Effect of Magnetic Field on Marangoni Convection in Relatively Hotter or Cooler Liquid Layer

Transcription

1 Iteratioal Joural of Advaed Researh i Physial Siee (IJARPS) Volume, Issue, Jauary 05, PP 7-3 ISSN (Prit) & ISSN (Olie) ffet of Mageti Field o Maragoi Covetio i Relatively Hotter or Cooler Liquid Layer Deepti Surya Researh Sholar, Himahal Pradesh Uiversity, Cetre for veig Studies, Shimla, Idia. Abstrat: The effet of uiform vertial mageti field o the oset of surfae tesio drive ovetio i a relatively hotter or ooler layer of eletrially odutig liquid is osidered. A Fourier series method is used to obtai the harateristi value equatio for the Maragoi umber M. Whe istability sets i as statioary ovetio it is established umerially that both the ritial Maragoi umber ad wave umber irease with itesity of the mageti field, irrespetive of whether the liquid layer is relatively hotter or ooler. The asymptoti behavior of the ritial Maragoi umber for large values of the Chadrasekhar umber is also obtaied. Keywords: Codutig, Covetio, Liearstability, Statioary, Surfae-tesio.. INTRODUCTION The pheomeo of the oset of surfae tesio idued ovetio i a thi horizotal liquid layer heated from below with free upper surfae was first established theoretially by Pearso [] who demostrated that if the surfae tesio of the free surfae is liearly depedet o temperature the ovetive istability a our aalogous to those desribed by Rayleigh [] i terms of buoyay (disussed i detail by Chadrasekhar [3]). Nield [4] osidered the ombied effets of surfae tesio ad buoyay o the oset of ovetio i a fluid layer heated from below with free upper surfae ad foud that the two effets ausig istability reifore oe aother ad that as the thikess of the fluid layer dereases, the surfae tesio effets beome more domiat. Further otributios made by may researhers amely, Srive ad Sterlig [5], Smith [6], Davis [7], Takashima [8, 9] ad Gupta ad Surya [0] have refied Pearso s model by iorporatig more realisti oditios. For a detailed study of Maragoi ovetio oe may be referred to the work of Normad et al. [], Koshmieder [] ad Shatz et al. [3]. I view of the stabilizig ature of mageti field, a fat that has already bee established by Chadrasekhar [3] for the buoyay drive ovetio, by Nield [4] for the ovetive istability idued by both surfae tesio ad buoyay. The ifluee of mageti field o the pure Maragoi ovetio i a eletrially odutig liquid layer heated from below have bee disussed by Rudraiah et al [5], Maekawa ad Taasawa [6], ad by Wilso [7]. Reetly, the oset of Maragoi ovetio i a relatively hotter or ooler liquid layer has bee aalyzed by Gupta ad Shadil [8] ad established that irrespetive of the thermal ature (odutig or isulatig) of the lower boudary, the ritial Maragoi umber sigifiatly depeds o whether the liquid layer is relatively hotter or ooler, ad hotter the liquid layer more the postpoemet of the oset of ovetio. The problem osidered here is a geeralizatio of Gupta ad Shadil [8] work to ilude the effet of vertial mageti field. A Fourier series method is used to obtai the harateristi value equatio aalytially. The umerial results are obtaied for a wide rage of values of the Chadrasekhar umber Q. It is show umerially that both the ritial Maragoi umber M ad the ritial wave umber a irease mootoially with Q for a fixed value of the parameter T0 where T 0 ad beig the appropriately hose mea temperature ad oeffiiet of speifi heat (at ostat volume) variatio due to temperature variatio respetively of the fluid layer. The asymptoti behavior of the ritial Maragoi umber for large values of the Chadrasekhar umber is also obtaied. ARC Page 7

