Effect of MFD Viscosity on the Onset of Ferromagnetic Fluid Layer Heated from Below and Cooled from Above with Constant Heat Flux

Transcription

1 478/v48-9--x MEASUEMENT SCIENCE EVIEW Volume 9 No 3 9 Effet of MFD Visosity on the Onset of Feomagneti Fluid Laye Heated fom Below and Cooled fom Above with Constant Heat Flux C E Nanjundappa I S Shivakumaa K Sikuma 3 Depatment of Mathematis D Ambedka Institute of Tehnology Bangaloe INDIA e mail: enanju@hotmailom UGC-Cente fo Advaned Studies in Fluid Mehanis Depatment of Mathematis Bangaloe Univesity Bangaloe 56 INDIA e mail: shivakumaais@gmailom 3 Depatment of Mathematis C M Institute of Tehnology Bangaloe INDIA The effet of magneti field dependent (MFD) visosity on the onset of onvetion in a feomagneti fluid laye heated fom below and ooled fom above in the pesene of vetial magneti field is investigated theoetially The bounding sufaes ae onsideed to be eithe igid-feomagneti o stess fee with onstant heat flux onditions The esulting eigenvalue poblem is solved using the Galekin tehnique and also by egula petubation tehnique It is found that inease in MFD visosity and deease in magneti numbe is to delay the onset of feoonvetion while the nonlineaity of fluid magnetiation has no influene on the stability of the system Keywods: feomagneti fluid onstant heat flux Galekin tehnique MFD visosity INTODUCTION EOFLUIDS ae stable olloidal suspensions onsisting F of single-domain magneti patiles oated with a sufatant and immesed in a aie fluid The patiles ae ommonly magnetite and of ode nm in diamete while the aie liquid is typially an oil o wate base Feofluids ae of patiula inteest sine thee ae no known natual ouenes of these fluids and in distintion to magnetohydodynamis the flow phenomena ou without the need fo eletial uent and thus in the absene of oesponding Loent foes Theeafte feofluids of vaious kinds have been developed and thei appliation to engineeing uses attempted Pesently magneti fluids ae in wide use in seals beaings magnetostati suppot jet pintes sepaation of nonmagneti patiles flow ontol and dag edution dampes atuatos sensos tansdues and medial appliations An authoitative intodution to this fasinating subjet is amply povided in the liteatue [] [] Thus feofluids have eeived muh attention in the sientifi ommunity The magnetiation of feofluids depends on the magneti field tempeatue and density Hene any vaiations of these quantities indue hange of body foe distibution in the fluid and eventually give ise to onvetion in feofluids in the pesene of a gadient of magneti field Thee have been numeous studies on themal onvetion in a feofluid laye alled feoonvetion analogous to ayleigh-benad onvetion in odinay visous fluids The theoy of onvetive instability in a hoiontal laye of feofluid began with [3] and extensively ontinued ove the yeas [4]-[] and efeenes theein A vaiety of veloity and tempeatue bounday onditions on the onset of feoonvetion in an initially quiesent feofluid laye in the pesene of a unifom magneti field has been investigated eently [] Besides studies have been made to undestand the effet of diffeent paametes on themomagneti onvetion of feofluids in a ylindial geomety using the moden Taguhi tehnique [] and heat tansfe phenomena in a keosene based feofluid in two ylindes with diffeent dimensions using omputation fluid dynamis tehnique [3] Investigations have also been aied out to analye magneti onvetion of feofluids in enlosues [4]-[6] Themal onvetion in feomagneti fluids is gaining muh impotane due to astounding physial popeties One suh popety is visosity of the feomagneti fluid Fluids with feomagneti popeties may be fomed by olloidal suspension of solid magneti patiles suh as magnetite in a paent liquid The visosity of the magneti fluid is pedited by dimensional analysis to be a funtion of the atio of hydodynami stess to magneti stess [7] The effet of a homogeneous