Anomalous transport regimes and asymptotic concentration distributions in the presence of advection and diffusion on a comb structure

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1 Anomalos ranspor regimes and asympoic concenraion disribions in he presence of advecion and diffsion on a comb srcre Olga A. Dvoreskaya and Peer S. Kondraenko Nclear Safey Insie, Rssian Academy of Sciences, 5 Bolshaya Tl skaya S., 59 Moscow, Rssia We sdy a ranspor of impriy paricles on a comb srcre in he presence of advecion. The main body concenraion and asympoic concenraion disribions are obained. Seven differen ranspor regimes occr on he comb srcre wih finie eeh: classical diffsion, advecion, qasidiffsion, sbdiffsion, slow classical diffsion and wo kinds of slow advecion. Qasidiffsion deserves special aenion. I is characerized by a linear growh of he mean sqared displacemen. However, qasidiffsion is an anomalos ranspor regime. We esablished ha a change of ranspor regimes in ime leads o a change of regimes in he space. Concenraion ails have a cascade srcre, namely consising of several pars. PACS nmber(s): Fb, g, Mh I. INTRODUCTION Anomalos ranspor in highly heerogeneos media is a sbjec of exensive sdies for decades []. De o he complexiy and variey of real heerogeneos media, general heory of he ranspor is no ye available for hem. For his reason, sdying anomalos ranspor by simple physical models is a maer of excepional imporance. A comb srcre is one of sch models. This model has mch in common wih a percolaion clser. A backbone and eeh of he comb srcre are like respecively o a backbone and dead (dangling) ends of he percolaion clser. The simples version of he comb srcre based on he classical diffsion eqaion was analyzed in []. A random walk on he comb and comblike srcre was sdied in Refs. [3], [4]. Sbdiffsion wih a power / 4 was obained in []. Anoher ranspor regime has been fond and ermed as a qasidiffsion in Ref. [5]. This regime is a resl of he paricles deparre ino he eeh of he srcre and advecion. Classical diffsion as a physical ranspor mechanism for a backbone of he comb srcre has been spplemened by longidinal advecion in Refs. [6, 7]. However, he ahors coldn have obained some ineresing resls as he backbone and he eeh had an infinie hickness and lengh in hese works. A finie hickness of he comb srcre and finie lengh of he eeh case addiional ranspor regimes and power rains. So, he resls obained in Refs. [-7] do no exhas all problem aspecs of he impriy ranspor on he comb srcre. A paern of ranspor regimes is sill an open qesion in he case of finie lengh eeh, as well as a fine srcre of he asympoic concenraion disribion a all ime inervals.

2 The prpose of his paper is a deailed analysis of he impriy ranspor on he comb srcre. In pariclar, we obained ranspor regimes in addiion o he known previosly and fond ha he asympoic concenraion disribion ofen differs from Gassian form. Also, we esablished ha a ransiion from one ime inerval o anoher may be accompanied by a change of he ranspor regime in media wih a conras disribion of characerisics. Concenraion ails have a cascade srcre (see also Refs. [8-0]). The srcre of he paper is as follows: secion presens he problem and basic relaions; secion 3 focses on obaining resls sch as ime evolion of he impriies concenraion in he backbone. The oal nmber of impriies in he backbone is fond in secion 4. Main conclsions of he paper are smmarized in he final secion. II. PROBLEM FORMULATION AND BASIC RELATIONS FIG.. Comb srcre The comb srcre is illsraed in Fig.. I consiss of a backbone and a periodic sysem of eeh. The backbone is a sraigh cylinder of an infinie lengh along he x axis. The crosssecional area of he backbone is eqal o S. Each ooh is also a sraigh cylinder wih a lengh h. The bondary beween each ooh and he backbone is plane wih he size of S. L is a period of he comb srcre. A ranspor of impriy paricles in he backbone resls from advecion wih a velociy direced along he x axis and isoropic diffsion wih coefficien D. The ranspor occrs by diffsion wih coefficien d in each ooh. A concenraion of impriy paricles is denoed by n in he backbone and c - in a ooh. The impriy paricles resided in he backbone are called acive. A bondary condiion on he oer srface of he comb srcre is a normal componen of zero impriy flx. A condiion on he bondary beween he backbone and eeh consiss in coniniy of he concenraion and normal componen of he flx densiy. I is assmed ha he

