A Nonlinear Transport Problem of Monochromatic Photons in Resonance with a Gas

Transcription

1 A Nonlnear ransport Problem of onochromatc Photons n Resonance wth a Gas G. Lauro, R. onaco and. Pandolf Banch Dpartmento d atematca Applcata G. Sansone, Unverstà d Frenze, Frenze, Italy Poltecnco d orno, orno, Italy Abstract. A transport problem arsng from the dynamcs of a gas n a radatoeld, recently modelled n knetc theory, s formulated and the trend to equlbrum of the gas-photon system s studed. A computatonal technque matchng relevant mathematcal aspects of dfferental quadrature and spectral methods s appled. he numercal results are then compared wth those of other models known n lterature. INRODUCION Am of the present paper s to provde a numercal analyss of a nonlnear transport problem arsng n the dynamcs of a gas mbedded n a monochromatc radatoeld. he gas partcles are endowed wth two levels of nternal energy, the fundamental and the ected one. Besdes the elastc collsons, the gas partcles may eperence nelastc nteractons, passng from anergy level to the other. Absorpton, stmulated and spontaneous emsson processes are taken nto account n the nteracton between photons and gas partcles. In ths paper, the set of moment equatons, namely the macroscopc conservatoquatons, derved from the knetc equatons gven n Ref. [], s consdered under the assumpton that the characterstc relaaton tme of elastc collsons s much smaller than the one relevant to nelastc and gas-radaton nteracton. An ntal boundary value problem n a slab s formulated for the nonlnear system of the moment equatons and radatve transfer equaton, wth the am of studyng the trend to equlbrum of the gas-photon system. In unbounded domans such a trend has been shown n Ref. [2]. A numercal technque, proposed n Ref. [3], based on the spectral appromaton of the solutopanded n terms of Legendre polynomals, transforms the orgnal set of partal dfferental equatons nto a set of ordnary dfferental equatons to be numercally solved wth pertnent ntal conons. Numercal results are then gven and compared wth those obtaned for the same physcal problem, treated n Refs. [4], [5], [6] by means of smplfed versons of the moment equatons. GOVERNING EQUAIONS Consder, n slab geometry, a gas wth partcles endowed wth two nternal energy levels n presence of a monochromatc radatoeld. Wth reference to papers [2], [4], the closed set of moment equatons of the gas system s derved n a dmensonless and rescaled form: u nf u ϑ η 4π ϑ 2 () u ne u ϑ η 4π ϑ 2 (2)

2 du u u u σ 2 3 u 2 3 σ log ϑ Besdes the moment equatons, the model ncludes the radatve transfer equaton: I µ I (3) ϑ 2 (4) I (5) he dmensonless state varables u I are, respectvely, the number densty of gas molecules wth nternal energy at the fundamental ( f ) and ected ( e ) level, the mean velocty of the gas, the absolute temperature and the radaton ntensty of the monochromatc feld of photons. he parameters η, ϑ 2, and σ are gven by η I r n r chν ϑ γ βchν σ hν mc 2 (6) where n r I r are gven reference values of the total number densty of the gas partcles and the radaton ntensty, respectvely. In partcular, accordng to paper [4], let I α r β, α β beng the so-called Ensten coeffcents whch account for absorpton and emsson rates. oreover, hν s the photonergy, c the speed of lght, and γ, γ 2 are the nelastc frequences of the atom-atom collsons, whch are postve constants under the assumpton of awellan molecules nteracton law. Note that,, u and depend on t R, and R whereas I depends also on µ cosθ, θ 2π beng the angle between the -as and the velocty of photons. In adon s the ntegrated radaton ntensty, defned as t 2π I t µ dµ (7) Observe that the presence of n Eqs. (), (2) mples that the model equatons actually consttute an ntegro-dfferental system and thus dffer from the ones of papers [2], [4] and [5], where the Eddngton appromaton [7] has been used. As shown n paper [], Eqs. (-5) admt aqulbrum soluton gven by whch corresponds to the thermodynamcal equlbrum of the system. I (8) INIIAL-BOUNDARY VALUE PROBLE An ntal-boundary value problem, n lne wth the one studed n the above mentoned papers [4], [5] and there solved for a smplfed verson of model (-5), can now be formulated n the slab for the gas-photon system governed by Eqs. (-5), wth assgned conons as follows. Intal conons t : (9) () u () (2) I µ I 4π µ (3)