magneti field on the visosity of a fluid with solid patiles possessing intinsi magneti moments has been investigated [8] The effet of magneti field dependent (MFD) visosity on the onset of feoonvetion in a otating medium [9] with o without dust patiles [] and the nonlinea stability analysis [] have been examined All these studies ae dealt with isothemal bounday onditions The pesent study deals with the effet of MFD visosity on the onset of onvetion in a feomagneti fluid laye in the pesene of a unifom vetial magneti field fo diffeent ombinations of veloity bounday onditions with pesibed heat flux at the boundaies Suh a study is useful in the pope design of numeous devies and poesses suh as levitation aeleometes magnetifluids seals diet enegy onvetes and visous dampes [7] The esulting eigenvalue poblem is solved using the Galekin tehnique and also egula petubation tehnique fo both boundaies igid (igid-igid) both boundaies fee (fee-fee) and lowe bounday igid and uppe bounday fee (igid - fee) bounday ombinations To ahieve the above objetives the pape is oganied as follows Setion is devoted to mathematial fomulation The method of solution is disussed in Setion 3 In Setion 4 the numeial esults pesented ae disussed and some impotant onlusions follow in Setion 5 75

2 MEASUEMENT SCIENCE EVIEW Volume 9 No 3 9 MATHEMATICAL FOMULATION The system unde onsideation is an initially quiesent hoiontal feofluid laye of haateisti thikness d in the pesene of an applied magneti field H o in the vetial dietion The lowe and the uppe boundaies ae maintained at onstant tempeatue T and T( < T) espetively and thus onstant tempeatue diffeene T( = T T) is maintained between the boundaies A Catesian o-odinate system (x y ) is used with the oigin at the bottom of the laye and -axis is dieted vetially upwad Gavity ats in the negative -dietion g = gkˆ whee ˆk is the unit veto in the -dietion The fluid is assumed to be inompessible having vaiable visosity given by η = η ( + δ B ) whee δ is the vaiation oeffiient of magneti field dependent visosity and is onsideed to be isotopi [9] η is taken as visosity of the fluid when the applied magneti field is absent and B = ( Bx By B ) is the magneti indution Expeimentally it has been demonstated that the magneti visosity has got exponential vaiation with espet to magneti field [] As a fist appoximation fo small field vaiation linea vaiation of magneti visosity has been used The equations govening the flow of an inompessible feofluid ae [] : q = () q ρ [ + ( q ) q] = p + ρ g t () + [ η D] + µ ( M ) H M DT CVH H T Dt VH (3) M DH + µ T = kt T T Dt VH ρ µ ρ = ρ [ α t ( T T)] (4) B = H = o H = ϕ (5ab) B = µ ( M + H ) (6) H M = M H T H ( ) (7) M = M + χ( H H) K( T T) (8) Hee q = ( u v w) is the veloity veto M the magnetiation H the magneti intensity of the fluid p the T pessue t the time D = [ q+ ( q) ]/ the ate of stain tenso ρ the fluid density ρ the density at T = T µ the magneti pemeability of vauum kt the themal ondutivity of the fluid C VH the speifi heat at onstant volume and magneti field α t the themal expansion oeffiient φ the magneti potential χ = ( M / H) H T the magneti suseptibility K = ( M / T) H T the pyomagneti oeffiient = / x + / y + / the Laplaian opeato and M = M( H T) It is lea that thee exists the following solution fo the basi state: qb = pb( ) = p ρg ραtgβ µ Mκ β µ κ β ( + χ) ( + χ) (9) Tb ( ) = T β Kβ Hb ( ) = H kˆ ( χ) + Kβ M ( ) ˆ b = M + k ( + χ) whee β = T / d is the tempeatue gadient and the subsipt b denotes the basi state To investigate the onditions unde whih the quiesent solution is stable against small distubanes we onside a petubed state suh that q = q p = pb ( ) + p' η = ηb ( ) + η' T Tb ( ) T () = + H = Hb ( ) + H M = Mb ( ) + M whee q' p' η ' T' H' and M' ae petubed vaiables and ae assumed to be small Substituting Eq () into Eqs (6) and (7) and using Eq(5) we obtain (afte dopping the pimes) M Hx + Mx = + H x H M H y + My = + Hy H H + M = ( + χ ) H K T () Again substituting Eq () into the momentum Eq () lineaiing eliminating the pessue tem by opeating ul twie and using Eq () the -omponent of the esulting equation an be obtained as ( afte dopping the pimes ) 76

