DENSITY DISTRIBUTION FOR THE MOLECULES OF A LIQUID IN A SEMI- INFINITE SPACE

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1 DENSITY DISTRIUTION FOR THE MOLECULES OF LIQUID IN SEMI- INFINITE SPCE VINCENZO MOLINRI lm Me Sudioum-Uivesià di olog Moecuccolio Lbooy of Nucle Egieeig vi dei Colli 6 6 olog (ITLY) vicezo.molii@fswebe.i RRY D. GNPOL Uivesiy of izo Depme of eospce d Mechicl Egieeig Tucso z. 857 (US) Gpol@cowboy.me.izo.edu DOMIZINO MOSTCCI * lm Me Sudioum-Uivesià di olog Moecuccolio Lbooy of Nucle Egieeig vi dei Colli 6 6 olog (ITLY) domizio.moscci@uibo.i The Suheld ppoximio o he v de Wls foces is pplied o he deivio of selfcosise Vlsov-ype field i liquid fillig hlf spce bodeig vcuum. The esuig Vlsov equio is he deived d solved o pedic he behvio of he desiy d i he viciiy of he liquid-vcuum iefce. umeicl soluio o he Vlsov equio is lso poduced d he desiy pofile show d discussed. Keywods: Suheld poeil; Vlsov self-cosise foce field; liquid-vcuum iefce; liquid desiy e discoiuiy.. Ioducio The viio of he umbe desiy i liquid s cosequece of boudy discoiuiy is ofe disegded i my pplicios. Neveheless his subec is of gowig iees i my fields such s i he sudy of hi films e.g. i he peoleum idusy o i biology whee ivesigios e becomig moe d moe deiled wih decesig dimesios see e.g. efs.-; lso efs.-6. The im of ou pese wo is o obi esime of he umbe desiy disibuio i liquid s fucio of disce fom discoiuiy whe he liquid is es d i hemodymic equilibium i.e. * coespodig uho - e-mil: domizio.moscci@uibo.i; el ; fx:

2 V Molii D Gpol D Moscci he velociy disibuio fucio is Mxwelli wih vege velociy zeo. The seig cosideed will be liquid bodeig vcuum. The poblem of liquid i coc wih is vpou hs bee eed exhusively i ef. 7. The pese ivesigio is coduced i he fmewo of ieic heoy imig o obi specified liquid popeies s bough bou by discoiuiies. Whe delig wih liquid phse he effecs of iecio bewee molecules c eve be disegded; hee esime of his effec is obied esoig o self-cosise Vlsov field. The self-cosise field is ppopie o descibig he dymic behviou of sysem whee evey molecule iecs simuleously wih lge umbe of suoudig molecules d he coelio bewee pis of molecules c be disegded sice he effec of he le becomes egligible wih espec o he collecive iecio. This physicl siuio occus whe he ge of he iemolecul foce is lge i compiso o he vege disce bewee molecules s is he cse of plsm o liquid ses. Fom his poi of view iecios i liquid sysem e eed somewh lie is doe i plsm physics whe hee e eough picles i he so-clled Debye sphee d simuleous muliple collisios pevil. To clcule he Vlsov self-cosise field he iemolecul iecio fom is eeded: sevel pheomeologicl poeil models hve bee poposed o descibe he el muliple foce iecio mechism. I his wo he Suheld pi poeil model is used becuse i is mhemiclly simple d he pedicios o he liquid popeies e expeced o be quliively cosise wih he behviou of my el fluids.. Goveig Equios The fis equio of he of GKY hiechy descibig he micoscopic empol behviou of he picles of he sysem i ems of posiio d velociy is obied fom he Liouville equio 8 s f f F f F f v ddv m v () m v whee f ( v ) deoes he sigle picle (o simple) disibuio fucio f ( v v ) is he wo picle (o double) disibuio fucio F deoes he exel foce o picle of mss m loced ( ) d F is he ie-picle foce bewee wo picles loced posiios d. Sice he collisio iegl o he igh hd side depeds o f he coelio i posiio d velociy bewee collidig picles is explicily cosideed. This equio is vlid up o he ode ssocied wih he model used fo he iemolecul poeil s descibed below. Coside sysem i sedy hemodymic equilibium d whose picles e o subec o exel foce so h f d F. I ddiio ssume h he ie-picle foce F is give by cel poeil idepede of velociy. I his cse ssumig molecul chos he double disibuio fucio f is give by f ( v v ) fm ( v ) fm ( v ) ()