3 concenraion disribion a he iniial momen 0 is specified and locaed in he region wih a size 0 wihin he backbone. We will be ineresed in he concenraion disribion a imes max S,L () 4D In his case, here is almos a homogeneos concenraion of acive paricles n x, in he backbone. In rn, a concenraion inside each ooh depends on is longidinal coordinae y, ime and coordinae x as a parameer. Averaging hree-dimensional advecion-diffsion eqaion over cross-secional area and period of he comb srcre, we obain he eqaion for he concenraion of acive paricles where q( x, ) n x, n x, n x, D q x,, x x is a magnide of he impriy flx from he backbone ino he eeh per ni of () lengh along he x axis S c q d LS y y 0 - (3) The eqaions for he concenraion in he ooh and he corresponding bondary condiions are c x,y; c x,y; d y. (4) c y y h 0, c 0,;x n x,. (5) Now rn o he Forier-Laplace space(k,p) in Eqs. () - (5). Then he solion of Eq. (4) wih condiions (5) leads o he following expression for he impriy flx wih kp kp q n p anh p, (6) LS h, d S d. (7) We frher assme ha and saisfy he ineqaliy. is he ime when he nmber of paricles in he eeh is compared wih ha in he backbone, while characerisic ime of diffsion a disances of he order of ino he eeh. Using Eqs. (), (6), we find he concenraion of acive paricles in he space(k,p). h is a 3

4 where 0 n kp 0 nkp n kp, p p anh p ik Dk is a Forier- Laplace ransform of he iniial concenraion disribion averaged on (8) he cross-secional area of he backbone 0 n x n x,0. o Performing he inverse Forier-Laplace ransformaion in Eq. (8) and inegraing wih respec k, we find where Green s fncion wih and Gx, 0 n x, dx G x x, n x, is given by b i x dp exp p; x, Gx, exp x ; Reb 0 D (0) i p b i x p;x, p p, () D p p p anh p 4D. Evidenly, is he ime, where a displacemen de o advecion becomes comparable wih a diffsion lengh. The ranspor regime is deermined by wo imporan qaniies: an average of he displacemen x relaed o advecion and he impriy variance of he displacemen (9) () x dx xnx,, N dxx x n N x,, (3) where N dxnx,is he oal nmber of acive paricles a ime. Noe ha he concenraion ails correspond o his condiion: assme ha 0. Ths we have N 0 n x, x x. Frher G x,. (4) S 4

5 Here he iniial oal nmber of he acive paricles is denoed by 0 0 N S dxn x and he reference poin of coordinae x is chosen in he area of he iniial concenraion disribion. Using Eq. (8), we obain Hereafer, we consider Gx (, ) N N dxg x, 0 (5) a posiive x. In order o find he expression for Green s fncion a negaive x, one can se Eq. (0). III. TRANSPORT REGIMES AND ASYMPTOTIC CONCENTRATION DISTRIBUTION The Green fncion behavior and asympoic concenraion srcre depend on a relaion beween characerisic imes, and. Le s analyze he problem separaely for each of hese relaion and characerisic ime inervals. We sress ha he main body concenraion is deermined by p in Eq. (0). In rn, he concenraion ails correspond o p.... This case is obained as a limi 0,. Therefore Green's fncion is given by / x Gx,4 D exp 4D (6) This is a well known classical diffsion expression... h We have p following expressions for for he main body concenraion. Therefore, we make se of he p and p;x, of Eqs. (), () wih p 4 p p p, x p p;x, 4 p p, x (recall ha x 0 ). (7) (8) 5

6 We show below ha he G - fncion behavior essenially depends on wheher a crren ime is more or less han he characerisic ime 3 3 Le s analyze he cases and separaely. 3 /. Formally, his is deermined by which of he wo erms in parenheses of Eq. (8) is a dominan nder inegraion in Eq. (0)..a. 3 3 Firs we consider he main body concenraion as a dependence on he spaial variable x. We sppose ha significan vales of p in Eq. (0) are deermined by he erm in he exponen / from Eq. (8) a x. Then we have p~. A hese vales of p, he erm in Eq. (8) proporional o ~ p is esimaed as variable x saisfies he ineqaliy x srong ineqaliy 3/ 4 x p x ~ 3. Frher calclaions reveal ha a spaial in he main body concenraion. Ths we obain he x p 3 I confirms he assmpion made above relaive o predominance of erm 3/ 4. (9) ~p, hence allowing s o neglec he erm ~ p in Eq. (8) while calclaing he inegral in Eq. (0). As a resl, we ge / x Gx,4 D exp. 4D This expression corresponds o he classical advecion. Here he average of he displacemen and he impriy variance of he displacemen are x, D and so x. The Eq. (0) is valid a he disances no oo far from he peak of G -fncion. To evalae Green s fncion a he large disances, we ake advanage of saddle-poin echniqe while inegraing in Eq. (0) wih respec o p. The saddle-poin is given by eqaion p 0 p;x, 0 (0) 0 and akes he vale p0. Noe ha p 0 has a real vale and a sign, opposie o a sign of. Expression (0) remains valid in he righ wing of G -fncion (i.e. where x ), becase he original conor of inegraion enconers no singlariies in Eq. (0) while shifing oward he saddle-poin. Anoher siaion occrs in he lef wing (where 0.). Here he saddle-poin is negaive. Therefore shifing he inegraion conor o saddle-poin, we mee he branch poin p 0 of he inegrand in Eq. (0). I resls from erms ~ p in 6

