UNSTEADY HELICAL FLOWS OF A MAXWELL FLUID

Transcription

1 PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Seies A, OF THE ROMANIAN ACADEMY Volue 5, Nube /4,. - UNSTEADY HELICAL FLOWS OF A MAXWELL FLUID Cosai FETECAU, Coia FETECAU Techical Uivesiy of Iasi, Roaia E-ail: cfeecau@yahoo.de The exac soluios coesodig o soe useady helical flows of a icoessible Maxwell fluid, saisfyig o-sli bouday codiios, ae deeied by eas of he exasio heoe of Seklov. The siila soluios fo a Navie-Sokes fluid aea as a liiig case of hese soluios. The seady sae soluios ae also obaied fo.. INTRODUCTION The siles cosiuive equaio fo a fluid is he Newoia oe. Fo he icoessible case i is of he fo T = I S, S = ì A, acea =, ( whee T is he sess eso, he hydosaic essue, S he exa-sess eso, A he fis Rivli- Eickse eso ad µ he dyaic viscosiy. May coo fluids show his Newoia behavio ad i fac he whole discilie of classical fluid echaics is based uo his equaio. Howeve, he cosiuive equaio ( does o show ay of he oal sess effecs o elaxaio heoea ad fo ay fluids, like dilue olyeic fluids, i is fo ha easo o acceable. I cases of ie deede flows, fo isace due o abu chages i he flow geoey, o o ie deedecie of bouday codiios, elaxaio heoea should be icluded. The siles way o do his is o use a equaio of Maxwell ye []: whee λ is he elaxaio ie ad ue coveced deivaive äs S ë = ì A, ( ä δ / δ deoes a covecive deivaive. The os oula choice is he δ S = S LS SL T, δ ( whee L is he velociy gadie ad he do deoes aeial ie diffeeiaio. The associaed ue coveced Maxwell-odel has he advaage ha i is cosise wih soe ioa icoscoical odels of olyes ad ha is edicios of he oal-sess diffeeces ae qualiaively acceable. Recely, he Maxwell odel has eceived secial aeio. Thus, soe exisece ad uiqueess esuls coesodig o diffee seady flows of a class of fluids icludig he Maxwell odel ae obaied i [, ]. The exisece of a lage class of soluios, which aise fo saially eiodic eubaios of uifo shea flow, is oved i [4]. The fis exac soluios obaied fo he flow of a Maxwell fluid see o be hose fo [] ad [5]. Ohe aalyical esuls ae obaied i [6-8]. The eseach eoed hee is devoed o he sudy of a helical flow of a icoessible Maxwell fluid bewee wo ifiie coaxial cicula cylides. The flow is due o he cylides, which ae assued o oae abou hei axis ad slide i he diecio of he sae axis wih escibed velociies. Fially, he secial case of he flow i a cylide is also cosideed. The siila soluios coesodig o he Navie-Sokes fluid aea as a liiig case. Recoeded by Lazã DARGOª, Mebe of he Roaia Acadey

2 Cosai FETECÃU, Coia FETECÃU. HELICAL FLOW BETWEEN CONCENTRIC CYLINDERS We coside hee a useady helical flow bewee wo ifiie coaxial cylides locaed a = R ad = R ( > R i he cylidical coodiae syse (, θ, z. Such a flow whose hysical cooes of he velociy field ae give by [5, 9] v =, v =ω (,, v = u (,, (4 θ is called helical because, i geeal, is sealies ae helices. Sice he velociy field is ideede of θ ad z, he exa-sess eso S will also be ideede of θ ad z ad he icoessibiliy codiio is auoaically saisfied. Moeove, sice he fluid was a es u o he oe = z ω (, = u (, = ad S(, =. (5 The flow is oduced by he wo cylides which a = suddely begi o oae abou hei coo axis ( = wih he agula velociies Ω ad Ω ad o slide i he z-diecio wih he velociies U ad U. Assuig ha he fluid adhees o he walls we have he bouday codiios ω ( R, = RΩ, ω ( R, = R Ω ; >, ur (, = U, ur (, = U ; >. Subsiuig (4 io ( ad ( ad akig io accou ( 5 we fid ha S = ad ( λ τ =µ ωω ( /, ( λ τ =µ u, ( λ τ =λ[( ωω/ τ ( u τ ], ( λ σ = λ ( ωω/ τ, ( λ σ = ( λ u τ, whee τ = S θ, τ = Sz, τ = Sθz, σ = Sθθ ad σ = Szz. The equaios of oio, i he absece of body foces, educe o (6 (7 ω σ =ρ, τ τ=ρ ω, τ τ =ρ u, (8 whee ρ is he desiy of he fluid, θ = due o he oaioal syey ad z = fo he assuio ha hee is o alied essue gadie alog he axial diecio (cf. []. Now, we obseve ha Eqs. (7 5 ad (8 fo τ, σ, σ ad ae o couled wih Eqs. (7, ad (8,, eaig ha oe ca solve he syse of he lae fou equaios fis ad he calculae τ, σ, σ ad. Eliiaig τ ad τ bewee Eqs. (7, ad (8, we aai o he ex wo aial diffeeial equaios λ ω (, ω (, =ν (,, ω λ u (, u (, =ν u (,, whee ν = µ / ρ is he kieaic viscosiy of he fluid. I is also woh ehasizig ha hese equaios ae of a highe ode ha he siila Navie-Sokes equaios. I ode o obai exac soluios he addiioal iiial codiios [5] have o be saisfied. ω (, = u (, =, ( (9