3 f M whee ( v) deoes Mxwelli disibuio d ( ) umbe desiy Obsevig h ( ) f dvdv Desiy Disibuio i Semi-Ifiie Spce deoes he wo-picle. () f v M m v K T f M ( v ) () whee K is he olzm cos d T is he (uifom) empeue Eq. () becomes K whee ( ) is he sigle picle desiy T F d (5) ( ) d. (6) Whe coelio bewee pis of picles c be disegded he wo-picle umbe desiy simplifies sigificly:. (7) The self-cosise field is impo o descibe he dymic behviou of sysem whee evey molecule iecs simuleously wih lge umbe of suoudig molecules d he coelio bewee pis of molecules c be disegded sice is effec becomes egligible comped o he collecive iecio. This physicl siuio occus whe he ge of he iemolecul foce F is lge i compiso o he vege disce bewee molecules s is he cse of plsm o liquid se. Fom his poi of view iecios i liquid sysem e eed somewh lie is doe i plsm physics whee specific pmee he plsm pmee esblishes whe hee e eough picles i he so-clled Debye sphee d simuleous muliple collisios pevil. The fo he liquid se we c ssume Eq. (7) o be vlid d he collisio iegl becomes ' F d F d (8) F ' whee F is he uow self-cosise field h mus be deemied o he bsis of he piwise iecio poeil. Wih his poeil Eq. (5) c be ewie s. T K F' ( ) (9)

4 V Molii D Gpol D Moscci.. Self-cosise field The ex sep is o fid he self-cosise field F ' fom F () ϕ ( ) ( ) d V whee ϕ is he piwise iecio poeil bewee wo molecules loced posiios d especively. To clcule he Vlsov self-cosise field sevel pheomeologicl poeil models hve bee poposed which moe o less descibe he el muliple foce iecio mechism 9-. I his wo he Suheld pi poeil model is used fo is mhemicl simpliciy d becuse he pedicios of he liquid popeies e expeced o be quliively cosise wih he behviou of my el fluids. The pheomeologicl Suheld pi poeil model give by 9 α σ ϕ ( ) ε ϕ ( ) fo fo σ < σ () wih he coespodig foce F α σ ϕ εα ˆ. () α F The Suheld pmees ε d σ coespod especively o he deph of he poeil well (divided by ) d o he disce of closes ppoch i he Suheld ppoximio esseilly eled o he molecul dius. oh pmees e buled fo umeous molecules see e.g. ef. 9. To clcule F v ' molecule loced he poi ( z ) will be cosideed d he foce fom ll he suoudig liquid will be deemied usig he Suheld poeil ssumig slb symmey i.e. desiy depeds oly o he z coodie d cosideig elemey volume dv locio defied by spheicl coodie sysem ceeed o he molecule of iees. y expessig he Cesi foce compoes d iegig he foces ove he ll spce fe some lgeb he selfcosise field fo he semi-ifiie medium is foud o be F' ( ζ) ( z ζ) ( ζ) dζ 5 ( ζ z) z σ 6 8 πεσ dζ. () 5 zσ Fo compleeess he sme pocedue pplied o slb of hicess would yield F' ( ζ) ( z ζ) ( ζ) dζ 5 ( ζ z) zσ 6 8 πεσ dζ. (b) 5 z σ

5 If he desiy viio is mild ( z) oly he fis few ems gives ( ζ) ( z) d Desiy Disibuio i Semi-Ifiie Spce c be expded i Tylo seies d eiig ( z) ( ) d ( z) ( ζ z) d ( z) ( ζ z) d ( z) ( ζ z) ζ z!! O [( ζ z) ] 5. () Neglecig ems of ode d highe d subsiuig io Eqs. () (fe some moe lgeb) he followig equios e obied: F ' ( z ) 6 8πεσ z d σ z d z (5) fo he semi-ifiie cse d gi fo compleeess 6 F' ( z) 8πεσ d z z σ ( z ) ( z) ( z ) ( z) fo slb of hicess.. The Semi-Ifiie Liquid d (5b) Coside liquid occupyig semi-ifiie spce z. The ohe p of he spce z < c be vcuum o meil i gseous liquid o solid phse. Oly he fis cse will be cosideed hee: z < is vcuum. The equio fo he umbe desiy disibuio Eq. (9) fo he oe dimesiol cse - d doppig fom ow o he subscip - becomes d subsiuig he expessio of d F' ( ) (6) KT F ' i Eq. (6) gives fe some lgeb K T " ' z 6 (7) σ z πεσ z h is olie sigul diffeeil equio of he secod ode. This equio c be ecs io self-simil fom defiig he followig dimesioless quiies x z σ ; χ z o give whee ( ) 8 K T χ χχ" χ' x χ. (7b) x πεσ x

6 V Molii D Gpol D Moscci Fo he usul vlues of σ d ε he quiy K T c be disegded ( C e.g. πε σ fo we i is.9 fo ehol. fo ceoe.) d Eq. (7b) becomes d χ 8 dχ x x χ x (8) dx dx ξ he ( ) Wih he ddiiol vible sfomios: x χ ξ ξ he coflue hypegeomeic equio is obied which hs ow soluio. Reveig o he oigil vibles he soluio o Eq. (8) is foud o be χ x ( x) C x e Φ( ; ) x whee Φ is he coflue hypegeomeic fucio wih ( ) d is he lge soluio of (9) () () 9 To fid C he followig sympoic behviou is used (bmowiz 96) χ ( x) C Γ ( ) ( ) Γ ( ) x () d sice χ( x ) oe fids ( ) Γ( ) Γ C. (). Numeicl Soluio The umeicl scheme begis wih Eq. (8) i he moe coveie fom ( ) ( ) d d ( ) ( ) () d d wih subsiuios x χ x d /9 8/9 d /9. Sice he ODE is sigul specil eio mus be give o he siguliy he oigi which comes bou becuse of he discoiuiy i meil popeies. Oce he sigul ue of Eq. () hs bee ccommoded hough he ledig o-lyic fco fo < he soluio is expeced o be ifiiely smooh d heefoe Tylo seies should hold. Hece we pply he mehod of coiuous lyicl coiuio (CC) o povide highly ccue soluio fo gee h zeo.