7 p;x, and p. Hence we shold ake ino accon a conribion b G from he inegraion along he banks of he c from he branch poin. To find above conribion, we can neglec he erm ~p in p;x,. Then we sbsie Eqs. (7), (8) in Eq. (0) and expand o he firs order. Finally, we find G x, b x 3 4 This conribion de o he nsal behavior of he concenraion disribion a imes () 3. Possessing a power decrease x / he conribion b G has advanage 3 over exponenially decreasing expression (0) a he relaively far disances from he peak. By comparison of Eq. (0) and Eq. (), we conclde ha power conribion () dominaes a he 3 condiion ln. FIG.. Qaliaive behavior of Green s fncion a imes 3 A imes 3 we have a regime similar o he classical advecion wih nearly symmerical shape of he concenraion disribion, slighly "spoiled" by he presence of a power rain [see Eq. ()]. Tha behavior of Green s fncion is illsraed in Fig..b. 3 In accordance wih he esimae (9), erms ~p and p in Eq. (8) exchange roles nder a ransiion from he inerval o he inerval 3 3. Therefore, we ge he following approximaion A sbsiion of Eq. () in Eq. (0) gives he expression x p p;x' p. () 7

8 x x Gx, exp, 4 D 4D (3) wih D. A similar behavior of he concenraion disribion (o or knowledge) has no been fond before. To find asympoic concenraion profiles we ake advanage of saddle-poin echniqe. There are wo saddle-poin when ' 0. Clearly, i is worh i o ake ino accon sch a saddle-poin ha leads o a smaller exponen vale. I follows ha sch saddle-poin is / 3. This conribion is redced o expression (3). p x 0 4D a imes Since only one saddle-poin 3 G -fncion for his vale of p 0 p 0 / remains in he region 3 G x, exp 3 /, we have 4 (4) 3 Also one saddle-poin p 0 akes place in he case 0 and. Hence his conribion is deermined by Eq. (0). ' 3 Expression (3) applies o he whole ime inerval 3. However here is fndamenal difference beween case 3 and. A imes 3 he average of he displacemen is x, and he impriy variance of / he displacemen (widh of he peak) is ~. Obviosly, in ha case x. Conseqenly one can replace he nmeraor in exponen of Eq. (3) by. Ths we have x G x, exp. 4 D 4 (5) 8

9 FIG. 3. Qaliaive behavior of Green s fncion a imes 3 In ha way, we have advecion wih sharply asymmeric spaial concenraion disribion a 3. This is illsraed in Fig.3. Namely he lef wing of he concenraion disribion is characerized by a power low and he righ wing corresponds o a rapid exponenial decay of Eq. (5) followed by Gassian decrease. The presence of power rains is nexpeced in he cases where eeh canno ye become significan a hese imes., becase prima facie However, sch nsal behavior of he concenraion disribion has a physical meaning and qaliaive explanaion. The peak of disribion had reached he ooh and some of acive paricles had been going ino he ooh. Then he peak of he disribion moved behind he ooh and paricles came back o he backbone. Ths he presence of power rains is de o deparre of acive paricles ino he eeh and sbseqen comeback ino he backbone. A imes displacemen have he same order by find he average of he displacemen and he impriy variance of he x ~ ~ D and D. Hence one can replace applying he main body concenraion and he firs sage of he ail in Eq. (3). Finally, we x x G x, exp. (6) 4 D 4D Similar ranspor regime has been fond in [5] and ermed as qasidiffsion. The second and hird sages of he ail coincide wih he firs and he second sage in case..b. Noice ha / akes place in qasidiffsion similar o classical diffsion b he oal nmber of acive paricles is no reained. So qasidiffsion is an anomalos ranspor regime..3. 9

