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Nombre de Froude, & , exprims en h6k" v*  :  6gv> 9Exemples simples# Exemples moins simples4 ( () Principes (le retour)$   En crivant les modles : I Supposer que le petit paramtre e , associ aux htrognits ou oscillations des proprits, est infiniment petit ( trs grand nombre d htrogneits, d oscillations des proprits, d obstacles, & ) d!$   O X,9 m5-  Principes$  I  Trouver la limite II  Etudier la limite (en un sens adquat ! ) lorsque du modle (quation) ~{    /  X6  Problmes& (I  Quelle limite? 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H p 0޽h ? ̙33___PPT10i.`fʭ+D=' = @B +  0 rj(     08,S" wTt<  , ~  s *,<Y ,  0 TA  ? ?0    ,f  s *A ? ?   f  s *A ? ? z &F  f   s *A ? ? z (   0 TA L? ?~j  L ,H  0޽h ? ̙3380___PPT10.`fʭr M6<VלMY.E( H r vEquation Equation.30,Microsoft Equation 3.0  Equation Equation.DSMT40*MathType 4.0 Equation" E ՜.+,0t    Affichage l'cranmcs]MG Times New RomanFrench Script MTSymbolModle par dfautMicrosoft Equation 3.0MathType 4.0 EquationWMODELISATION MATHEMATIQUE - CHANGEMENT DECHELLE Homognisation Prsentation GnraleMathematicians are like Frenchmen: whatever you say to them, they translate into their own language and forthwith it is something entirely different. Dfinition Principes PrincipesExemples simplesExemples moins simplesPrincipes (le retour) Principes ProblmesDiapositive 11Diapositive 12Diapositive 13Diapositive 14Diverses techniquesAPPLICATIONS CLASSIQUESAPPLICATIONS CLASSIQUESCONCLUSIONS : + / - Polices utilisesModle de conceptionServeurs OLE incorporsTitres des diapositives _]foglianiatAlain Bourgeat2$+{z*Jű 2$Ҷ +:/W2w6 2$OGfK62$ܔ$7_ͻo_K8g!p]`Y2$݂S1qc2$_!V2$1L"Po F 0AA@3@ʚ;ʚ;g46d6d 08ppp@ <4!d!d w 0T0~<4dddd w 0T0~ <4BdBd x 0T80___PPT10 pp?$-27/10/24O  =MODELISATION MATHEMATIQUE - CHANGEMENT D ECHELLE Homognisation Prsentation Gnrale0W03&6 (Alain Bourgeat)Z(UMathematicians are like Frenchmen: whatever you say to them, they translate into their own language and forthwith it is something entirely different.6V     ' J. W. Goethe$(( ( Dfinition 4L'homognisation consiste gommer les htrognits inhrentes aux microstructures afin d'obtenir une loi de comportement plus simple du milieu, asymptotiquement quivalente, dite homognise. On ne dispose pas en gnral de formule explicite de la loi homognise sauf dans des cas trs particuliers . Principes (lA partir d une modlisation mathmatique adimensionnalise , crite l chelle microscopique Hypothses et domaine de validit clairement dfinis Met en vidence les importances relatives des phnomnes (par exemple : Diffusion /Dispersion /Convection/ Dcroissance radioactive/& ) Permet de considrer conjointement les grandeurs physiques importantes (diffrents Flux, Energie, & ) un choix du temps caractristique interessant BbZZZbP >*  Principes ( Mettre en vidence le petit paramtre e associ aux htrogneits ou oscillations des proprits Taille relative de la priode spatiale ou temporelle Mesurer toutes les rapports: de taille, de puissance des phnomnes, en fonction de ce petit paramtre e Taille des grains, porosit, permabilit, & Nombre de Peclet, Nombre de Reynolds, Nombre de Froude, & , exprims en h6k" v*  :  6gv> 9Exemples simples# Exemples moins simples4 ( () Principes (le retour)$   En crivant les modles : I Supposer que le petit paramtre e , associ aux htrognits ou oscillations des proprits, est infiniment petit ( trs grand nombre d htrogneits, d oscillations des proprits, d obstacles, & ) d!$   O X,9 m5-  Principes$  I  Trouver la limite II  Etudier la limite (en un sens adquat ! ) lorsque du modle (quation) ~{    /  X6  Problmes& (I  Quelle limite? En quel sens? ex: II  Comment obtenir cette limite? pV    : Diverses techniques<APPLICATIONS CLASSIQUES(Validation des hypothses sous jacentes, domaine de validit; moyens de calculer exactement; Micro  Mso Navier-Stokes Darcy, Brinkman,Forcheimer,& Navier-Stokes +Elasticit Biot Ecoulements non newtoniens Lois de filtration ad-hoc Navier-Stokes ou Diffusion tailles critiques des msopores ou fractures Diffusion diffusion dispersion RjX\   Xj     / ;  5  VAPPLICATIONS CLASSIQUES(AValidation des hypothses sous jacentes, domaine de validit; moyens de calculer exactement; Mso - Macro Transport, diffusion lois globales de milieux fissurs (vs. Taille des fractures) Transport, diffusion modles champ lointain de site de stockage (avec possible endommagement) Ecoulements multiphasiques lois doubles porosit de milieux fissurs Ecoulements multiphasiques Lois macroscopiques de comportements Pc, Kr; K Navier-Stokes lois de contact fluide libre- milieu poreux, Fl\ PE I L ?<  0 SK d(  d d  0[S" Tt  [  d0 NA ? ?s 9   [ d0 NA ? ?9 j  [  d 0,   \,Contenant ventuellement des couches limites- 2-  d 0[  u  ]II - Dveloppement:. 2  (  d 0#[S  DI - Notion de limite faible, de limite double chelles, & (en considrant l Energie) La limite de l Energie donne le comportement grande chelle (ou global)4 2 4 ! 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