dimesioless by hoosig d d / v d / v d v / k ad H as uits of legth time veloity temperature ad mageti field sales respetively; o puttig D d d dz a ad W wd d p sd P r v P m v K K H ad omittig asterisks() for oveiee we obtai Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Page 6

4 A.K. Gupta & S.K. Dhima D a D a p W D D a K a (7) a T P p T W (8) D r D a P p K DW m (9) 4 4 where P P g d / ad H d r m / 4 v are respetively the thermal Pradtl umber mageti Pradtl umber ayleigh umber ad Chadrasekhar umber. We restrit our aalysis to the ase whe the priiple of exhage of stabilities is valid for the preset problem so that istability first sets i as statioary ovetio. I that ase we a put p= i equatios (7) (9) ad whe K is elimiated from the resultig equatios we obtai D a D W a () a T W () D I terms of ew variables the o-dimesioal form of boudary oditios (5a b ) (6a b ) a be writte as W DW ad (a b ) evaluated o the lower boudary z = ad W D W a D L (3a b ) evaluated o the upper boudary z =. Here d / v is the aragoi umber ad L qd / k is the Biot umber. We ote that the boudary oditios o mageti field do ot affet the stability oditio. Equatios () () together with boudary oditios (a b ) (3a b ) ostitute a eigevalue problem of order six with as a eigevalue for give values of the remaiig parameters. 3. SOLUTION OF THE POBLE The Fourier series method as preseted by Nield [] is oveiet for the problem uder osideratio. The method osists of expadig eah of the two variables W(z) ad z i two equivalet forms: As the sum of a polyomial ad a Fourier series whih a be differetiated the required umber of times ad As a sigle Fourier series. Thus we let W 3 z D W D W z D W z 6 W W D W D W z W A si z (4a) 3 6 A D W D W 3 3 W z si z (4b) W W ad z z B si z (5a) z B si z. (5b) The forms (4a) ad (5a) are suitable for satisfyig the boudary oditios. The omplete Fourier expasios for W ad give by (4b) ad (5b) are substituted i the differetial Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Page 7

5 Effet of Uiform ageti Field o ayleigh Bèard aragoi Covetio i a elatively Hotter or Cooler Layer of Liquid equatios ad the oeffiiets of si z are equated. The resultig system of equatios a be solved for A ad B whih are the substituted i the boudary oditios. Fially elimiatio of D W ad D W gives the eigevalue equatio whih may be writte as Where (6) (7) We a obtai from the eigevalue equatio (6) i terms of a L T ad as the ratio for two determiats give by 4. NUEICAL ESULTS AND DISCUSSION The umerial alulatios may be arried out as follows. For fixed values of T ad L the eigevalue equatio (8) gives the relatios betwee ad with a as a parameter. If we fix the values of T ad L the expressio (8) determies the aragoi umber as a futio of the wave umber a. The miimum value of is the ritial aragoi umber ad the value of a at whih attais the miimum is the ritial wave umber a. Alteratively for fixed values of T ad L we may determie the ayleigh umber as a futio of the wave umber a. The miimum value of is the ritial ayleigh umber ad the value of a at whih attais the miimum is the ritial wave umber a. By either meas we a plot the ( )-urve for fixed values of T ad L orrespodig to the margial stability. Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Page 8 (8) The umerial values of ad a ( = ) omputed from expressio (8) whe L = ad 6 L ( ) are respetively preseted i Table ad Table for various give values of T ad. The values of are hose so as to ompare the results obtaied by us with the results of Nield []. We fid that whe T values of ad a for various values of obtaied here are i lose agreemet with orrespodig values obtaied by Nield []. Whe = the results obtaied here for various values of T agree preisely with orrespodig values obtaied by Gupta et al. [8].

