Numerical simulation of elastic membrane - fluid interaction in a closed reservoir

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Elisabeth Doyle
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1 mrca smaon of asc mmbran - fd nracon n a cosd rsror Yda ra Mc Brcor * Jn 8 3 bsrac In [5] was nrodcd probm of 3D asc mmbran-fd nracon n a cosd rsror. In s wor w sod nmrca probm mnond sng a smpr asc mod n D. To so ransn probm w nwonan-scos fd wo prmnar sps wr an. Frs song sad sa probm s probm s an ordnar dffrna qaon a was sod anaca aso. T scond sp was o so ransn probm w an da fd. Snc fd s da srs on as a om consraor. Wn song probm w a nwonan-scos fd fd srs no on as a om consraor b aso carr forcs on mmbran. robm dfnon Foowng [5] and rdcng 3D mod o a D mod w g a rcang wos ongr dg s para o -as and s ng s L mrs ng of sorr dg s H mr. T ppr dg para o -as s a fb mmbran Fg. Snc mod s D ns masrng qan of fd ar ara m rar an om m 3. Wn fng bo w an ncomprssb fd of ara > L*H bcas mmbran s dsndd. * Hbrw nrs Scoo of Compr Scnc and Eng. Safra Camps Jrsam 994 Isra arabrco@cs..ac.

2 Γ mmbran Ω Γ rgd Fgr Framwor of mod dscrbng sod-qd nracon. bo w 3 rgd facs boom f and rg and an asc mmbran op. D s dspacmn D s dspacmn a -as D s dspacmn a - rgd mmbran ar compaona doman bondar. Ts wor w da w foowng probms: robm - Sad-sa. Gn proprs of mmbran and ara of fd fnd sadsa sap of mmbran wn s no sbc o rna forcs. robm -. Gn f a dscrbs forcs appd o mmbran a. Ida fd fnd sap of fb fac as a fncon of m ndr condon of da fd. b. scos fd - fnd oc fd and sap of fb fac as a fncon of m ndr condon of scos fd. Soon scms Easc mod Wn assmng an soropc mmbran a s wn dspacmn dos no dpnd on drcon of oad ar asc qaons ma b appd. D D ρ w G Φ.

3 3 D E D ρ w Φ. Wr D and D sand for dspacmns n and drcons rspc for mmbran wa cnss for sar corrcon facor G for sar mods E for Yong mods for osson rao wc s qa o ½ for ncomprssb mara ρ for msc wa mass and Φ [Φ Φ ] T for forc rm wc s d o rna forcs ncdng srss ndcd b fd. For smpc s amn baor of s mod ndr condon wr orzona dspacmn s mc smar an rca dspacmn. In s cas qaon. can b ngcd and qaon. rdcs o: p w η E Φ η wr Φ Φ Eqaon s an ordnar dffrna qaon of scond ordr and s nown as Indpndn Rngs Mod [ ]. Tr s owr a mod of nrmda comp bwn and wc as no accon nson of mmbran: η η β ση f 3 wr η D and wr D s ngcd Wn β mod rdcs o ndpndn rngs mod. Ts wor w s qaon 3 as asc mod.

4 4 robm sad-sa ndr sad-sa mmbran dspacmn η dpnds on on spaa coordnas and rfor rdcs o η and on forc mposd on mmbran. sad-sa probm s dfnd b: d η β ση f 4 Wr qaon 4 dscrbs mmbran dspacmn η η ; f for rna forcs and for Lagrang mpr for consran 5. η 5 wr qaon 5 accons for ara consraon -L*H s na ara. T bondar condons ar η η. Snc r ar no rna forcs f s on nrna forc a s drosac prssr rd b fd on mmbran. s consan or fow wod b ndcd bcas of prssr dffrnc cangng mmbran dspacmn wc oas sad-sa assmpon. Moror snc prssr srs as a Lagrang mpr for qaon 5 ms b a scaar. W w now so qaon 4 anaca and n prsn a nmrca soon. T soon of a scond ordr ordnar nar dffrna d d a a b 6 nra I s gn b [3] r r r r ψ c c [ ] b d 7 r r Wr r and r ar roos of caracrsc ponoma of qaon 5 c and c ar consans and I.