3 MEASUEMENT SCIENCE EVIEW Volume 9 No 3 9 ρ η b w µ K β = hϕ t µ K β + ht + ρα tg ht + χ () whee h = / x + / y is the hoiontal Laplaian opeato and ηb = η[ + δ µ ( M + H)] The enegy Eq (3) afte using Eq () and lineaiing takes the fom (afte dopping the pimes) T ϕ ρ C µ TK = t t µ TK ρc wβ + k T + χ (3) whee ρ C = ρ C VH + µ H K Equations 5(a b) afte substituting Eq () and using Eq () may be witten as M ϕ + ϕ+ ( + χ h ) H T K = (4) Sine the piniple of exhange of stability is valid [3] the nomal mode expansion of the dependent vaiables is assumed in the fom { wt ϕ } = { W( ) Θ( ) Φ ( ) } exp [ i( x+ my) ] l (5) whee l and m ae wave numbes in the x and y dietions espetively On substituting Eq (5) into Eqs () - (4) and non-dimesionaliing the vaiables by setting x y ( x y ) = d d d d W = W ν κ Θ = Θ β vd + χ) κ Φ = Φ and Κ β vd δ = µ H (+ χ) δ (6) whee v = η / ρ is the kinemati visosity and κ = kt / ρc is the effetive themal diffusivity we obtain ( + δ )( D a ) W + a MDΦ (7) a ( + M) Θ= ( D a ) Θ+ ( M ) W = (8) ( D a M3) Φ DΘ= (9) Hee D = d / d is the diffeential opeato a = l + m is the oveall hoiontal wavenumbe = αt gβ d 4 / νκ is the themal ayleigh numbe M = µ K β /( + χ) α t ρg is the magneti numbe M = µ TK / ρc( + χ) is the magneti paamete and M 3 = ( + M / H ) /( + χ) is the measue of nonlineaity of magnetiation The typial value of M fo magneti fluids with diffeent aie liquids tuns out to be of the ode of -6 and hene its effet is negleted as ompaed to unity The bounding sufaes (igid o fee) of the feofluid laye ae onsideed to be feomagneti and insulated to tempeatue petubations Thus on the igid bounday W = DW =Φ= DΘ= and on the stess-fee bounday W = D W = DΦ = DΘ= 3 METHOD OF SOLUTION Equations (7) (9) togethe with the oesponding bounday onditions onstitute an eigenvalue poblem with as an eigenvalue The method of solution is desibed in the following sub-setions 3 Solution by galekin tehnique The Galekin method is used to solve this poblem as explained [3] In this method the test (weighted) funtions ae the same as the base (tial) funtions Aodingly W Θ and Φ ae witten as n W = Ai Wi( ) i= n Θ ( ) = Ci Θi( ) () i= n Φ ( ) = Di Φi( ) i= whee A i C i and detemined The base funtions ( ) i D i ae the unknown onstants to be W Θ i ( ) and Φ i ( ) ae geneally hosen suh that they satisfy the oesponding bounday onditions but not the diffeential equations Fo igid-igid igid-fee and fee-fee boundaies the base the base funtions ae espetively and 4 3 Wi =( + ) T i Θ i = ( / 3) T i Φ i =( )( ) T i 4 3 Wi =( ) T i Φ i = ( / 3) T i Φ i = ( / 3) T i () () 77

4 MEASUEMENT SCIENCE EVIEW Volume 9 No Wi =( + ) T i Θ i = ( / 3) T i Φ i = ( / 3) Ti (3) whee Ti ' s ae the modified Chebyshev polynomials The above tial funtions satisfy all the bounday onditions Multiplying Eq (7) by Wi ( ) Eq (8) by Θ i ( ) and Eq (9) by Φ i ( ) ; pefoming the integation by pats with espet to between = and = and using the bounday onditions we obtain a system of linea homogeneous algebai equations in A i C i and D i A non-tivial solution to these equations leads to a elation involving the physial paametes δ M M 3 and the wave numbe a in the fom f( δ M M3 a) = The itial value of (ie ) is detemined numeially with espet to a fo diffeent values of δ M and M 3 3 Solution by egula petubation tehnique Sine the itial wave numbe is negligibly small when the boundaies ae insulated to tempeatue petubations (ie DΘ = at = ) the eigenvalue poblem is also solved analytially using egula petubation tehnique with wave numbe a as a petubation paamete Aodingly the vaiables W Θ and Φ ae expanded in powes