Multiquadrics method for Couette flow of a yield-stress fluid under imposed torques

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1 Ieol Wokshop o MeshFee Mehods 003 Mulqudcs mehod fo Couee flo of yeld-sess flud ude mposed oques M. E-R, A., C. ou, O. Seo-Gullume Absc: A L descpo of he Couee flo beee o col cyldes, of vscoplsc flud (.e. ehb yeld sess) ude mposed oque s peseed. Beyod vlue of he she sess, he vscosy vo s ppomed by lye of o flud eos h dffee vscoses such h µ / µ <<. So he heolocl behvou s descbed by he model of bvscosy hch ppoches he Bhm model. I hs ok meshless dl bss fuco mehod s used o buld ppomo of he PDEs ove he Couee flo. The used echque s bsed o he pplco of loblly mulqudcs dl bss fuco o compue he velocy feld d he fee sufce sep he o phses. Ioduco The ol fomulo of he cosuve equos fo vscoplsc mels s ve by Oldoyd [5] d he hs bee suded eesvely fo dffee eomey [- 7]. Hoeve vey fe umecl, heoecl o epemel oks hve bee dedced o flos of yeld-sess fluds hch s due o he dffcules boud up h he fee sufce sep he sold d el phses. We sudy he Couee flo of vscoplsc flud h flls he p beee o ccul cyldes he cos oque C s suddely ppled o he e cylde, he oue e be kep mooless. The flud s ssumed o be es d pese self lke elsc mel (el) l me. By sep che of oque follos ul velocy he eo cylde. The flud lyes he ehbouhood of he ll of he e cylde e o o udeo ul dsoo. Whe hs eceeds ccl vlue, he mechcl lks beee chs of polymes e desoyed, d che of phse s obseved: psse of el se o he lqud se. Lbooe de Mécque de Mleu Hééoèes, F.S.T de Te, BP 46, Moc LEMTA CRS, veue de l Foê de Hye BP 60, Vdeouve, 54600, Fce

2 M. E-R, A., C. ou, O. Seo-Gullume ω lqud el s efce Fue : Cofuo of he poblem The efce sep he lqud d el phses s o o evolve he p ul ech soy se. The suo s skeched F I. We ve hee he evoluo of he efce d velocy feld pofles he o phses. Mhemcl fomulo The flo s eeed by he oo of he eo cylde h smll ul velocy. Fo he bvscosy model, he she sess s skeched belo F II, hee τ s s he yeld-sess d τ c s he ccl vlue. d deoed he ul velocy he lqud phse d el phse especvely. She sess ll be oed τ l d τ. τ (she sess). τ τ s µ γ τ c τ s. τ µ γ. γ c γ. (she e) Fue II: Model of bvscosy I h follos e ll ve he equos ove hs ype of flo dmesoless vbles, hee s he dl poso:

3 Ieol Wokshop o MeshFee Mehods Lqud phse The equos h oves he Lqude phse s ( τl ) fo < < s () hee τl τs () he boudy codos e s follos: -A he ll of he e cylde ( ): I C π Lτl (3) hee I s he ul momeum d L s chcesc leh of cyldes.. Gel phse Fo he el phse e hve : ( τ ) fo s < < (4) hee τ µ µ, µ µ (5) Wh he boudy codos : - A he ll of he oue cylde ( ): 0 (6) -A he efce lqud-el ( s), oe mus hve couy of velocy d she sess.:, τ l τ s, τ τ s. (7) Ths ls elo becomes: 0. By pply he follo che of vbles he o phses: ( ) d ( y ), hee s, s equo () becomes: [ 3 ( )] τs (8) Th my be e s : A B C D (9) Whle he boudy codos e:

4 4 M. E-R, A., C. ou, O. Seo-Gullume ( ) π τ 0, s L m m m C I (0) Theefoe ssfy he sysem : [ ] ) ( 3 µ D () Th my be e s : R Q P () subec o he boudy codos : ( ) 0 µ τ y s y s (3) Couy of velocy felds, he efce, ves : ( ) ( ) (4) Soluo h Rdl bss fuco mehod I hs seco mplc scheme s devsed us he MQ mehod o solve he equos (8) d () subec o he boudy codos (0), (3). Us he oo ( ),,,, hee, hee s he me sep sze d follo he de of MQ mehod [8-0], s ppome, ech eo, by: ( ) β β (5) Whee ( ) ( ) e he loblly mulqudque fuco, () ( ) / c d c > 0 s ve shpe pmee. We oce h : ( ) β ) (, ( ) β ) ( (6) So oduc he fom (5) d (6) he equos (9) d () d pply hem o colloco pos, e hve: ( ) ( ) D C B A β β (7)

5 Ieol Wokshop o MeshFee Mehods P Q β ( ) R β ( ) (8) Addol codo β 0 mus be dded o equo (7) d (8) o ssue he esece d uquess of he soluo. The efce c be clculed ech me eo by us he equo (4). The Dchoome mehod s used s he opmso echque fo locls he efce poso. Dscusso d cocluso A umecl scheme fo compu he velocy feld d he efce lqud-el poso of he Couee flo of yeld sess flud s esblshed. The echque s bsed o he loblly mulqudcs dl bss fuco. Besde he oley of he poblem s fom he uko poso of he efce, s lso obseved h hee s physcl suly he s of he flo oed o he mposed oque hch complced moe he umecl modell. Fue III descb he efce loco s(), llused hs physcl suly d shos he dffcules o coveece of he scheme. I s ecessy o oe h hs cuve ed ods lm poso s l. Fo he shpe pmee c, e dop he sueso fom Cho e. l. [9] by choos c o be cos mes he mmum dsce beee o colloco pos. I ould be ese o mpleme lohm fo compu he efce hle us eule descpo of he poblem. So ll be ecessy o dd kemcl codo cy o volume fco Fue III: Evolv efce poso us 50 colloco pos.

6 6 M. E-R, A., C. ou, O. Seo-Gullume Refeeces [] Oldoyd J.G. : A ol fomulo of he equos of plsc flo fo Bhm sold. Poc. Cmb. Phlos. Soc. 43, 947. [] Alkhb M.A.M, Wlso S.D.R : The developme of Poseulle flo of yeld-sess flud J. o-eo.flud.mech,00, pp.-8, 00 [3] Bed D., ou C.: Tse Couee flos of Oldoyd's vscoelsc fluds ude mposed oques, Joul of o eo Flud Mechcs 77, 998 [4] Bevely C.R, Te R.I: umecl lyss of hee-dmesol Bhm plsc flo. J. o-eo Flud Mech, 4, 99 [5] Lpscomb G.C, De M.M : Flo of Bhm fluds comple eomees. J. o- eo.flud.mech,4,. pp , 984 [6] Pu J.M: Flo of yeld-sess flud lo dom. Applco o flo o cled ple. J.Rheol., 40, (4), pp. 7-73, 996 [7] Wlso S.D.R : The developme of Poseulle flo. J. Flud Mech. 38,969 [8] Ks E.J : Mulqudcs- A sceed d ppomo scheme h pplcos o compuol dymcs. II Soluos o pbolc, hypebolc, d ellpc pl dffeel equos. Compu. Mh. Appl, 9, (6-8), pp [9] Cho Y.C,, Cheu K.F, Mo X.Z, Ks E.J: A mulqudc soluo fo he shllo e equos. Joul of Hydoloy, 5, (5), pp [0] Fsshue G.E : Solv pl dffeel equos by colloco h dl bss fucos. I Sufce F d Mulesoluo Mehods, LeMéhué A.,Rbu C, Shumke L, eds Vdebl Uvesy Pess, 997

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