3 hese data correspond to the followng physcal stuaton. At t, the gas, wth number denstes and ne, s n absolute equlbrum at a temperature ; the boundares of the slab are perfectly reflectng walls for the gas-partcles and perfectly reflectng mrrors for the radatoeld, so that ths one s qulbrum wth the gas at the ntensty I. In data (9-3) the epressons of and I are n agreement wth the hypothess of absolute equlbrum. Boundary conons t : u u (4) (5) log I (6) I µ I µ I (7) hese data correspond to the followng. For t the mrrors are removed and successvely the gas, t, s subjected to a radaton ntensty I µ, µ, on the wall at, and to zero radaton ntensty, µ, on the other wall at ; conversely, t, the gas-partcles can never cross the walls, as stated by (4). ore n detal, conons (5),(6) epress that the the wall at s thermcally nsulated, snce no radaton source s present at as stated n (7), whereas the wall at s accomodated at the temperature n equlbrum wth the radaton source of ntensty I present at. Such a physcal problem has ts orgn n paper [8] and has been consdered n book [6], as well. APPROXIAION EHOD he method here proposed to construct an appromated soluton to the ntal-boundary value problem (-5), (9-7) adapts the dfferental quadrature techqnque, recently revewed n [9], to spectral methods, for what attans the truncated seres epanson of the soluton n the bass of Legendre orthogonal polynomals. In order to apply the method, t s convenent frst to rewrte system (-5) n vector form. Let g denote the state varable u I ; the system of Eqs. (-5), recallng defnton (7) of the ntegrated radaton ntensty, can be rewrtten as wth ntal data g g g G t g he appromated soluton to Eq. (8) has the spectral representaton (8) (9) g t m c m t L m (2) where L m m, are the orthogonal Legendre polynomals of degree m, and the vectors c m t are the epanson coeffcents, gven by c f m c e m c u m c m c k m c m t 2m L 2 m g t d m (2) he dmenson of the slab, namely the nterval, s dscretzed wth N sub-ntervals by N equally spaced nodes N Note that the number N of the nodes does not depend on the mamum degree of the Legendre polynomals. he space dervatve of the state varable g s appromated n the nodes by where a m defne a N g g dl m t m d c m t matr A gven by a m m c m t (22)

4 A a m dl m d (23) whch can be computed once forall. Hence, Eq. (8), by takng nto account Eqs. (2-23), s transformed nto the set of ordnary dfferental equatons dg G t g g N N c t c t A N (24) where g g t and t. he tme dependng coeffcents c t c t must be computed at each tme step through ther defnton (2). he ntal data for system (24) are suppled by conon (9) dscretzed ach node. On the other hand, the boundary conons (4-7) wll be naturally ncluded n the ntal value problem, accordng to a procedure whch wll be shown n the net secton. COPUAIONAL SCHEE he procedure outlned above s now appled to problem (-5), (9-7), leadng to the formulaton of fve ordnary dfferental systems wth pertnent ntal data. As t wll be shown, the number of equatons of each system depend on the assgned boundary conons. Equatons for the number denstes and. d u m c f ma m ϑ ϑ 2 m c u ma m η 4π (25) d u m c e ma m ϑ ϑ 2 m c u ma m η 4π he ntal data to be joned to Eqs. (25), (26), takng nto account conons (9-), are (26) (27) Snce no boundary conons are prescrbed for Eqs. (-2), the nde actually ranges from to N. Equatoor the mean velocty u. du u m c u ma m σ he ntal data to be joned to ths equaton are u c m ma m c m f m c e m a m (28) Snce the boundary conons (4) are appled to the frst node and to the last one N, solutons u and u N are drectly gven by u t, u N t t hus the nde n Eqs.(28) ranges, ths tme, from 2 to N. Equatoor the temperature. d u m c ma m m c u ma m ϑ ϑ 2 (29)

5 wth ntal data Snce the boundary conon (6) s appled to the last node N, t s mmedate to wrte N t log t (3) so that the nde vares from to N only. Conversely, boundary conon (5), whch s of Neumann type, has no drect nfluence on the number of solutons to be computed, because t needs to be treated n a dfferent way. By takng nto account the dervatve epanson (22), we can wrte d d c m ma m c m ma m I c a c a c m ma m (3) Let us underlne that, n ths case, the boundary conon mples that the unknown coeffcents to be computed are only c c. In adon, note that formula (3) has been arranged n such a way, snce n the Legendre bass one has a. Equatoor the radaton ntensty. Frst of all let us dscretze the varable θ, θ 2π, n aven number K of equally spaced angles θ k, k K. In order to avod partcles grazng the walls, namely partcles movng n the drecton of the y-as, the dscretzaton of θ must be performed n such a way that the varables µ k cosθ k never vansh. Accordngly the radaton ntensty feld wll be dscretzed by I t µ k I k t k K hen the form of Eq. (5), ach node, s he ntal data (3) assume the form di k µ k c m k m a m I k I k t I (32) For what concerns the boundary conons (7), let us frst ntroduce two sets of ndees S IN and S IN such that k S µ k k K k S µ k k K 2 2 K hus t s mmedate to wrte t k S : I k t ; k S : I k N t I (34) he nde n Eqs. (32) runs from 2 to N when k S and from to N (33) whenk S. Consequently the number of equatons (32) s K N. he knowledge ach node of I k t allows to compute, by a numercal quadrature on the varable µ, the ntegrated radaton ntensty ach node as whch appears n Eqs. (25), (26). t t 2π I t µ dµ (35) NUERICAL RESULS Numercal results have been obtaned usng a 4 th -order Runge-Kutta routne to ntegrate Eqs. (25), (26), (28), (29), and ( 32). Computaton of the ntegral terms (35), concernng the radaton ntensty, have been performed va a Gauss- Legendre formula, based on the same dscretzaton ponts, as those n Eq. (32). he purposes of ths secton consst both n valdatng the proposed numercal method, through the qualtatve evoluton of the system descrbed n Ref. [8], and n comparng the quanttatve results of the present model wth those obtaned