of a as ( W Θ Φ ) = ( W Θ Φ ) + a ( W Θ Φ ) + (4) Substituting Eq (4) into Eqs(7) - (9) as well as in the bounday onditions the tems of diffeent odes ae olleted and solved The solution to the eo-th ode equations fo igid-igid and igid- fee boundaies is found to be W = Θ = and Φ = while the solution fo feefee boundaies is given by W = Θ = and Φ = The geneal solution fo veloity at the fist ode fo igid-igid igid-fee and fee-fee bounday onditions espetively is given by 4 3 ( + M) ( + ) W = (5a) 4 ( + δ ) 4 3 ( + M) ( ) W = (5b) 4 ( + δ ) and 4 3 ( + M) ( + ) W = (5) 4 ( + δ ) An expession fo the itial ayleigh numbe fo igid-igid igid-fee and fee-fee bounday onditions is obtained fom the solvability ondition = Wd (6) espetively in the fom 7 ( + δ ) = ( + M) (7a) 3 ( + δ ) = (7b) ( + M) and 3 ( + δ ) = (7) ( + M) It is inteesting to hek the above elations fo some speial ases Letting δ = Eqs (7ab) espetively edue to 7 = (8a) ( + M) 3 = ( + M) (8b) = (8) ( + M) and thus eoveing the esults fo the ase of onstant visosity [] When M = (ie odinay visous fluid) Eqs (8ab) edue to the itial ayleigh numbes of = 7 3 and whih ae the known exat values fo the igid-igid igid-fee and fee-fee boundaies espetively Fom Eqs (7ab) it is inteesting to note that the nonlineaity of fluid magnetiation (ie M 3 ) has no effet on the onset of onvetion; a esult whih is evealed by numeial omputations aied out in the pevious setion Sine at the onset of onvetion a = (vey lage wave length) one would expet that M 3 has no effet on the stability of the system (see Eq9) The numeially omputed values of fo diffeent values of δ and M ae ompaed in Table with those obtained using egula petubation tehnique We note that the esults obtained fom simple egula petubation tehnique oinide exatly with those obtained fom time onsuming numeial methods and thus povide a justifiation fo the esults obtained analytially 4 ESULTS AND DISCUSSION The effet of MFD visosity δ on the onset of onvetion in a feofluid laye in the pesene of a unifom vetial magneti field is studied The boundaies ae onsideed to be insulated to tempeatue petubations and the esulting eigenvalue poblem is solved numeially using the Galekin tehnique as well as analytially by employing egula petubation tehnique with wave numbe as a petubation paamete The itial eigenvalue obtained numeially and analytially fo diffeent values of M δ and fo diffeent bounday onditions ae pesented in Table Sine the igid boundaies suppess the petubations to a geate extent ompaed to the fee boundaies we note that the itial ayleigh numbe fo igid-igid boundaies is the geatest followed by igid-fee boundaies and the least 78

5 MEASUEMENT SCIENCE EVIEW Volume 9 No 3 9 fo fee-fee boundaies The table also pedits the effet of δ whih epesents the effet of visosity vaiation with magneti field on the iteion fo the onset of onvetion We note that inease in δ though the effet is not so signifiant has a stabiliing effet on the system That is the effet of inease in the value of δ is to delay the onset of feoonvetion An inease in the value of M is to deease the value of and thus its effet is to augment feoonvetion due to an inease in the magneti foe The vetial veloity eigenfuntions W( ) fo diffeent bounday ombinations ae pesented in Fig and fo diffeent values of δ and M espetively As an be seen inease in the value of MFD visosity δ (see Fig) is to inhibit the feofluid flow and hene its effet is to delay the onset of feoonvetion Howeve inease in the value of M (see Fig) is to aeleate the feofluid flow and hene its effet is to hasten the onset of feoonvetion 8 δ = δ = 5 δ = 9 6 igid-igid igid-fee 4 Fee-Fee W () 3 4 Fig Vetial veloity eigenfuntion fo diffeent values of δ when M = 8 Fee-Fee igid-fee igid-igid M = W () 6 8 Fig Vetial veloity eigenfuntion fo diffeent values of M when δ = M Table Citial ayleigh numbe igid-igid boundaies δ = δ = 5 δ = fo vaious values of M δ and fo diffeent boundaies igid-fee boundaies Fee-Fee boundaies δ = δ = 5 δ = 9 δ = δ = 5 δ =

