Symmetric form of governing equations for capillary fluids

Transcription

1 Symmeric form of governing eqaions for capillary flids Sergey Gavrilyk, Henri Goin To cie his version: Sergey Gavrilyk, Henri Goin. Symmeric form of governing eqaions for capillary flids. Gerard Iooss, Olivier Ges, Anne Nori. Monographs and Srveys in Pre and Applied Mahemaics, 1999, France. Chapman and Hall/CRC, 106 ISBN X, pp.pages , 1999, Trends in Applicaions of Mahemaics o Mechanics, Monographs and Srveys in Pre and Applied Mahemaics Vol 106. <hal-00537> HAL Id: hal hps://hal.archives-overes.fr/hal Sbmied on 1 Feb 008 HAL is a mli-disciplinary open access archive for he deposi and disseminaion of scienific research docmens, wheher hey are pblished or no. The docmens may come from eaching and research insiions in France or abroad, or from pblic or privae research ceners. L archive overe plridisciplinaire HAL, es desinée a dépô e à la diffsion de docmens scienifiqes de nivea recherche, pbliés o non, émanan des éablissemens d enseignemen e de recherche français o érangers, des laboraoires pblics o privés.

2 Symmeric form of governing eqaions for capillary flids Sergey Gavrilyk and Henri Goin Laboraoire de Modélisaion en Mécaniqe e Thermodynamiqe, EA596, Universié d Aix-Marseille, Marseille Cedex 0, France hal-00537, version 1-1 Feb 008 Absrac In classical coninm mechanics, qasi-linear sysems of conservaion laws can be symmeried if hey admi an addiional convex conservaion law. In pariclar, his implies he hyperboliciy of governing eqaions. For capillary flids, he inernal energy depends no only on he densiy b also on is derivaives wih respec o space variables. Conseqenly, he governing eqaions belong o he class of dispersive sysems. In ha case we propose a symmeric form of governing eqaions which is differen from he classical Godnov - Friedrichs - Lax represenaion. This new symmeric form implies he sabiliy of consan solions. 1 Inrodcion Qasi-linear sysems of conservaion laws can be symmeried, if hey admi an addiional convex conservaion law Godnov, 1961, Friedrichs and Lax, The symmeric form implies hyperboliciy of governing eqaions. For conservaion laws wih vanishing righ-hand side, he hyperboliciy is eqivalen o sabiliy of consan solions wih respec o perrbaions of he form e ik x λ, i = 1, k = k 1,, k n, x = x 1,, x n, where denoes he ransposiion. Indeed, he following symmeric form of governing eqaions for an nknown vecor variable v A v n + B i v x i 1 i=1 where marix A = A is posiive definie, B i = B i, implies he dispersion relaion n deb λa, B = B i k i sergey.gavrilyk@niv-ceanne.fr henri.goin@niv-ceanne.fr i=1 1

3 which deermines real vales of λ for any real wave vecor k. In his noe we ge an analog of symmeric form 1 for eqaions of capillary flids ha belong o he class of dispersive sysems, becase he inernal energy depends no only on he densiy b also on is derivaives wih respec o space variables. We will see ha he analog of eqaion is deb + ic λa 3 where C = C is an anisymmeric marix depending on he wave vecor k. Since B + ic is Hermiian marix and he symmeric marix A is posiive definie, all he freqencies λ are also real. For a capillary flid he marix C is of he form C = C = O ρ e kk 0 ρ e kk O where ρ e is he eqilibrim flid densiy and O is he ero-marix 3 3. Here and laer, for any vecors a,b we se he noaion a b for he scalar prodc he line is mliplied by he colmn vecor and a b for he ensor prodc or a b he colmn vecor is mliplied by he line vecor. Divergence of a linear ransformaion A is he covecor diva sch ha, for any consan vecor a, diva a = divaa. The idenical ransformaion is denoed by I. In secion we presen he mli-dimensional case in Elerian coordinaes for a pariclar form of he inernal energy. In secion 3 we consider in Lagrangian coordinaes a one-dimensional case for he general form of inernal energy. Governing eqaions in Elerian coordinaes The inernal energy per ni volme of a capillary flid is aken in he form eρ, η,w = ερ, η+ c w where ρ is he flid densiy, w = gradρ or w = ρ, η is he enropy per ni volme, c is he capillariy coefficien which is assmed o be a consan [Rocard, 195, Rowlinson and Widom, 1984]. The homogeneos energy ε saisfies he Gibbs ideniy dε = µ dρ + θ dη 5 where µ = ε + P θη/ρ is he chemical poenial, θ is he emperare, P = ρ ε ε is he hermodynamic pressre. By sing Hamilon s principle, ρ governing eqaions of sch a flid were obained by Casal 197 see also Casal and Goin, 1985, Gavrilyk and Shgrin, They are in he form 4 ρ + div j = 0 η η + div ρ j jj 6 j + div ρ + p I + cww