3 Useady helical flows of a Maxweel fluid whee Makig he chage of ukow fucios Ω Ω Ω ( = Ω.. Calculaio of he velociy field ω (, =Ω ( v (,, u (, = U( v (,, ( R R R ( R U U ad U ( = U l( R/ we easily ge fo (9, l( R / R (6 ad (5, he ex wo obles wih iiial ad bouday codiios λ v (, v (, = L v (,; ( R, R, >, ( v (, =V (, v (, = ; ( R, R, ( v ( R, = v ( R, = ;, (4 whee V( = U (, V( =Ω (, L = ad =,. I ode o solve hese obles we shall use, as i [], he well-kow exasio heoe of Seklov. I view of his heoe ou soluios v (, whose aial deivaives v ad v have o be iecewise coiuous, ca be wie, fo each >, as Fouie-Bessel seies absoluely ad uifoly covege i es of he eigefucios B A J( R ( = J( Y(, Y( R of he eigevalues obles Lv λ v =, v ( R = v( R = ; =, i.e., (5 v (, = v (B (. (6 Hee, J ( ad Y( ae Bessel fucios of ode of he fis ad secod kid, ae he osiive oos of he ascedeal equaios B( R = ad he cosas A ae chose so ha he oalizaio codiios R R [ ] B( d = ; =,, o be saisfied. Now, ioducig (6 i (, ulilyig he by B ( ad iegaig we esec o fo R o R, we fid ha Fo ( i also esuls ( ( (,. (7 λ v v ν v = > (8 v ( = V, v ( =, (9 whee V ae he fiie Hakel asfos of V (, []. Solvig (8 ude iiial codiios (9 ad havig i id (6 ad ( we ge fo ω (, ad u (, he exessios:

4 Cosai FETECÃU, Coia FETECÃU 4 esecively, s ex( s s ex( s ω (, =Ω( Ω B( P s s β β Ω ex cos si B(, λ λ β λ s ex( s s ex( s u (, = U ( U B( P s s β β ex cos si B(, U λ λ β λ ( ( ± 4νλ whee s,s =, β = 4νλ, < q λ νλ wih =, ad q =... Calculaio of he ageial esios τ ad τ The soluios of he odiay diffeeial equaios (7, wih he iiial codiios (5 ae ad µ τ ω(, τ τ (, = ex ex ω (, τ d τ, λ λ λ µ τ τ (, = ex ex u (, τ d τ. λ λ λ Ioducig ( ad ( i ( we ge By akig Ω Ω RR τ (, = µ ex λ R R µ ex( s ex( s Ω [ B( B( ] 4νλ µ Ω ex β si B( B( λ β λ [ ] U U τ (, =µ ex λ l( R/ R ex( s ex( s µ U B( µ 4νλ Ω β ex si B(. λ β λ i (,(, ( ad (4 we ge ( ( (4

5 5 Useady helical flows of a Maxweel fluid ad Ω Ω R ( R U U ω ( = Ω, u ( = U l( R / R R l( R/ R (5 which eese he seady sae soluios. Ω Ω ( RR, ( U τ = µ τ =µ U, R R l( R / R (6. HELICAL FLOW THROUGH A CIRCULAR CYLINDER Takig he lii of Eqs. (5 ad (7 whe R we fid he eigefucios J( /[ RJ( R ] ad J( /[ RJ ( R ] coesodig o he helical flow hough a ifiie cicula cylide. The bouday codiios (6 us be chaged by ω (, <, ω ( R, = RΩ ; u(, <, ur (, = U; > (7 ad he cooes of he velociy ω (, ad u (, ake he fos s ex( s s ex( s J( ω (, = ΩΩ J( s s R β β J( Ω ex cos si λ λ β λ J( R (ideically wih (.4 of [5] whee J ( has o be chaged by J ( ad U s ex( s s ex( s J( u (, = U R s s J( R U β β J( R R The associaed ageial esios ex cos si. = λ λ β λ J( (8 (9 ad ex( s ex( s J( τ (, =µω 4νλ J( R = β J( µω 4 ex si λ β λ J( R ( µ U ex( s ex( s J( τ (, = R 4νλ J( R ae also obaied as liiig cases of ( ad (4. 4µ U β J( R λ β λ J( R ex si, (