2 Deepti Surya. FORMULATION OF TH PROBLM We wish to examie the stability of a ifiite horizotal eletrially odutig liquid layer of uiform thikess d subjet to stregth exterally imposed vertial mageti field of H, whih is heated from below with the upper free surfae ope to the ambiet air, where surfae tesio gradiets arise due to temperature perturbatios. We hoose a Cartesia oordiate system of axes with the x ad y axis i the plae of the lower surfae ad the z- axis alog the vertially upward diretio so that the fluid is ofied betwee the plaes at z = 0 ad z = d. A temperature gradiet is maitaied aross the layer by maitaiig the lower boudary at a ostat temperature T 0 ad the upper boudary at T (< T 0 ). It is assumed that surfae tesio is give by the simple liear law ( T T) where the ostat τ is the uperturbed value of τ at the uperturbed surfae temperature T = T ad ( / T ) T T represets the rate of hage of surfae tesio with temperature, evaluated at temperature T, ad surfae tesio beig a mootoially dereasig futio of temperature, σ is positive. Followig Gupta ad Shadil [8], we a write the liearized perturbatio equatios for a eletrially odutig liquid i the presee of mageti field are H w h t 4 z ( T ) w t 0 z () () t w hz H. z Where w, ad hz deote respetively the z-ompoet of veloity perturbatio ad temperature perturbatio from uiform vertial temperature gradiet, ad the z-ompoet of the perturbatio from the basi vertial mageti field H, is the kiemati visosity; is the thermal diffusivity, permeability ad resistivity are eah assumed ostat. is the temperature gradiet whih is maitaied, T 0 is the uperturbed temperature of the lower surfae ad eah is assumed ostat. Further, that oeffiiet α (due to variatio i the temperature) is a ostat that rages from 0 to 0 3 for the liquid with whih we are oered., ad t deotes x y z time. I seekig solutios of the equatios (), () ad (3), we must satisfy ertai boudary oditios, The boudary oditios at the lower rigid ad odutig surfae z = 0 are straightforward ad give by w = 0 (4) (3) The boudary oditios at the upper free surfae z = d are more ompliated. Beause of the o-defletig surfae, the ormal ompoet of the veloity must vaish, that is, w 0 (7) The stress-balae oditio satisfy the equatio w z Here is the desity ad x. The boudary oditio (8) is usually referred to as y Iteratioal Joural of Advaed Researh i Physial Siee (IJARPS) Page 8 (5) (6) (8)

3 ffet of Mageti Field o Maragoi Covetio i Relatively Hotter or Cooler Liquid Layer the Maragoi boudary oditio (Pearso [6]). Fially, if we osider oservatio of heat trasport aross the upper free surfae, the we have k q z where k is the thermal odutivity of the fluid ad q is the heat trasfer oeffiiet. We ow suppose that the perturbatios w, ad h z are of the form (9) [ w( x, y, z, t), ( x, y, z, t), h ( x, y, z, t)] [ w( z), (z), h (z)]exp{ i( a x a y) pt} z z x y (0) where a a a is the wave umber of the disturbae ad p is a time ostat (whih a be x y omplex). The express the equatios obtaied by substitutig expressio (0) i equatios ()- z hz (3) i dimesioless form by takig z*, a* ad, K*, * d H p pd wd, W*, * d d ad D d. dz Restritig to the ase whe istability sets i as statioary ovetio, that is, whe the margial state is haraterized by settig p 0 i the resultig equatios. limiatig K from these equatios, ad omittig asterisk for simpliity, we obtai ad the orrespodig boudary oditios are evaluated o the lower rigid boudary z 0, ad () () W(0) 0 DW (0) 0 (3a, b, ) (0) 0 W() 0 D W() a M() D() L() (4a, b, ) evaluated o the upper free surfae z, where d M is the Maragoi umber ad Hd 4 Q is the Chadrasekhar umber, qd L is the Biot umber. k Solutio to equatios () ad () is sought subjet to boudary oditios (3a, b, ) (4a, b, ). Thus we have a eigevalue problem of order six. It is evidet that whe Q = 0 the system redues to the ase whih has bee aalyzed by Gupta ad Shadil [8]. 3. SOLUTION OF TH PROBLM The Fourier series method as preseted by Nield [4] is oveiet for the problem uder osideratio. The ostats to be elimiated are deoted by D W(0), D W(), (). 3 We let (5) Iteratioal Joural of Advaed Researh i Physial Siee (IJARPS) Page 9