6 a) b) FIGURE. a Profles of and b profles of versus, at t 5, for a) b) FIGURE 2. a Profles of and b profles of versus, at t 45, for n prevous studes. As proven n Ref. [5], and shown n Ref. [8], the ntal boundary value problem formulated n ths paper presents avoluton towards a statonary equlbrum state for all the relevant macroscopc observables,.e.,, and. In partcular, when the statonary state s reached, temperature and ntegrated radaton ntensty have an ncreasng monotone profle from the left boundary at to the rght one at. Conversely, the mamum values of and, due to a small negatve mean gas velocty, shft from the rght to the left boundary and when total densty has ts mamum at, then the process becomes statonary and u vanshes agan. Such a behavour s well represented by Fg. a and Fg. 2a for temperature and ntensty and by Fg. b and Fg. 2b for numercal denstes of the two populatons,. In partcular, Fgs. are prnted at tme t 5 (transent behavour), whle Fgs. 2 show the statonary state at t 45. he ntal data used to obtan both Fgs. and Fgs. 2 are: 7,. Relaaton to equlbrum s a bt faster when s hgher (and, consequently, I s smaller) and the profles reach a slghtly dfferent shape. hs stuaton s shown by Fg. 3a, prnted at t 2 for 5. he ntal datum on does not affect the rapy of relaaton. All fgures have been obtaned for the same value I 2 at the boundary. As mentoned before, n paper [5] a smplfed verson of macroscopc equatons has been used to solve the same

7 a) b) FIGURE 3. a Profles of versus, at t 2, for 5; b ths model (sold lne) compared wth that of [5] (dot lne) t ) and that for physcal problem. Such a verson dd not consder the equatoor the gas mean velocty (.e. u, snce t was supposed that for each, n, beng n. In paper [5] the results of that model were compared wth those obtaned n book [6], at least for the radaton ntensty, snce Chandrasekhar model dd not consder avolutoquatoor the temperature. In Fg. 3b, for the same data as n Fgs. 2, we report the statonary profles of and, for the present model (sold lnes) and for that of [5] (dot lnes). ACKNOWLEDGENS he present work has been supported by the Cofn Project Problem atematc Nonlnear d Propagazone e Stabltà ne odell del Contnuo (coord.. Rugger). REFERENCES. Rossan, A., Spga, G.P. and onaco, R., Knetc approach for two-level atoms nteractng wth monochromatc photons, ech. Research Comm., 24, (997), p Bove, A., Deartno, S. and Lauro, G., rend to equlbrum dynamcs of a gas nteractng wth a radatoeld, ath. Comput. Smul., 54, (2) pp onaco, R. and Pandolf Banch,., On a class of methods for the appromated solutons to nonlnear ntal boundary value problems, ech. Rep. IPR-9923, echncal Unversty of Graz, Austra, (999). 4. onaco, R., Polewczak,. and Rossan, A., Knetc theory of atoms and photons: an applcaton to the lne-chandrasekhar problem, ransp. heory Stat. Phys., 27, (999), pp onaco, R., Polewczak,. and Rossan, A., Knetc theory of atoms and photons: on the soluton to the lne-chandrasekhar problem, ransp. heory Stat. Phys., 28, (999), pp Chandrasekhar, S., Radatve ransfer. Dover Pubs., New York, (96). 7. Pomranng, G.C., he Equatons of Radaton Hydrodynamcs, Pergamon Press, Oford, (973). 8. lne, E.A., he dffuson of mprsoned radaton through a gas,. London ath. Soc.,, (926), pp Bert, C.W. and alk,.,dfferental quadrature method n computatonal mechancs: a revew, ASE, Appl. ech. Rev., 49, (996), pp. -27.