6 A.K. Gupta & S.K. Dhima Table. Values of ad a for various values of ad T whe L = T.3.5 T a a a Table. Values of ad a for various values of ad T whe L =6 T T T.3.5 a a a 3.7 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x T Fig. argial stability urves at the oset of Béard-aragoi ovetio for various values of whe (dotted urves) ad.3 (thik urves). T T The umerial values of ad a (=) omputed from expressio (8) whe L= ad 6 L are respetively preseted i Table.3 ad Table.4 for various give values of T ad. Whe T values of ad a for various values of obtaied here agree preisely with orrespodig values as obtaied by Nield []. Whe = the results obtaied here for various values of T agree preisely with orrespodig values obtaied by Gupta et al. [8]. Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Page 9

7 Effet of Uiform ageti Field o ayleigh Bèard aragoi Covetio i a elatively Hotter or Cooler Layer of Liquid Fig3. argial stability urves at the oset of ayleigh-béard ovetio for various values of whe (dotted urves) ad.3 (thik urves). T T For fixed values of T ad it may be metioed here that as L ireases values of both ad irease. I the limit L we fid that value of beomes asymptotially proportioal to L while that of teds to a fiite limit. I either ase the orrespodig ritial wave umber remais fiite. The ase whe L ireases from to orrespods to the situatio wherei thermal boudary oditios at the free surfae hages from ostat heat flux D to ostat temperature. Therefore whe L is small it is easier for temperature perturbatios to be set-up but whe L is large ay temperature variatios aross the free surfae deay rapidly. Thus as L beomes large the values of ted to ifiity sie it beomes diffiult for the surfae tesio to be operative while those of remai fiite sie the buoyay fore is still operative. The eutral stability urves for surfae tesio drive ovetio (=) ad for buoyay drive ovetio (=) are plotted respetively i Fig. ad Fig.3 for various values of ad T whe L=. Poits below a urve represet stable state. The upward movig eutral stability urves i Fig or 3 for ireasig values of whe value of T is fixed illustrate that mageti field has stabilizig effet o the oset of ovetio i eah ase. Further it is also see that ireasig values of T for a fixed value of has stabilizig effet i eah ase. We fid that presee of mageti field suppresses ovetio more effetively i a relatively hotter layer of liquid tha a ooler oe uder idetial oditios otherwise irrespetive of whether the agey ausig istability is surfae tesio or buoyay. Table3. Values of ad a for various values of ad T whe L = T.3.5 T a a a Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Page T

8 A.K. Gupta & S.K. Dhima Table4. Values of ad a for various values of ad T whe L =6 T.3.5 T a a a T Fig4. Variatio of aragoi ad ayleigh umbers (ormalized to give uit iterepts) at the oset of ayleigh-béard-aragoi ovetio for various values of whe (dotted urves) ad T.3 (thik urves). T Fig5. Wave umber orrespodig to margial stability plotted agaist ormalized ayleigh umber at the oset of ayleigh-béard-aragoi ovetio for various values of whe.3. The asymptoti behavior of as depeds ritially o both T ad L whereas the asymptoti behavior of a depeds oly o L. Whe L= we fid that T Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Page

9 Effet of Uiform ageti Field o ayleigh Bèard aragoi Covetio i a elatively Hotter or Cooler Layer of Liquid ad 4 T a.8 whih are i lose agreemet with the orreted values ad a 4.79 as obtaied by Wilso [8] (whe T ) after orretio of the iaurate asymptoti results of both Nield [] ad aekawa ad Taasawa []. Whe L is large we fid that L T a.393 ad 3 whih are i lose agreemet with the results of Nield [] (whe ). O the other had the asymptoti behaviour of as asymptoti behaviour of both ad ad T a 4 6 T depeds oly o T whereas the a is idepedet of L. For ay value of L we fid that whih are i lose agreemet with the asymptoti results of both Chadrasekhar [6] ad Nield [] whe T. The ( )-loi orrespodig to eutral stability urves for the ombied surfae tesio ad buoyay effets ormalized for ritial values of ad are plotted i Fig.4 for various values of ad T for the limitig ase whe L= (isulatig boudary). The values plotted are the ritial values at the orrespodig wave umbers. Fig.4 illustrates that the ihibitig effet of mageti field o the oset of ovetio remais uhaged for give values of T. As ireases the lous goes away from the lie T whih (as explaied i Gupta et al.[8]) orrespods to perfet ouplig showig that ouplig betwee the two agaris ausig istability beomes weaker. Fig 5 illustrates the variatio of the orrespodig ritial wave umbers at the margial stability for various values of. We ote that the value of ritial wave umber remais uhaged for differet values of T. 5. CONCLUSION The liear stability aalysis of the ayleigh-béard-aragoi ovetio i a relatively hotter or ooler layer of liquid i the presee of mageti field has bee studied theoretially ad we olude that a relatively hotter layer of liquid is more stable tha a ooler oe uder idetial oditios otherwise irrespetive of whether the two mehaisms ausig istability at idividually or simultaeously. ACKNOWLEDGEENT The authors are very grateful to Professor.G. Shadil (etd.) Departmet of athematis H.P. Uiversity Shimla-75 for his ivaluable advie throughout the ourse of preset work EFEENCES [].B. Baerjee ad J.. Gupta Studies i Hydrodyami ad Hydromageti Stability Silver Lie Publiatios Shimla (99). [] Baerjee.B. Gupta J.. Shadil.G. Sharma K.C. ad Katoh D.C. A modified aalysis of thermal ad thermohalie istability of a liquid layer heated uderside. J. ath. Phys. Si. 7 pp (983). [3] Béard H. Les tourbillos ellulaires das ue appe liquide trasportat de la haleur par ovetio e regime permaet. evue geerale des Siees pures at appliqués pp.6-7 (9). Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Page