5 5 Transformng qaon 5 o form of qaon 6 wod d d η σ η β f 8 β β σ fr sng and b snc r ar no rna forcs. Hnc w oban β β d η η b 9 Eqaon 9 caracrsc ponom s λ nd s roos ar r / ± ow sbsng n qaon 7 c c [ ] b Cacang ngra n ds d c c b 3 Snc b s consan. Sng and sbsng ngraon bondars w oban b c c 4 Rarrangng mns ds b b ψ c c 5 Sng bondar condons ψ ψ ds b b ψ c c c c 6.

6 6 b b c c ψ 6. Song qaons 6 ds b c 7. c c 7. ψ 8 b b b b ψ 9 b b and from 8 w oban raon bwn prssr and na ara

7 7 b b 3 β 4 Trfor wn s gn can b cacad sng 4 and mmbran dspacmn s cacad sng 5. To so qaons 4 and 5 nmrca w nd o posa a raon bwn and r dspacmn or prssr. Howr w a a mor sragforward approac. W frs caca ara ndr dspacd mmbran. snc ara ndr mmbran s proporona o prssr a bnar sarc w prssr as paramr s prformd n rqsd ara s racd. Ts probm rprsnd b qaons 4 and 5 rdcs o qaon 4 abo: gn a prssr fnd corrspondng ara ndr dspacd mmbran.

8 8 ppromang soon w Fn mn mod w rqr ransformng 4 o wa form. Frs mpng w s fncon sc a wr s } : { H and } : { L L H and } : { < L f d η σ η β 5 Wr f f ν ν. Tn dscrzng spac and appromang η w bass fncon η f d σ β 6 f d σ β 7 nd sbsng bass fncons.. n ransforms 7 o f d σ β.. 8 B ngrang b pars qaon 8 and snc f d d σ β 9 nd n mar form: F B σ β 3

9 9 Wr f F B d d Coosng s fncons as nar fncons: > < < < < < 3 Sng - for ± and B ar: d d f <- or > d d d d d d B f <- or >

10 So > ± < and > ± < 3 6 B nd sad-sa soon s: F B σ β 3 W qaons 4 5 w now a fna agorm for cacang mmbran dspacmn. gorm.. S ow na gss prssr an nmbr grar an g Mc Largr nmbr an.. s. 3. W c > Toranc a. Caca dspacmn of mmbran dspacmn sng nmrca scm w F and b. S c c. If c > Toranc. S ow /. g d. If c < -Toranc. S g /. ow.

11 o T np o gorm s na ara and oranc oranc wc drmns rror of agorm robm.a - da fd n da fd as no frcon; rfor srs on as ara consrar. T qaons a conro mmbran sap ar: f ση η β η 33 η 34 Wr η η s mmbran dspacmn s na ara β and σ ar proprs of mmbran f s rna and s Lagrang mpr for consran 34. Transformng o wa form b mpng w a s fncon ngrang and sng η wr ds: f η σ η β η 35 f σ β 36

12 f σ β 37 Rpacng w bass fncon.. ds β.. 38 f σ nd afr ngraon b pars w oban f σ β 39 Dscrzng m and sng mpc forward-bacward scm o so scond dra ds F B B σ β δ 4 wr f f F wr s m a m sp. Lmpng mass mar wod g a good appromaon and wod smpf cacaons. So qaons ar: I I F I B δ σ δ β δ 4

13 3 Rorganzng qaon ds I I F I δ δ σ β δ 4 sng: I C σ β δ ds I I F C δ δ 43 Snc n 6 bo and ar nnowns an addona qaon mass consraon qaon qaon 5 s aso mpod. ow raon bwn wo ms b formad. Rarrangng mns n rg wng of qaon 6 and nrodcng wo nw arabs rss n ITERL rna I I f C δ δ 44 Trfor ITERL rna. 45 Snc prssr srs as a Lagrang mpr for qaon 5 s a scaar w consan a a ac nod or a consan cor α.