10 A hese imes he mos significan vales of Laplace variable are p for he main body concenraion and he firs sage of ail. Ths we have approximaion 3 anh p p p. Using Eqs. () and (), we ge 3 xd p;x, p p, p 3, x here we denoe, D D,. 3 Sbsiing Eq. (7) ino Eq. (0), we obain G x, D exp x 4D 4 (7). (8) This is classical advecion wih modified advecion velociy and modified diffsion coefficien. So we called i slow advecion. The average of he displacemen x variance of he displacemen D. Hence, x. and he impriy Along wih he main body concenraion Eq. (9) describes also he acive paricles disribion a he firs sage of he ail nil he saddle-poin saisfies he ineqaliy p 0. The second sage of he ail is deermined by p 0 and given by Eq. (6). I corresponds o qasidiffsion. An approximae border beween ail sages mees he disance x ~ and G - fncion is The hird sage begins from disances 3 / 3 G x, exp. (9) ~ / and has a form (0), where. While calclaing Green s fncion a imes smaller hen one can se an approximaion p p D p,. (30).. This case is enirely similar o he.... 0

11 The erm ~ p shold be negleced nder he roo. Combining Eq. (30) and Eq. (0), we obain / 4 x Gx, F 3,. D D a i ds / 4 / 4 F s exp s s, s p, Rea 0. i a i This expression corresponding o sbdiffsion was fond early in Refs. [, 6, 7, 0, ]. The impriy variance of he displacemen has an esimaion ~ D firs sage of asympoic Green s fncion Gx, (3) I was also fond in []. The second sage of he ail ( diffsion expression (6). / Gx, ~ 4 D exp / second and he firs sage of he ail (where ~ 4 / 3 )..3. (3). Using Eq. (3), we ge he / 4 / 3 4/ 3 exp D 4 4 3/ 4 p 0 ) corresponds o he classical a sample bondary beween he In his ime inerval qasidiffsion expression (6) holds for he main body concenraion and he firs sage of he ail. The second sage of he ail corresponds o Eq. (3) and he hird sage is described by classical diffsion [see Eq. (6)]..4. Here he dedcion formally coincides wih he case of.3 for he main body concenraion and firs sage of he ail, leading o he expression for he slow advecion [see Eq. (8)]. A hese imes he ail consiss of for sages. The second, hird and forh sages are deermined by Eq. (3), (0) and Eq. (6) respecively. A deailed analysis showed ha he "ails" have a cascade srcre and he following reglariy akes place: wih increasing disances sch a ranspor regime occrs wha was realized in he main body of concenraion a an earlier ime inerval. Earlier hese properies of he ails were esablished in Refs. [8-0]. In he nex case a ails consideraion is omied, becase he above-menioned reglariy also is valid corresponds o case is enirely similar o he..

12 3.3. In his case, one can se approximaion concenraion and firs sage of ail. I now follows ha anh p p, 0 o he main body G x, 4 D x exp, (33) 4D where D D. This expression corresponds o he slow classical diffsion This case formally is similar o he.3. So, he Green fncion akes a form (8) for he main body concenraion and he firs sage of he ail. B he effecive diffsion coefficien shold be replaced by D D. IV. TOTAL NUMBER OF ACTIVE PARTICLES In order o find he oal nmber of acive paricles N( ), we ake advanage of obvios relaions where n kp is defined by Eq. (8). b i dp 0 N nkp, N0 N 0 0 nk, (34) i k k0 b i N( ) is given by simple expressions in hree cases N( ) N, ; 0 N( ) N 0, ; N( ) N,. 0 (35) This means ha a imes he relaive nmber of paricles ino he eeh has been very small ye. Therefore, he oal nmber of acive paricles almos coincides wih is iniial vale N 0. In he case, mos of he impriy paricles are locaed in he eeh and a raio / N0 N is inversely proporional o he volme of he eeh occpied by impriy paricles. A imes / N paricles go ino he eeh very inensively and. A similar noaion was