4 Deepti Surya (6) where the boudary oditios (3a), (3) ad (4a) have already bee used while writig equatio (5) ad (6) The, we have D W z [A ( ) { ( ) }]si (7) A ( D W )si z (8) D B z ( )si (9) The differetial equatios () ad () are satisfied by substitutig the omplete Fourier expasios for W, ad their derivatives from equatios (5)-(9) ad equatig the oeffiiets of si z, we obtai [( a ) Q ] A [ 3 3 ( ) ]{ a ( a ) Q } (0) ( T 0) a ( T 0) A ( a ) B [ 3 3 ( ) ] ( ) 3, () The remaiig boudary oditios require that ( ) A 0, am3 0, () (3) B L 3 ( ) ( ) 0, From equatios (0)-(), A ad B a be expressed i terms of, ad 3 whih are whe substituted i equatios (), (4) ad o makig use of equatio (3), the yield two homogeeous equatios i ad. limiatio of these ostats gives the eigevalue equatios as (4) L ( T0 ) ( ) F ( T0 ) ( ) F ( a H ) Ma ( ) 0 (5) ( a ) where, F, H R R ( a ) with R a a Q ( )[( ) ]. From the eigevalue equatio (5), M a be determied as a futio of a, Q, T0 ad L as the ratio of two determiats Iteratioal Joural of Advaed Researh i Physial Siee (IJARPS) Page 0

5 ffet of Mageti Field o Maragoi Covetio i Relatively Hotter or Cooler Liquid Layer M( a, T, Q, L) = 0 a ( T ) 0 ( ) F a H ( ) 0 ( ) F F L (6) 4. NUMRICAL RSULTS AND DISCUSSION The umerial alulatios may be arried out as follows. For fixed values of T, 0 Q ad L, expressio (6) gives the Maragoi umber M as a futio of the wave umber a. The miimum value of M is the ritial Maragoi umber M ad the value of a at whih M attais the miimum is the ritial wave umber a. For give values of T0 ad Q it may be metioed here that the system was foud to be less stable for a isulatig upper free surfae (L = 0), whih is as expeted, sie a odutig upper free surfae ( L ) yields a ompletely stable system so far as the surfae tesio drive ovetio is oered. Values of ritial Maragoi umber M ad the ritial wave umber a omputed from expressio (6) for assiged values of T0 ad Q, are listed i Table ad Table respetively for L = 0 ad L. I either ase it is evidet that for a give value of T0both M ad a irease mootoially with Q, ad that for a fixed value of Q, a irease i the value of T0 leads to a ireased value of M while the value of ritial wave umber a remais uhaged. Whe T0 = 0, the ritial Maragoi umber ad the orrespodig wave umber obtaied here are idetial to the results obtaied by Nield [4] (for the ase whe the Rayleigh umber R = 0). Table. Numerial values of M ad a, for various values of Q whe L=0 T0 0 T0 0.3 T0 0.5 Q M a M a M a For a fixed value of T0, the asymptoti behavior of M ad a whe Q deped ritially o the thermal oditios at the upper surfae. We foud from the eigevalue equatio (8) that whe L = 0 M 4 T 0. 8 ad a 0. 8Q Q, Table. Numerial values of M ad a, for various values of Q whe L Q T0 0 T0 0.3 T0 0.5 M / L a M / L a M / L a Iteratioal Joural of Advaed Researh i Physial Siee (IJARPS) Page