14 4 Sng and o b ITERL rna rna C δ f I I 46 ITERL α δ C 47 Sng n rna δ C and from 46 and 47 : ITERL α n rna. 48 from 45 α rna n rna 49 and rna n rna α 5 T nnown α can b cacad from qaon 5 rna α 5 n rna

15 5 W qaons 4 5 w now a fna agorm for cacang mmbran dspacmn. gorm :. S n rna δ C. For ac m sp. a. Caca Forc sng 43. b. Fnd α sng 5. c. Caca sng 49. o gorm rcs as np wo conformaons of mmbran wr s mmbran conformaon a frs m sp and s mmbran conformaon a scond m sp. To so probm.a s sad-sa soon.

16 6 robm.b - scos fd Snc scos fd as frcon w mmbran fd srs no on as a ara consraor b aso carr forc on mmbran and s nrna forcs ar a combnaon of prssr and srss forc. T srss nsor of nwonan fd s drd from oc fd. Snc fd sc o mmbran s momn cangs oc fd. T qaons a gorn fow of an ncomprssb scos fd ar ar-sos qaons: ρ p 5 53 Wr p s prssr and s oc cor s dnamc scos and ρ s dns. W wo bondar condons: η for mmbran Γ s Fgr 54 for Γsod s Fgr 55 Wrη s mmbran dspacmn and s as. Song probm s don b spng probm no wo sb probms: Sb probm : Fnd mmbran dspacmn gn rna and nrna forcs. Sb probm : Fnd oc fd a a gn mmbran dspacmn.

17 7 Sb probm Song sb probm s smar o song qaon 4 and 5 probm a w cpon a now nrna forc on mmbran s racon n T σ 56 Wr σ s sss-nsor. S pi σ. I s dn mar pi s prssr nsor and S s ra-srss nsor. wonan fd S s nar w sran-nsor a s T d S. Hr w a rna forcs o b as n probm a a gn fncon of m and spac. Sb probm Snc probm s D n sng and dscrzng ar-sos wod prod p p ρ ρ 57 Sng and p p ψ and mpng w s fncons g D wr } : { Ω H D and } : { D D D L L L H wr } : { < Ω L D and ngrang on doman Ω ds

18 8 g g g g g g g g g g M ψ ρ ψ ρ 58 Sng g... and ψ.. ds ψ ψ ψ ρ ψ ρ 59

19 9 Ingrang b pars 43 ads o 44 bcas s fncons qa on mmbran Γ sod Γ ψ ψ ψ ρ ψ ρ 6 T mar noaon of 6 wod b: C C C K K M C K K M 6 Wr Ω Ω Ω Ω Ω Ω Ω d K d K d d C d d C d M ρ ψ ρ ψ ρ

20 Rcocng arabs w oban C C T C K K T C K K M M 6 ac m sp dspacmn of mmbran s cacad sng forcs mposd on drng prcdng m sp and n ocs ar cacad sng 63 Fg. Fgr Cacang ocs of nods on mmbran T oc on nod n s n n δ 63 Snc asc probm s mor sns an fow probm asc probm s sod w smar m sps an fow probm pdang srss nsor on onc n K m sps.

21 W qaons w now a fna agorm for cacang mmbran dspacmn. gorm 3: S sad-sa soon.. S srss forcs F.. s mporar 3. For ac m sp. a. For o K. Caca sng gorm w mporar as mmbran poson paramrs srss forcs F and rna forcs s nos. b. S K mporar c. Caca nod ocs w 63 sng K and d. So fd probm w bondar condons sng 6 sbc o bondar condons 54 and 55 w ocs cacad a b.. Caca srss forcs and pda F. os T m sp arab sars a. W assm a rna forc sasfs condon f and a s connos. Snc racon forcs ar fncons of oc and a sad-sa oc s n srss forcs a ar zro. frs m sp forc mposd b fd s on drosac. Ts s cacad from sad-sa condons.

