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Multiscale modelling of complex fluids: A mathematical initiation.

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(1)Multiscale modelling of complex fluids: A mathematical initiation. Claude Le Bris, Tony Lelièvre. To cite this version: Claude Le Bris, Tony Lelièvre. Multiscale modelling of complex fluids: A mathematical initiation.. [Research Report] RR-6275, INRIA. 2007. �inria-00165171v3�. HAL Id: inria-00165171 https://hal.inria.fr/inria-00165171v3 Submitted on 31 Aug 2007. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Multiscale modelling of complex fluids: A mathematical initiation. C. Le Bris — T. Lelièvre. N° 6275 Août 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6275--FR+ENG. Thème NUM.

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(126) MOh[t[N =zL— =z  — 3 \^ZŽZsho‘`[\^c‘ho_kMmÏXœxNN•[qL‘cfl c‘\d’X =zL—   

(127) N \d\dc’_L‚ ‚Fh9\dh šrmj_utmFZ^Zs`uM9c’_L‚ œrho`L_ut[N;t›IuN_ c’ZJœxhF`L_rt[NGt§šu›N(hoœ[\^mFc‘_ H =zL—  α. . λ. . . ∂τ p + u · ∇τ p − ∇uτ p − τ p (∇u)T ∂t. . ξ ˙ p + τ p γ) ˙ = ηp γ, ˙ + Z(tr(τ p ))τ p + λ (γτ 2. Z. ξ. . φ.

(128)

(129)

(130) 1 + φλ tr(τ p )

(131)  ηp  , Z(tr(τ p )) =

(132)

(133)

(134) exp φλ tr(τ p )

(135) ηp. . . .  .    ∂τ p  T  + u · ∇τ p − ∇uτ p − τ p (∇u) + Z(tr(τ p ))τ p  λ ∂t      ∂ ηp   ˙ + u · ∇ ln (Z(tr(τ p ))) = ηp γ, −λ τ p +  λ ∂t d Z(tr(τ p )) = 1 + b. d. . tr(τ p ) 1+λ d ηp b. λi. . −∞. τ. τ (t) =. Z. t. −∞. ,. ηp,i. t − t0 τ (t) = τ (t0 ) exp − λ. t0. . . +. Z. t. t0.   t−s η exp − γ(s) ˙ ds. λ λ. γ˙  η t−s exp − γ(s) ˙ ds. λ λ .

(136) ;y 5TN_uhj\dc’_L‚OœYX H. "  '

(137)      "$ A "  . . Dz[— z.  d   γ(t0 , t) = γ(t) ˙ dt ,   γ(t0 , t0 ) = 0.  . mF_utc‘_B\dNG‚Fb,mŠ\^c‘_L‚9œYX qumFbs\,ZšY›N(hFœ[\,mjc’_m“”¦hobdM N;eB`Lc‘˜ŠmF‘NG_o\\^h = z[— 6H. =zL— IuHJc’IuZs\dc’HJZ

(138) hobd”¦ILhFXN-b^hFM ho”_LN \^N IL•[©³t[NqLc’’M9t[c lN c´NG”¦\^_uho‘X(bdZdMc‘hoZdIL_umjhŠ\dmj›*c’ªhFl Z§_£ho\^—@M9IuHJmŠquIL\;`[šjNŽ\^mFmj”¦Z.`L\dmjc’_uhF_Ll _ _L\dc’hoqxhF`LN_ _ubd”¦lhFNGb^t(MON;NGmjtŸb^‘mFc’Nœrb;hŠš˜o\^NTILNJlGc’mjl Z_ hFhF_rœx”âZ]\dN-\^NGb^_"‚FmFNGc‘lG_B_Lmj\N’b,mj‘N;mj\

(139) t’c\d;Gc’mMONGtO!N \^"h“ tLIuN$c‘qx‚oN+0IL_uN1#tLb Z. t[ho c’_(MO! $\dNIL ._[N —© Zdc’hF_uZmF_ut XYc’NftLZ/H 3]t =zL— 2  \^› IL" Iu1N N$#b^ZdN hj©³ #lmF!" ‘$#’cfNG Zt m"! " MO ' N$ MO " HhF b^Xn! " ”¦`u' _u$#l\^c‘=homŠ_£\ š \^c‘MOcfZN m"‚F— c’˜FHJNG_ ILN7b^NG’mFmj4\s\d˜ŠNGmjb ’`LcfZŸNGt \^IL”¦`LN7_rc’l_Y\d˜Fc’hFNG_

(140) b^Zdš¢N"mF\d_uNGt _uZshobŸhj”(\^ILN t[NG _L hj \^ NG+Z Dz[— '|  ›IuNb^N =zL— y  cfmjZ

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