7 Desiy Disibuio i Semi-Ifiie Spce Followig he mehod of Fobeius e he oigi he lgoihm begis wih he soluio f (5) whee ( ) f. (5b) y subsiuio he coefficies c be show o be ( ) ( ) (6) wih. (6b) is e s uiy sice he soluio is o be omlized o uiy fo lge. Sice he discoiuiy is oly i he fis ievl he soluio is expeced o follow he Tylo seies (7) heefe i y ievl. The coefficies e foud by subsiuio of Eq. (7) io Eq. () o give (fe some lgeb) fo ( ) [ ] 6 (8) d fo ( ) [ ] ( )( ) [ ] (8b) Filly he fucio d is deivive he ed of he ievl give he iiil wo coefficies (9c) d d (9d)

8 V Molii D Gpol D Moscci which coec o he ex ievl compleig oe eie ievl clculio. ecuece is iiied fom he fis ievl wih The ( ) ( ) (9d) To povide cofidece i he umeicl pocedue Tble gives he compiso fo lge vlues of fo he u-omlized soluio. s obseved wih icesig he sympoic fom ideed emeges. This clculio equied h ech ievl be fuhe divided io coveged sub-ievls fo 9-plce pecisio. Tble. Compiso o sympoic Soluio ( ) sy ( ) Ideicl Digis 5.E.8785E E 5.E E E 5.5E.86958E.86955E 6.E.86985E.8698E 6.5E.86999E E 6.E.8699E E 6.E.86959E E 6 5.E.869E.86996E 7 6.E.86979E.86978E 7 5. Coclusios Fis ecll h he cse cosideed is h of iefce bewee liquid d vcuum. Molecules i he bul of he homogeeous liquid e subec o bidig foces hvig o pefeeil diecio o i ohe wods isoopic foces. Hece desiy is o be expeced cos i egios f eough fom iefces. Howeve s iefce is ppoched (i.e. disces of he ode of he es of molecul dimees) bidig foces become peed owd he bul of he liquid d i c be expeced h his pevilig iwd foce would led o icese i he desiy which is he opposie of wh is commoly ssumed. The im of he pese wo ws o veify he exisece of his pheomeo: Figue cofims his pedicio isof s he cive foces pese e of he v de Wls ype. This behviou will be of picul impoce i ivesigig sufce esio. If iefce wih solid sufce is cosideed o he ohe hd his behviou c be sigificly chged o eve evesed depedig upo he elive foces bewee he molecules belogig o he wo phses. This is he subec of fuhe wo ledy i pogess.

9 Desiy Disibuio i Semi-Ifiie Spce Figue. Dimesioless desiy vs. dimesioless disce fom he iefce. cowledgemes The secod uho hs he Uivesiy of olog Isiue fo dvced Sudy fo hosig his sy duig he compleio of his wo Refeeces. Plech U. Klemd d J. Peisl J. Phys.: Codes. Me () 556. R. Tdmo J. Phys.: Codes. Me () L95. D. o D. Ross E. ed K. Rgil N. Shhideh D. ose d J. Meuie Phys. 6: () 79. J. Klei U.Rviv S. Pei N. Kmpf L. Chi d S. Gisso J. Phys.: Codes. Me 6 () S57 5. N. Shhideh D. o J. Meuie d. Mvo Phys. Rev. 6 () 9 6. T. Vilmi d E. Rphël Phys. Rev. Le. 97 (6) Fezzoi L. Gibelli F. Loezi Phys. Fl. 7 (5) 8. J.L. Delcoix. es Physique des plsms (Ieédiios Pis 99) (i Fech) 9. J.O. Hischfelde C.F. Cuis d R.. id Molecul Theoy of Gses d Liquids (Wiley New Yo 95). R.K. Phi Sisicl Mechics (Pegmo Pess Oxfod UK 97). D. Moscci V. Molii d M. Pemud Eu. Phys. J. 7 (9) 7. M. bmowiz d I.. Segu Hdboo of mhemicl fucios (Dove Publicios New Yo 96)

One of the common descriptions of curvilinear motion uses path variables, which are measurements made along the tangent t and normal n to the path of

One of the common descriptions of curvilinear motion uses path variables, which are measurements made along the tangent t and normal n to the path of Oe of he commo descipios of cuilie moio uses ph ibles, which e mesuemes mde log he ge d oml o he ph of he picles. d e wo ohogol xes cosideed sepely fo eey is of moio. These coodies poide ul descipio fo