22 Rss probms wr sod nmrca sng FID rson 8.6. for nmrca soon of fd probm and MTLB R for asc probm. T ns ssm s KGS. T doman dmnsons ar L mr wd and H. mr g. T dnamc T mn wd s.m. T asc m sp s. sc and fow m sp s.sc. robm sad-sa Fgr 3 sows soons of sad-sa probm of mmbrans w dffrn proprs as was cacad b agorm w anaca soon of qaon 5. T prssr was adsd o manan an ara of m. Mmbran proprs rssr a rqrd mananng m a consan ara of m. β σ B β σ Rd. β σ Grn Tab T prssr rqrd o manan an ara of m ndr mmbran w aros proprs.

23 3 Dspacmn m -as m Fgr 3 Smaon rss for probm wn a consan ara of m was manand ndr mmbran w aros proprs as n Tab. Tab and Fgr 7 sow a β gras nfnc s n drmnng forc a as o b appd on mmbran w σ drmns mmbrans sap.

24 4 robm da and scos fds Fgrs 4-7 sow ncs of mmbran dspacmn a r dffrn ss afr mposng rna prssr f n. a gn b qaon m f 48 s ac s mmbran was fd w fd w dffrn scos:. Ts T fd as no scos da fd.. Ts - T fd as scos of Sc. m 3. Ts 3 - T fd as scos of Sc. m T forcs s nfcd on mmbran a sad-sa dspacmn afr bng dsndd b.m of fd. Dspacmn -as m Fgr 4 Mmbran sap a sc for ss b rd and 3 grn.

25 5 Dspacmn -as m Fgr 5 Mmbrans sap a sc for ss and 3 as n fgr Dspacmn -as m Fgr 6 Mmbrans sap a 3sc for ss and 3 as n fgr

26 6 Dspacmn -as m Fgr 7 Mmbran sap a m 4 for ss and 3 as n fgr Hg scos dsspas rna forc rd on fd and rfor a g scos mmbran dspacmn s smar Fgrs 4-7. Fgrs8-7 sow oc fd and sram ns ndcd b mmbran dformaon a dffrn m pons n s.

27 7 Fgr 8 oc fd a sc

28 8 Fgr 9 sram ns asc

29 9 Fgr cor fd a.5 sc

30 3 Fgr sram ns a.5sc

31 3 Fgr oc fd a 3sc

32 3 Fgr 3 Sram ns a 3sc

33 33 Fgr 4 oc fd a 4sc

34 34 Fgr 5 Sram ns a 4sc

35 35 Fgr 6 oc fd a 5sc

36 36 Fgr 7 Sram ns a 5sc

37 37 Fr wor Ts wor prsnd a nmrca mod o so asc mmbran fd nracon n a cosd rsror. T n ncssar sp s o dop sab and conrgnc anass of s mod. n nrsng nson o crrn mod w b sng a mor compcad asc or scos mods. Rfrncs Hbr D n ffcn ar Sos sor and s appcaons o fd fow n asc bs Cooqa Socas Janos Boa 5: rod K Rappsc G Mamaca modng of oca arra fow and ss mcancs. In Cro J. Oaon R. ds. Compaona Mods for Fd Srcr Inracon. man Rsarc os n Mamacs o. 36 pp Harow: Longman Coddngon E n nrodcon o ordnar dffrna qaons Engwood Cffs.J. rnrc-a Inc an-tn Low T H On Fow of a non-wonan qd ndcd b nsn conracons orna of bocmca ngnrng o / Dsardns B Esban MJ Grandmon C L Tac Wa soons for a fd-asc srcr nracon mod. Rsa Mamáca.C.M. 4 p Dsardns B Esban MJ On wa soons for fd-rgd srcr nracon: comprssb and ncomprssb mods Comm..D.E p

OUTLINE FOR Chapter 2-2. Basic Laws

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