10 A.K. Gupta & S.K. Dhima [4] Béard H. Les tourbillos ellulaires das ue apple liquide trasportat de la haleur par ovetio e re gime permaet. A. Chimie (Paris) 3 pp.6-44 (9). [5] Blok.J. Surfae tesio as the ause of Béard ells ad surfae deformatio i a liquid film. Nature 78 pp (956). [6] S. Chadrasekhar Hydrodyami ad Hydromageti Stability Dover Publiatios New York (98). [7] Gupta A.K. ad Shadil.G. aragoi ovetio i a relatively hotter or ooler liquid layer. Pro. Natl. Aad. Si. Idia Set. A Phys. Si. 8() pp.3-6 (). [8] Gupta A.G. Shadil.G. Kumar S. ad Surya D. O revisitig surfae tesio ad buoyay drive ovetio Coferee Proeedig Natioal Coferee o Advaes i athematis ad its Appliatios pp.57-6 (3). [9] E.L. Koshmieder Béard Cells ad Taylor Vorties. Cambridge Uiversity Press Cambridge (993). [] aekawa T. ad Taasawa I. Effet of mageti field o the oset of aragoi ovetio. It. J. Heat ad ass Trasfer 3 pp (988). [] Nield D.A. Surfae tesio ad buoyay effets i ellular ovetio. J. Fluid eh. 9 pp (964). [] Nield D.A. Surfae tesio ad buoyay effets i the ellular ovetio of a eletrially odutig liquid i a mageti field. Z. Agew. ath. Phys. 7 pp.3-39 (966). [3] Normad C. Pomeau Y. ad Velarde.G. Covetive istability: A physiist s approah. ev. od. Phys. 49 pp (977). [4] Pearso J..A. O ovetio ells idued by surfae tesio J. Fluid eh. 4 pp (958). [5] ayleigh L. o ovetio urrets i a horizotal layer of fluid whe the higher temperature is o the uderside. Phil. ag. 3 pp (96). [6] Shatz.F. ad Neitzel G.P. Experimets o the rmoapillary istabilities Au. ev. Fluid eh. 33 pp.93-7 (). [7] Sparrow E.. Goldstei.J. ad Josso V.K. Thermal istability i a horizotal fluid layer: effet of boudary oditios ad o-liear temperature profile. J. Fluid eh. 8 pp (964). [8] Wilso S.K. The effet of a uiform mageti field o the oset of aragoi ovetio i a layer of odutig fluid.. J. eh. ath. 46() pp.-48 (993). AUTHO S BIOGAPHY Ail Kumar Gupta is Professor i the Departmet of athematis Himahal Pradesh Uiversity Cetre for Eveig Studies Shimla-. His researh iterests are i the field of Hydrodyami ad Hydro mageti Istability. Additioally he is also author of six books i the subjet of athematis for professioal ourses. He has published a umber of researh papers i various reputed jourals ad guidig.phil ad Ph.D studets i the field of Cotiuum ehais. Iteratioal Joural of Advaed esearh i Physial Siee (IJAPS) Page 3