6 Cosai FETECÃU, Coia FETECÃU 6 4. LIMITING CASE λ = Takig he liis of Eqs. (, (, (, (4, (8, (9, ( ad ( as λ, we obai ( ω (, =Ω( Ω B( ex( ν, ( = u (, = U ( U B( ex( ν, Ω Ω RR µ τ (, = µ Ω B( B( ex( ν, (4 [ ] R R U U τ (, =µ µ U B( ex( ν, (5 l( R/ R J( ω (, = ΩΩ ex( ν, (6 J( R esecively, U J( u (, = U ex( ν, R (7 J( J( τ (, = µω ex( ν, (8 J( R µ U J( τ (, = ex( ν, (9 R J( R which ae he siila soluios coesodig o a Navie-Sokes fluid. Fially, by akig i ayoe of he above exessios we ge he seady sae soluios. They ae he sae fo boh yes of fluid. 5. CONCLUSIONS I his ae we have esablished he exac soluios coesodig o a helical flow of a Maxwell fluid bewee wo ifiie coaxial cicula cylides. By leig R ad R R i hese soluios we aai o he siila soluios coesodig o a helical flow hough a ifiie cicula cylide. All hese soluios, give by (, (, (, (4, esecively, (8, (9, ( ad (, coai sie ad cosie es. Tha idicaes ha by coas wih he Newoia fluid, whose soluios ( (9 do o coai such es, oscillaios ae se u i he fluid. The aliudes of hese oscillaios decay exoeially i ie, he daig beig ooioal o ex( / λ o ex( / λ. Diec couaios show ha ω(,, u (,, τ(, ad τ (, saisfy boh he associae aial diffeeial equaios ad all iosed iiial ad bouday codiios, he diffeeiaio e by e i ad beig clealy eissible. I he secial case, whe he elaxaio ie λ, ou soluios educe o hose coesodig o a Newoia fluid. The seady sae soluios ae also obaied as a liiig case fo. They ae he sae fo boh yes of fluid.

7 7 Useady helical flows of a Maxweel fluid REFERENCES. BÖHME, G., Söugsechaik ich-ewosche fluide, B. G. Teube, Suga-Leizig-Wiesbade,.. NOVOTNY, A., SEQUEIRA, A. ad VIDEMAN, J. H., Seady oios of viscoelasic fluids i hee-diesioal exeio doais. Exisece, uiqueess ad asyoic behavio, Ach. Raioal Mech. Aal. 49, , FONTELOS, M., FRIEDMAN, A., Saioay o-newoia fluid flows i chael-like ad ie-like doais, Ach. Raioal Mech. Aal. 5,. -4,. 4. RENARDY, M., Wall bouday layes fo Maxwell fluids, Ach. Raioal Mech. Aal. 5, 9-,. 5. SRIVASTAVA, P. N., No-seady helical flow of a visco-elasic liquid, Ach. Mech. Sos. 8 (,. 45-5, FETECAU, C., FETECAU, C., A ew exac soluio fo he flow of a Maxwell fluid as a ifiie lae, I. J. No-Liea Mech. 8 (,. 47-4,. 7. FETECAU, C., FETECAU, C., Decay of a oeial voex i a Maxwell fluid, I. J. No-Liea Mech. 8 (7, ,. 8. FETECAU, C., ZIEREP, J., The Rayleigh-Sokes-Poble fo a Maxwell fluid, ZAMP 54,. -8,. 9. TRUESDELL, C., NOLL, W., The o-liea field heoies of echaics, Hadbuch de Physik, Vol. III/, Sige-Velag, Beli-Heidelbeg-New Yok, RAJAGOPAL, K. R., BHATNAGAR, R. K., Exac soluios fo soe sile flows of a Oldoyd-B fluid, Aca Mech.,. -9, FETECAU, C., FETECAU, C., O he uiqueess of soe helical flows of a secod gade fluid, Aca Mech. 57,. 47-5, SNEDDON, I. N., Fouie asfos, McGRAW-HILL Book Coay, New Yok-Tooo-Lodo, 95. Received Novebe 6,