4 where j = ρ, is he velociy vecor, index is he parial derivaive wih respec o ime, p = ρ δε δρ e = P c ρ divw + w, where δ means he δρ variaional derivaive wih respec o ρ. By sing 4 and 5 we can obain from 6 he energy conservaion law ε + j ρ + c w + div ε + j ρ + P + c w div j j div w Since div cww c ρ div w + c w I w w = cw div w + cw cw div w cρ div w cw x x = cρ div w he momenm eqaion reads j + div jj ρ + P I cρ div w The gradien of he mass conservaion law verifies anoher conservaion law If we add an iniial condiion sch ha w + div j w =0 = ρ =0 we can consider w as an independen variable. The fac ha w = ρ will be a conseqence of he governing eqaions. Finally, we obain eqaions 6 in he following eqivalen non-divergence form ρ + div j η + div η ρ j jj j + div ρ + P I cρ div w + div j w The heory of capillary flids is sally applied for van der Waals-like flids. For sch flids he energy ε ρ, η is no convex for all vales of ρ and η. We sppose ha we are in he viciniy of an eqilibrim sae ρ e, η e where he energy fncion is locally convex. Le s inrodce conjgae variables q, θ,, r by he formla de d ε + j ρ + c w µ qdρ + θdη + dj + r dw dρ + θdη + dj+c w dw

5 The Lagrange ransformaion of he oal energy E is defined by Π = ρq + ηθ + j + w r E = P + r c where he hermodynamic pressre P is considered as a fncion of q, θ and. Hence, in erms of he conjgae variables q, θ,, r defined by eqaion 9, eqaions 8 can be rewrien in he following form Π Π + div q q Π Π + div θ θ Π Π + div Π 10 r q x Π Π + div + Π r r q x If he capillary coefficien c is ero, Π = P and we ge he gas dynamics eqaion and he symmeric form of Godnov Mliplying eqaions 10 by q, θ, and r, smming p all of hem and sing he ideniy ro a b = [a,b] + adivb b diva where [a,b] = a x b b x a denoes he Poisson bracke, we ge he conservaion of he energy 7 in he form q Π q + θ Π θ + Π + Π r r Π + div q Π + θ Π + Π Π + r Π q θ r + Π q x r Π q r x The sysem 10 admis consan solions ρ e, η e, e,w e. Since he governing eqaions are invarian nder Galilean ransformaion, we can assme ha e. If we look for he solion of he linearied sysem proporional o e ik x λ, we ge eqaion 3, in which we have p A = v Π v C = C =, B = O ρ e kk 0 ρ e kk O n B i k i, B i = Π i, v v i=1 wih O = and v = q, θ,, r. Hence, eigenvales λ are real if A is posiive definie. 4

6 3 One-dimensional baroropic case In mass Lagrangian coordinaes, he governing eqaions are see Gavrilyk and Serre, 1995 v, + p, p = δe e δv = v e, e = ev, v v where v = 1 denoes he specific volme. This case is general: we do no sppose ρ a pariclar form 4 of he energy e. Consider an agmened sysem v w e v e w 11 Le s define π and he conjgae variables σ, r as π = e v v + e w σv + rw e w In erms of π and σ, r, he sysem 11 reads π σ π r σ r In marix form we ge where A = π σ π σ r A σ r π σ r π r B 1 σ r + C 1 σ r 0, B 1 = , C 1 =

7 Eqaions 1 and 13 imply a dispersion relaion of ype 3, if we p B = kb 1, C = k C 1. We noe also ha he sysem admis he energy conservaion law σ π σ + r π r π + σ + [r, ] where [r, ] = r r. Remark. Analogos symmeric forms may be obained for bbbly liqids, where he inernal energy is a fncion no only of he densiy b also of he oal derivaive of he densiy wih respec o ime. References [1] Casal, P. - La héorie d second gradien e la capillarié, C.R. Acad. Sc. Paris, , [] Casal, P. and Goin, H. - Connecion beween he energy eqaion and he moion eqaion in Koreweg s heory of capillariy, C.R. Acad. Sc. Paris, Série II, , [3] Friedrichs, K. O. and Lax, P. D. - Sysems of conservaion eqaions wih a convex exension, Proc. Na. Acad. Sci. USA, 68, No , [4] Gavrilyk, S. and Serre, D. - A model of plg-chain sysem near he hermodynamic criical poin: connecion wih he Koreweg heory of capillariy and modlaion eqaions. In IUTAM Symp. Waves in liqid/gas and liqid/vapor wo-phase sysems, 1994, Japan Morioka, S. and van Wijngaarden, L. eds., Klwer Academic Pblishers, 1995, pp [5] Gavrilyk, S. and Shgrin, S. - Media wih eqaions of sae ha depend on derivaives, J. Appl. Mech. Techn. Phys., , [6] Godnov, S. - An ineresing class of qasilinear sysems, Sov. Mah. Dokl, 1961, [7] Rocard Y. - Thermodynamiqe, Chaper V, Masson, Paris, 195. [8] Rowlinson, J. S. and Widom, B. - Moleclar heory of capillariy, Chaper III, Clarendon Press, Oxford