6 Deepti Surya ad whe L M T ad a Q LQ. Figure. Neutral stability urves for various values of Q whe T0 = 0 ad T0 = 0.3 with L = 0. I Figure, the eutral stability urves are plotted for isulatig upper free surfae (L = 0), ad T0 0, T0 0.3 usig relatio (8) for various values of Q. The regio below eah urve represets the stable state. From Fig., we observed that the eutral stability urves move upwards for ireasig values of Q, learly showig the stabilizig effet of Q, irrespetive of whether the liquid layer is relatively hotter ( T0 0.3 ) or ooler ( T0 0). RFRNCS [] Pearso J. R. A., O ovetio ells idued by surfae tesio, Joural of Fluid Mehais, Vol.4, pp , (958). [] Rayleigh L., o ovetio urrets i a horizotal layer of fluid, whe the higher temperature is o the uderside, Phil. Mag., Vol.3 (6), pp , (96). [3] Chadrasekhar S., Hydrodyami ad Hydromageti Stability, Lodo: Oxford Uiversity Press, (96). [4] Nield D. A., Surfae tesio ad buoyay effets i ellular ovetio, Joural of Fluid Mehais, vol.9, pp.34-35, (964). [5] Srive L.. ad Sterlig C.V., o ellular ovetio drive by surfae tesio gradiets: effets of mea surfae tesio ad surfae visosity, Joural of Fluid Mehais, Vol.9, pp.3-340, (964). [6] Smith K.A., O ovetive istability idued by surfae tesio gradiets, Joural of Fluid Mehais, Vol.4(), pp.40-44, (966). [7] Davis S. H., Buoyay-surfae tesio istability by the method of eergy, Joural of Fluid Mehais, vol.39 (), pp , (969). [8] Takashima M., Surfae tesio drive istability i a horizotal liquid layer with deformable free surfae: I. Statioary ovetio, Joural of Physial Soiety of Japa, Vol.50, pp , (98(a). [9] Takashima M., Surfae tesio drive istability i a horizotal liquid layer with a deformable free surfae. II. Overstability, Joural of Physial Soiety of Japa, Vol.50, pp , (98(b)). Iteratioal Joural of Advaed Researh i Physial Siee (IJARPS) Page

7 ffet of Mageti Field o Maragoi Covetio i Relatively Hotter or Cooler Liquid Layer [0] Gupta A.K. ad Surya D., Béard Maragoi ovetio with free slip bottom ad mixed thermal boudary oditios, MJIS, Vol.() pp. 4-54, (04). [] Normad C., Pomeau Y. ad Velarde M., Covetive istabilty: a physiists approah, Rev. Mod. Phys., Vol.49, pp.58-64, (977). [] Koshmieder.L., Béard ells ad Taylor vorties, Cambridge: Cambridge Uiversity Press, (993). [3] Shatz M.F., VaHook S.J., MCormik W.D., Swift J.B. ad Swiey H.L., Oset of surfae tesio drive Béard ovetio, Phys. Rev. Lett., Vol.75, pp , (995). [4] Nield D.A., Surfae tesio ad Buoyay effets i the ellular ovetio of a eletrially odutig liquid i a mageti field, Z.A.M.M., Vol. 7, pp. 3-39, (966). [5] Rudraiah N., Ramahadramurthy V. ad Chada O. P., ffet of Mageti Field ad o uiform temperature gradiet o Maragoi Covetio, Iteratioal Joural of Heat Mass Trasfer.,Vol. 8(8), pp. 6-64,(983). [6] Maekawa T. ad Taasawa I., ffet of Mageti Field o Oset of Maragoi Covetio, Iteratioal Joural of Heat ad Mass Trasfer.,Vol. 3, pp. 85-9, (988). [7] Wilso S.K., The ffet of a Uiform Mageti Field o the Oset of Maragoi Covetio i a Layer of Codutig Fluid, Q. Joural of Mehais ad Applied Math., Vol. 46, pp. -48, (993). [8] Gupta A.K. ad Shadil R.G., Maragoi ovetio i a relatively hotter or ooler liquid layer, Pro. Natl. Aad. Si., Idia, Set. A Phys. Si., Vol.8(), pp , (0). Iteratioal Joural of Advaed Researh i Physial Siee (IJARPS) Page 3