13 N made in Ref. []. The relaion / also was fond in Refs. [, 4, 8,, ]. Finally a he eeh are saraed wih he impriy paricles, and again become saionary b N( ) N0. We see ha he eeh of he comb srcre ac as raps. A similar effec occrs in he percolaion media [0, 3] V. CONCLUSION In he presen work we have sdied in deail he ranspor of impriy paricles on a comb srcre in he presence of advecion and diffsion in he backbone. All obained resls are easily generalized o a random comb srcre. Also, or resls are valid for a random saisically homogeneos comb srcre. We have obained a main body concenraion and a concenraion disribion a he large disances (concenraion ails). Seven differen ranspor regimes are realized. Each regime is deermined by he relaion beween characerisic imes and a considered ime inerval. Ths he following ranspor regimes occr: classical diffsion, sbdiffsion, slow classical diffsion, qasidiffsion, classical advecion and wo kind of slow advecion. The firs hree regimes exis de o he presence of diffsion, moreover he second and he hird significanly resl from he deparre of impriy paricles from he backbone ino he eeh. The nex for regimes are cased by ineracion of advecion and he paricles deparre ino he eeh. Three addiional regimes (wo kinds of slow advecion and slow classical diffsion) arise on he comb srcre wih finie eeh compared wih he srcre of infinie eeh. The impriy ranspor in he presence of diffsion only was sdied in Refs. [], [] where he ahors fond ypical ranspor regimes classical diffsion and sbdiffsion for he comb srcre wih infinie eeh. Besides, following noaion was also developed in Ref. []: a finie lengh of eeh resls in an addiional regime slow classical diffsion. Varios modificaions of he comb srcre were considered in []. I shold be noed ha ranspor regimes arising de o he presence of advecion have no been sdied as well as a fine srcre of concenraion ails in above menioned works. Or analysis showed ha he concenraion ails have a cascade srcre in all ranspor regimes excep for classical diffsion. The resls confirmed he reglariy, which earlier was esablished in Refs. [8-0]: wih increasing disances sch a ranspor regime occrs wha was realized in he main body of concenraion a an earlier ime inerval. Ths he change of ranspor regimes occrs in boh ime and space. 3

14 Some characerisics of advecion seem o be nexpeced a imes. In his ime inerval, he nmber of paricles locaed in he eeh is sill relaively small. However he inflence of he paricles deparre ino he eeh deermines o considerable exen he spaial widh of he concenraion disribion peak. Also his phenomenon resls in a power law decrease of he concenraion disribion in he lef wing. A faser decrease han Gassians occrs, namely G x, exp /4 x/ in he righ wing. I shold be noed ha many ahors defined anomalos diffsion as diffsion wih a nonlinear growh of he mean sqared displacemen [4-8]. Tha definiion is no fll. For example, qasidiffsion is an anomalos ranspor regime becase he oal nmber of acive paricles is no conserved, alhogh he variance of he displacemen in his regime depends on / ime as js as in classical diffsion. ACKNOWLEDGMENT This work sppored by he Rssian Fondaion of Basic Research (RFBR) nder projec a. [] P. S. Isichenko, Rev. Mod. Phys. 64, 96 (99) [] V. E. Arkhincheev and E. M. Baskin, Sov. Phys. JETP 73, 6 (99) [3] S. Havlin and G.H. Weiss, Physica 34A, (986) [4] Shlomo Havlin, James E. Kiefer, and George H. Weiss, Phys. Rev. A 36, 803 (987) [5] K. V. Chkbar, JETP 8, 05 (995) [6] A. Iomin and E. Baskin, Phys. Rev. Le. 93, 0603 (004) [7] A. Iomin and E. Baskin, Phys. Rev. E 7, 060 (005) [8] P. S. Kondraenko, L. V. Maveev, JETP 3, 3, 494 (007) [9] P. S. Kondraenko, L. V. Maveev, Phys. Rev. E 75, 050 (007) [0] A. M. Dykhne, P. S. Kondraenko, L. V. Maveev JETP Le. 80, 40 (004) [] K. V. Chkbar, A. S. Romanov, P. V. Popov, V. U. Zabrdaev, JETP 33, 5, 80 (008) [] V. E. Arkhincheev, JETP 5, 4, 85 (999) [3] Shlomo Havlin, Phys. Rev. A 34, 349 (986) [4] D. Hernández, C. Varea, and R. A. Barrio, Phys. Rev. E 79, 0609 (009) [5] T. Geisel and S. Thomae, Phys. Rev. Le. 5, 936 (984) [6] E. K. Lenzi R. S. Mendes and C. Tsallis, Phys. Rev E 67, 0304 (003) [7] E. Barkai and V. N. Flerov, Phys. Rev. E 58, (998) [8] L. C. Malacarne, R. S. Mendes, and I. T. Pedron, E. K. Lenzi, Phys. Rev. E 63, 0300 (00) 4