6 MEASUEMENT SCIENCE EVIEW Volume 9 Setion 3 No3 9 5 CONCLUSIONS Fom the foegoing study it is obseved that an inease in the value of δ and deease in M is to stabilie the feofluid motion against the onvetion In ontast to the isothemal boundaies ase the nonlineaity of fluid magnetiation M 3 is found to have no effet on the stability of the system The itial eigenvalues obtained fom the ombination of analytial and numeial tehniques omplement vey losely with Eah othe and note that ( ) igid-igid > ( ) igid-fee > ( ) fee-fee ACKNOWLEDGEMENTS The wok epoted in this pape was suppoted by UGC unde CAS Pogam Depatment of Mathematis Bangaloe Univesity The authos (CEN) and (KS) wish to thank the Management and Pinipal of D Ambedka Institute of Tehnology and C M Institute of Engineeing and Tehnology Bangaloe espetively fo thei enouagement EFEENCES [] osensweig E (985) Feohydodynamis Cambidge: Cambidge Univesity Pess [] Bashtovoy VG Bekovsky BN Vislovih AN (988) Intodution to Themomehanis of Magneti Fluids Washington: Hemisphee Publishing Cop [3] Finlayson BA (97) Convetive instability of feomagneti fluids Jounal of Fluid Mehanis [4] Odenbah S (3) Magnetifluids-suspensionsof magneti dipoles and thei manti ontols Jounal of Physis [5] Lalas DP Cami S (97) Themoonvetive stability of feofluids Physis of Fluids [6] Shliomis MI (974) Magnetifluids Soviet Physis Uspekhi [7] Gotoh K Yamada M (98) Themal onvetion in a hoiontal laye of magnetifluids Jounal of Physial Soiety of Japan [8] Shwab L Hildebandt U Stiestadt K (983) Magneti Bénad onvetion Jounal of Magnetism and Magneti Mateials [9] Stiles PJ Kagan MJ (99) Themoonvetive instability of a feofluid in a stong magneti field Jounal of Colloid and Intefae Siene [] Mahjan S Mahjan A (8) Nonlinea stability analysis fo magnetied feofluid heated fom below Poeedings of oyal Soiety of London A: Mathematial Physial & Engineeing Siene [] Nanjundappa CE Shivakumaa IS (8) Effet of veloity and tempeatue bounday onditions on onvetive instability in a feofluid laye Jounal of Heat Tansfe [] Jafai A Tynjälä T Mousavi SM Sakomaa P (8) CFD simulation and evaluation of ontollable paametes effet on themomagneti onvetion in feofluids using Taguhi tehnique Computes and Fluids 37 () [3] Jafai A Tynjälä T Mousavi SM Sakomaa P (8) Simulation of heat tansfe in a feofluid using omputational fluid dynamis tehnique Intenational Jounal of Heat and Fluid Flow 9 (4) 97- [4] Jue T-C (6) Analysis of ombined themal and magneti onvetion feofluid flow in a avity Intenational Communiations in Heat and Mass Tansfe 33 (7) [5] Tangthieng C Finlayson BA Maulbetsh J Cade T (999) Heat tansfe enhanement in feofluids subjeted to steady magneti fields Jounal of Magnetism and Magneti Mateials (-3) 5-55 [6] Snyde SM Cade T Finlayson BA (3) Finite element model of magnetoonvetion of a Feofluid Jounal of Magnetism and Magneti Mateials 6 () [7] osenwieg E Kaise Miskoly G (969) Visosity of magneti fluid in a magneti field Jounal of Colloid and Intefae Siene 9 (4) [8] Shliomis MI (97) Effet of visosity of magneti suspensions Soviet Physis JETP [9] Vaidyanathan G Seka amanathan A () Effet of magneti field dependent visosity on feoonvetion in otating medium Indian Jounal of Pue and Applied Physis [] Sunil Shama A Shandil G (8) Effet of magneti field dependent visosity on feoonvetion in the pesene of dust patiles Jounal of Applied Mathematis and Computing 7 7 [] Sunil Shama P Mahajan A (8) A nonlinea stability analysis fo themoonvetive magnetied feofluid with magneti field dependent visosity Intenational Communiations in Heat and Mass Tansfe 35 () 8-87 [] osenwieg E (966) Bouyany and stable levitation of a magneti body immesed in a magnetiable fluid Natue [3] Finlayson BA (97) Method of Weighted esiduals and Vaiational Piniples Aademi Pess 8