An immersed boundary method for mass transfer across permeable moving interfaces

Transcription

1 A immere boary meho for ma rafer aro permeable movig ierfae Haiog Hag eparme of Mahemai & Saii York Uiveriy, Toroo, Caaa hp:// oi work wih Xiaobo Gog a Zhaoi Gog Shaghai iao Tog Uiveriy Uiveriy of Pari VI ly 3, 05

2 Bakgro Srfae wih Veile ize Time ale m µ p f Meo ale Coare Grai Aomii Qam Mehai Å 00Å µm 0.mm Fli Mehai Mielle lifeime Membrae fio Coformaioal hage Molelar permeaio Legh ale Time a legh ale i rfaa olio (from Shelley e al., Crr. Opii. i Coll. & I. Si., 000) S aple Cora age RBC! bloo flow i apillary veel. Fli moai moel of bio-membrae hyrophili Time: ~ m Spae: ~ µ m hyrophobi

3 Moivaio For may problem ma rafer ro a boary of wo regio i give by he followig form: jf[g()] whe he boary alo evolve; A eample: io a waer flow ro ell membrae, er ormal a pahologial oiio h preaig epolarizaio; ivolve a lo of hemoai wih io a waer goig hrogh ell membrae i large qaiie. AMPA/ kaiae ormal ero i healhy brai I i I o a Ca a K SK 70 mv Ca I i H I O o 0 mv Ca Ca a R Swolle ero rig preaig epolarizaio Iffiie oim pmp Ca Ca a ECV 0% ECV 5% MAR opeifi aio hael reier, aral Meiie, 0 K a Ca K a Ca i k Pe ( k ) f k, f k k ( k z k k φ ), (β φ ) z k k, k,,p,φ (( k e k,,p,φ X,f k,f w, F ela k ( ) γ p z k k φ, 0, k ) ) (Pe X f k ) α(j k i,e a k ), (β φ ) i,e θc m [φ ], X Pe j w a w, [( Σ m (,p )β Σ e(φ ) ) ] F mem βθf ap. Mori e al, Phyia, 0

4 Objeive of hi alk ma rafer aro bio membrae i he miroirlaio yem a elllar level hp:// Ø Iveigae he effe of mehaial properie of eryhroye o oyge elivery for yig he mehaim of ieae relae o meaboli aiviie; Ø Sy he effe of he phyial & geomeri properie of lipoome o rg elivery effiiey.

5 Fli-rre ieraio Immere Boary Meho: Peki(977), Tryggvao e al.(99) Goverig Eqaio: 0, Saioary Gri Movig Ierfae Marker ρ Lagragia Trakig U ( T ) p µ f Fli Fli Membrae-Srre Ieraio f ( ΔF ) δ ( ) S G A ΔF :Sre mp δ ( ): G Smoohe ela Fio Veloiy Ierpolaio G V U δ ( )

6 Moel for ell membrae: hi plae Membrae re eor: T τ q ( Pozrikii, 00 ) I-plae hyper-elai re Beig re Sre jmp ro membrae [ ] ( S σ I ) ( τ q) ΔF σ : Hyroyami re eor S I I: Srfae projeor

7 Ma rapor Ma rafer aro permeable movig ierfae Sefa problem: i o H H Hery oeffiie merial meho: phae-fiel bae, aqmi, 999, CP 55:96-7 level-e bae, Gibo e al., 007, CP : ehalpy wih VOF, Sao e al., 03, CP 49:7-6 Rerie iffio: k[] k ma rafer oeffiie Referee: Immere Boary Meho bae pre iffio, Hag e al., 009, CP 8:537-53

8 Previo rel (Hag e al, 009) k [] Trapor eqaio for [ ] ( ) O he ierfae fl or Sigle rapor eqaio o Fl eqaio for Sigle fl eqaio o or k[] # k δ( & % ) ( $ ' [ ] k[ ] k # δ( ) & % ( ( ). $ k '

9 Previo rel (Hag e al, 009) k [] Trapor eqaio for [ ] ( ) O he ierfae fl or C k[] # k δ( ) & % ( $ ' # δ( ) & % ( ( ). $ k ' X

10 A more geeral problem (Gog e al, 04) Objeive: k [ ] Trapor eqaio for ( ) S evelopig a immere boary formlaio o hale rerie iffio where oeraio jmp omiae ma rafer over a eformig & movig ierfae. [ ] O he ierfae or iffive fl k[]

11 Mai iea. Repreeig a ioio fio ig he iiaor fio f f H f ( H ) ( f f )H f where H 0,,. Uig he properie of he iiaor fio arrie by a fli H H H 0 δ ( ) H H 3. Mahig p he highe orer of iglariie H 0 δ 0

12 erivaio (I) Bai eqaio for eah omai wih oio iribio of oeraio S ( ) efie H H where [] H [] H where 0, H, [] Coeraio jmp Time iffereial eqaio for oeraio of eire omai H ( H)

13 erivaio (II) Coeraio eqaio wih oeraio jmp S ( ) [ ] δ ( ) Boary oiio over he ierfae k[] k ( ) δ ( ) S where iffive fl Bil a iffive fl eqaio o avoi allaig he oeraio jmp

14 erivaio (III) efiiio: or Bil iffive fl eqaio bae o oeraio raporaio S S ) ( H H efiiio: H H H ) (

15 erivaio (IV) Fl eqaio wih oeraio jmp a ierfae S l k l k k δ δ δ - i veor of ageial ireio ) ( H S H S S reaio ore iie/oie ierfae

16 Oyge loaig over re ell H k a a O O T 50 γ χ δ γ γ!!! a HbO γ χ γ!! H a l ak l ak ak a O O O O O T 50 γ χ δ δ δ γ!! Oy-hemoglobi Oyge iffive fl a rai of ell wall hear rae of he flow γ! χ ioiaio rae oa of oyhemoglobi (Clark, 983)

17 merial meho: a pliig heme Ø Compe oveio & iffio of oyge O O k a a Δ Δ Δ δ γ γ!! * 3 3 Δ *** 3 *** *** ** a HbO Δ γ! Ø VOF approah for reaio kiei (oyhemoglobi iie re ell) m m m m m T Δ 50 γ χ τ! Ø Compe araio followig Hill eqaio wih a peo ime m m m m m Δ γ χ τ! give * & * H H H ** ** *

18 Eplii heme for iffive fl Δ S l k l k k δ δ δ -

19 Rel: igle ell

20 Rel: mliple ell

21 Oyge loaig rae Ø hemaori effe Ø ofigraio effe Ø iffe effe

22 iffive fl o veel wall iffive fl ormal o he veel wall er wo re bloo ell ofigraio: (a). ymmeri plae re bloo ell; a (b) aymmerially plae re bloo ell. The hemaori eqal 0.48.

23 Oyge loaig effiiey v. flow voriiy C

24 Colig remark! We have preee a ovel merial meho for he oveio-iffio of ma rafer aro mliple eformable movig ierfae er he framework of he immere boary meho! The merial rel how ha he geomerial a mehaial propery of re bloo ell affe ma rafer effiiey hrogh he hyroyami haraerii of he flow aoiae wih he membrae eformaio, h a he voriiy of he flow fiel O-goig a fre work! Eeio o membrae ha are permeable o fli;! Eeio o ioi peie i biologial ie;! Compario wih oher meho, e.g., phae-fiel.

25 Thak yo for yor ki aeio! Mai referee Gog e al. (04),. Comp. Phy., 78: Oher referee: Hag e al. (009),. Comp. Phy., 8(5): ; Gog e al. (009),. Biomeh. Eg., 3: