Impact of the Horizontal and Vertical Electromagnetic Waves on Oscillations of the Surface of Horizontal Plane Film Flow

Transcription

1 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce Ipac of he ozonal an Vecal Elecoagnec Waves on Oscllaons of he Suface of ozonal Plane Fl Flow IVAN V KAZACKOV Dep of Enegy Technology Royal Insue of Technology Sockhol 44 SWEDEN IvanKazachkov@enegykhse hp://wwwkhse//ns?l=en_uk Dep of Apple Maheacs Infoacs an Eucaonal Measueens Nzhyn Gogol Sae Unvesy UKRAINE hp://wwwnueuua Absac: - Poble on aheacal oelng of a plane fl flow affece by alenang elecoagnec fel (unnng hozonal EM wave) s consee The paaec excaon an suppesson of oscllaons on a fl suface a he essenal nea foces ae sue usng he evelope oel an copue sulaon In a lnea appoach soe neesng new phenoena of he elecoagnec wave speang n a plane fl flow an s pac on excaon an suppesson of oscllaons on a fl suface ae nvesgae Key-Wos: - Plane Fl Flow Elecoagnec ozonal Wave Insably Excaon Suppesson Oscllaons n he plane fl flow Eale nvesgae pac of he vecal elecoagnec wave on suface waves on he plane fl flow speang on a sol plae has shown he feaues of he syse (see Fg an Fg) [-3] Fg Confguaon of he nuce elecc cuen fo vecal EM wave n a plane fl flow bu Fg Ceaon of he plane fl flow: - half-wh of vecal e an flow velocy l- sance fo he nozzle o he hozonal plae The esablshe paaees of elecoagnec (EM) waves ( z )exp ( kx y) ae poan z fo analyss of he acon on conucve eu an selecon of he conol fels whch supply an acheveen of he equese behavo of he conolle lqu eal fl flow Analyss pefoe has shown ha sho elecoagnec wave wh consan by х у aplue s avalable f an only f agnec fel s aply eceasng n e so ha s sla o he pulse packe exp k / havng abup back fon of ype Soluon of he bounay poble by he oel obane has been one The appoxae soluon has shown fo k / ( k - wave nube - ensonless fl hckness): 4k a exp k exp F We k k expk F We ee s aplue of he fl suface wave Analyss of he soluon has shown ha vecal elecoagnec wave causes peubaon of he fl flow suface whch has a pa sla o he excng wave foce an he ohe one ffeen By eal k (spee of he wave speang s equal o he velocy of fl flow s ) he suface wave s alone (sla by fo o he excng foce) By he exponenal gowng of he Oh Ga aplue of suface oscllaons s avalable by nfluence of he suounngs s Oh Ga only quanave The aplue of fl suface ISSN: Volue 6

2 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce oscllaons exce by EM wave s popoonal o a an paaec esonance s acheve by: 8Be 8 / Ga k Oh k / Be 8 Ga / Oh ee Be Ga Oh ae he Bachelo Galleo an Ohnesoge nubes especvely The sngle ynac ceon s expesse hough he paaees of syse an wave nube k Be / Ga gbb / Oh b g / [3] Now he hozonal EM wave pac on he plane fl flow s n focus of he pesen pape Moel fo hozonal EM wave pac on plane fl flow oscllaon Conse EM wave wh only у-coponen (unnng by х fel o us alenang by f k=): y = (z) e kx () ceang poneoove foce (geneally also unnng) on axs z (unlke he pevous case of vecal EM wave whee was ece agans he unpeube flow) Accong o a sable sae [3] he funcon f x 4 / u x/ b us / z s z nouce whch s n he fae of oel consee onoonous an weekly-gaen In a fs appoach s possble o pu f an fuhe esae an eo of such splfcaon of a ask Then he syse of lneaze MD-equaons of he peube fl fo he consee case n nonnucve appoach can be pesene as follows: u + u p f = u u x x - y w u x z x z x w + p z = w w - y () x z z y + y f = y y x x z Coesponng bounay conons on sol plae: u z= u = w = ; z = z = w = + f x w p = + F z + - (3) y We x As naue of paaec oscllaons n a syse n lnea appoach oesn' epen on nal aplues of he peube fl he nal conons can be avoe seekng fo a lnea syse esponse n a wave oe () wh he ouble fequency an wave nube Theefoe fo () (3) he followng bounay ask fo he peubaon aplues s obane: w + p = ( k ) w u + ( fk) k u= w k u = k(p + ) + = k k f ; z= u=v=; z =- = ; (4) u z = = = w ( kf - ) p = + w + 4k F We Fo splfcaon he aplue peubaons of hyoynac paaees ae esgnae by he sae vaables lke naely hyoynac paaees as befoe Poneoove foce can be negave (pessng a fl o a plane) only n case of exsence of cuen n a fl Conseng showave flucuaons of a fl suface ( k ) an assue k k k (weak o oeae ncease of flucuaons on x) we cay ou he es assessen n he bounay ask (4) an ge s followng appoxae soluon: u= c exp ( ) q q z = exp = exp k( z ) k p= exp k ( z ) + exp(q z)+ w c (q z) c k q (5) q whee c =cons q - he oos of chaacesc equaon k - elecoagnec wave nubes 4 - he suaon nex ee: f k k = k f 4k k f k = k f 4k = c q = fk k n fk n = q g g3 g n ( g g3) q 5 ISSN: Volue 6

3 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce n q exp q q exp q q g g g 4 sh q g g q k = k + k (6) 4 3 = g5q qq sh q qch( q ) q exp( q ) 4 = g5q qq shq q chq q exp q whee q wh o nexes ae obane when n fon of squae oo s "+" wh even when "-" coespon he op sgns n he foulas 3 - o boo sgns Funcons g 4 n (6) ae: q g expq k 3exp q q expq 4k g5 exp k fk F We 4 Wh accoun of (4) he aplue of peubaons: expk k q shq chq q shq chq fk (7) whee fo s seen ha unlke he consee cases a a hozonal coss fel aplue of a supefcal wave s I can be explane ha econ of poneoove foce acon s vecal (up n case of lack of cuen n a fl excep nuce cuen) an heefoe wh ncease of neal foces also he conbuon o enegy pupng n waves Paaec esonance of suface waves Paaec esonance of he syse s possble by fk k f / when he phase spee of suface wave speang conces wh velocy of he unpeube (sable sae) fl flow The Egen oscllaons ae eene wh chaacesc equaon whee fo spesve equaon yels q k shq q exp q q chq qsh q q q q q q exp q q chq chq q exp q q q q q exp q (8) q ch q q q exp q 3 q q q q qch q qsh q shq q exp q exp q q exp q q q q q ch q chq chq q q q shq q shq qch q Ga 4k / Oh 3 q ch q sh q exp q fk ee can be seen ha he Egen oscllaons geneally spea on a fl wh speson as / k The esonan pa ( k ) as s easy o check geneally oesn' sasfy conon (8) e he paaec esonance s exce no a Egen fequency an beses he suface fl oscllaons ae no spesng n hs case on a conas o he Egen oscllaons because c / k f an c / k The las geneally speakng can be ncoec as fa as fcon of lqu on a plae epens on he oe of s flow; howeve whn he accepe oel s gh By / fo (8) afe he es esaon he followng spesve equaons esul: k fk 5 cos sn exp Ga 4 k / Oh k fk (9) g fk k fk whee I s assue fk fk I s easy o check ha a he above-sae paaees of syse noceable ncease of aplue of Egen oscllaons n e s possble only une conon ohewse he chaacesc e of peubaon evelopen s less han exsence e of a fl whch ege (he b fo unloang fo suface foces) beaks up owng o Raylegh nsably Theefoe fo (9) he followng syse s obane afe fuhe esaes an splfcaons: Acg n fk whee fo fk fk () 6; 346; 6437 ec (egula se) Soluon () s sasfe by k (sho-wave peubaons) By boh soluons conce an appoxaely coes o fk so ha n hs case One has o keep n n ha k canno be bg because n he sall-aplue lnea appoach he cuvaue of a suface canno be bg The phase an goup wave spees ae especvely fk c f k k fk C f / ISSN: Volue 6

4 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce whee fo follows ha he Egen oscllaons ae paccally no spesng an phase an goup spees ae appoxaely equal o unpeube velocy of fl flow (kneac waves) Acon of he consan elecoagnec fel By assue k wh accuacy of k/ f fo (6) follows q k q3 fk An hen conseng f esuls n k 4 f k k k k q3 fk k q4 k f Slaly fo (6) (7) wh accoun of aboveenone afe esaon of he es an fuhe splfcaon yels exp k 4 whee: exp k cos k exp k sn k k Ga 4k / Oh 4k k / Oh Now eal pa of he fl suface s expesse as exp k 4 () exp kx cos kx sn kx The expesson () hus obane shows ha by whle by Ga/ Oh s go Paaec esonance s possble n case of consan elecoagnec fel as seen fo () By k follows / k heefoe fo bg enough values k (sho waves) wh ncease of k he aplue of he excng elecoagnec fel us gow popoonally Bu hs case s paccally possble ue o echncal peens wh ceaon of alenang fel by х wh a sall wave lengh In geneal elecoagnec excaon of oscllaons of conucve fl suface une ansvesal hozonal alenang by х fel s eene by Ga Oh an wave nube k As an llusaon of hese esuls he calculaon pesene n Fg3 I was copue by () fo: 3 k k 5 u 443/s b k k k f 3 I coespons o Ga k / Oh an Ga k expesson () was splfe: 4k cos kx Ga sn kx 4 Ga exp k x whee s us quaac ynols nube copue by he spee of he fven waves as chaacesc flow velocy Appaenly fo Fg3 benng aoun oscllaoy fons have axes alos syec elavely he op an lowe bounaes Influence of agnec ynols nube n he consee oel saeen s nsgnfcan an s shown geneally on cos kx whch howeve s uch coeffcen a less han a coeffcen a sn kx ; hee ae Oh / 4 Ga 83 Fg3 Benng aoun oscllaoy fons of a fl: agans longunal coonae fo 3 values of fel =9* -4 9* -3 an 9* - Fallng of a fl flow velocy wh sance On he sances of х=8 an х=64 velocy of he unpeube fl flow eceases coesponngly by 5% an % heefoe wh accuacy suffcen fo pacce follows ha he assupon abou consan unpeube flow velocy s coec by < - when elecoagnec fel s weak enough Calculaon wh accoun of he funcon f(x) s pesene n Fg3 by ashe lnes In case of fl flow wh cuen n exenal agnec fel of he sae sengh when econs of he agnec an gavaonal foces conce he aplue of peubaon on a fl flow suface s lowe abou es copang o he only nucon suaon 3 Influence of he flu vscosy Fo nvsc appoach ( ) fo (3) eucon s appoxaely by one ecal oe n copason wh a vscous case an nvsc 9 appoach s coec a ha s alos uneal Thus he elecoagnec fel of he fo ISSN: Volue 6

5 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce expk k x causes pocess of wave foaon on a fl suface wh wavelengh wce bgge han by an sall shf on x Incease on x happens une he law exp kx Fo exaple he fel wh nal nucon of B 3T ( 9 ) causes peubaon whch aleay a x=95 eaches value (fl ecay) A snegaon of a fl he ops ae foe une he Raylegh law o a he acon of echancal flow ves wh foaon he ops by aee 4 The case of pogessve (unnng) waves Fo eal k k k / fo (6) follows q4 k q3 cos n 5 cos n 5 whee n=3 Then accounng (7) yels qg 5 exp q 4q q 3expq F q q F whee fo n case of s go sple expesson fo he peubaon aplue exp k F 8 F F An a eal pa of he fl suface peubaon s cos kx sn kx 4 4 F () The secon e of () n squae backes s sall; heefoe can be oe n engneeng calculaons whch splfes () as follows sn kx 4 whee fo follows ha / so ha wh ncease of he fel fequency he phase spee of he wave s gowng (he spees of he elecoagnec an suface hyoynac waves conce) an s aplue s fallng own The wo suface waves ffe n aplue (6 3 es) an n a phase (sall a 4 / F ) The paaec esonance s absen owng o wha conseng sae s possble o conclue ha a hozonal elecoagnec wave can exce snegaon of a fl flow only on he se wavelengh an unlke a case of a vecal wave an aplue of wave peubaon s efne only by fequency The exce oscllaons on a fl suface have a wavelengh nepenen of x hus hey on' 3 4 spese In a case of / F he pocess of wave foaon an wave speang n a fl s efne only by k an fo fl hckness epenence s lnea ( ) As seen fo () n hs case no fl flow ecay occu because an aplue of supefcal waves oesn' change n space-e an he waves of ae beyon he lnea heoy Excaon of oscllaons on he fl flow suface by vecal elecoagnec wave Densonless equaons fo hs case ae u u x x ( ) ( p ) u v v x y ( ) ( p ) v w w p ( ) w x x x x y vv (3) ; Bounay conons ae sae as follows: z= v v w x ; w u w v (4) e e x z y z w w p p ; F We x y z z z = v ; z = v ; (5) v p (6) whee u / F u / gb ub / ub / We bu / ae he fven Foue Webe ynols an agnec ynols nubes especvely; () an - ensonless oal hckness of he layes of conucve an nonconucve ea a/ b / / = We choose he chaacesc scales of lengh velocy e pessue an nensy of an elecoagnec fel especvely b u b/ u u an pu fo splcy u cons The fs bounay conon (5) coespons o a fl speang on a sol suface Fo fee fl flow nsea of hs conon by z= shoul be consee also bounay conon of conugae ISSN: Volue 6

6 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce ype on he neface wh suounngs Because ua / s a velocy of speang he fven waves he fven nube can be pesene as a ao of chaacesc veloces: u / u Soluon of he bounay poble Assung he sall-aplue peubaons we seek soluon of he bounay poble (3)-(6) as a lnea esponse of he syse o an exenal acon Followng he supeposon pncple he exce by elecoagnec wave z ( z )exp ( kx y) suface waves n a fl flow ae consee as popoonal o he funcon exp kx y wh coesponng aplues funcons of e Soluon of he bounay poble s convenen o seach wh use of he negal ansfoaon ehos [4] We apply o he PDE aay (3) he Laplace ansfoaon by e Accounng he above-enone an he nal conons (6) we coe o he followng syse of ffeenal equaons n he ansfos: Q 4 ( k ) Q s k q P Q W ( ku V) Q a (7) 4 ( k ) Q s k Q q P W 4 ( k ) W s k W P whee U V W ( s) P( s) () s ae he Laplaceansfos of funcons u v w ( ) p ( ) () funcon () s eene by exp k k / Accong o he above-enone assupons an foulae equaons s- paaee of negal ansfoaon The fs an he h equaons of (7) ae pesene fo splcy n a sybolc fo: Q V Q V by q an Q U Q U by q k The bounay conons (4) (5) n ansfos: z = U V W ; z = U V W ; (8) z= k W W P P 4 Z F We U Q Q qw qw (9) U V V W W s k Z whee Zs - he Laplace-ansfo of he aplue of fl flow peubaon The soluon of he bounay ask n ansfos (7)-(9) oesn' pesen pncpal ffcules howeve s so cubesoe ha he eun ansfoaon o he funcons ognals eans applcaon of he nuecal ehos Thus fo a fee fl (no esce by a sol suface) fo he fl flow suoune wh he lqu o gaseous eu fo he op an fo he boo he bounay conons (8) ae eplace wh conons of ype (9) owng o wha he soluon s songly coplcae Fo convenence of he analyss of he consee class of pobles by paaec excaon an suppesson of oscllaons on a suface of he conucve fl flows speang n non-conucve ea an fo eecon of he an egulaes of such physcal syses we conse he sho-wave peubaons an suppose k= Such case has poan applcaons n pacce of ceaon he pespecve fl MD-ganulaos [] The wen above allows sgnfcanly splfyng a ask an he syse (7) gves 4 W W ( q ) 8 4 k q W whee q 8 k ( s k ) Soluon of hs syse s W n g nz exp whee fo each value of he su s aken by nex n 4 The values g n ae g k g 3 g g q Consans of negaon n g 4 q () ae copue by subsuon he funcon W z n he bounay conons (8) (9) wh accoun of (7) In a lnea appoach yels U V W / an fo n he followng lnea algebac equaon aay (LAEA) of he 8-h oe s go: exp g s k Z 4 n n nexp gn s k Z ngn exp gn ngn ngn exp gn g n () g g 8 k exp g ngn n n n n n n Afe obanng he n n n he funcon Zs copue fo he las equaon (9) wh accoun of (7) Soluon of he syse () wh accoun of () s gong as follows Two equaons conanng s ISSN: Volue 6

7 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce boh n an ae solve an hen he eanng n syse of sx equaons s solve eang he values 4 as paaees afewas s eene 4 4 fo he syse of wo equaons Max of he fs sub-syse has he vew A whee А В ae nonzeo quaac aces of he h oe (wo B ohes ae zeo) an aonally С- colun of he fee es Such ax allows easy copung he an as well as aonal eenans hough eucon of he o he pouc of coesponng eenans of he h oe Thus he esul s: / 3; () A q k A q q q k q q k q exp whee he an an no eenans ae: g exp q ch g q shg g s k Z chg exp q q shg q g s k Z q exp g chq g shq q ch q g sh q q exp g q g s k Z exp q g q g s k Z (3) exp q gch g qsh g g exp q g shq q chq exp g q exp g q exp q shg g exp q exp q 4 chg 4 g s k Z 3 exp g qch g gsh q q exp g 4 s k exp q g g q Z exp q exp q q exp q A g exp g g 3 3 A 6k exp g exp g exp 8 exp q k q q k q In vew of bulkness of he eceve expessons () (3) a hs sage s expeen o cay ou he assessen of es allowng o splfy a ask fo each specfc physcal suaon an fuhe o o s soluon n each case sepaaely ha allows applyng he analycal ehos effecvely We conse one of such suaons fo he exaple Le / k / whee k - eal pa of he wave nube k Ths conon coves que bgh egon of he wave nubes an coesponng class of he wave pocesses has elaon o any paccal suaons The ange of vaaon s aken 4 3 heefoe k An as fa as k 5exp k hen ch q q Sla splfcaons ae n ohe expessons hen fo () (3): g s k / 4 4 / 3; 5 q q g 8 g q q g k q s k q g 4 q q s k Z 8 k g q q g s k g q 3 4 q q q g q gq s k Z g q exp g q g g s k Z (4) 4 q g q 4 g q s k Z q g s k Z q q g 3 4 q g exp q g s k Z q q g g exp q g q g exp g q 4 g s k Z 3 q g exp g q q 4 s k Z 4 4 Fuhe fo he las equaon of he syse (9) wh accoun of (4) he followng equaon fo Laplace-ansfo of he fl flow peubaon Zs s go: g q s k q s k g q g q 6k g q g q g q 8 k q g q g q g (5) s k s k Z Z F We The hn fls ae go by subsanally hgh flow velocy when neal foces songly peval he ag foces ue o fcon on a sol plane whee he 3 fl flow speas heefoe (by b 3 u = /s fo os eal els ) an expesson (5) afe esaon of he es yels: s k 4k Z s k F We s k ISSN: Volue 6

8 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce Inegal-ffeenal equaon fo fl suface oscllaons Now pefoe he evese Laplace-ansfoaon yels he negal-ffeenal equaon escbng he paaec oscllaons of he fl suface exce by elecoagnec fel havng z- coponen: 6 k k F We expk expk (6) Wh he assupons ae he funcon une negals n (6) ae hghly-oscllang heefoe hese negals ae sall an he equaon (6) s aveage by he secon schee [5] As a esul wh accoun of he EM wave fo he aveage ffeenal equaon s go he soluon of whch has he followng fo: 4k a exp k exp F We k k expk F We (7) The appoxae soluon (7) of he equaon (6) hus obane s close enough o he exac soluon on he nfne neval of e an s fely gows wh ncease of ae hee s k / ) k (accong o he assupons 3 Suppesson of he Egen oscllaons The soluon (7) accounng he above s ansfoe o he followng fo: a 4k exp exp 4 k / 4 k / exp k (8) whee fo one can see ha suppesson of he Egen oscllaons wh he wave nube k an fequency s avalable only by eans of he unnng fel havng he sbuon spee ohe han he spee of he oveen of a fl k s Ohewse boh excaon an suppesson of flucuaons ean bg enegy expenses (excep a esonance) 4 Fee fl flow n non-conucve eu Soluon of hese bounay ask s sla o he consee ask an oes no have any pncpal peens Bu ue o cubesoe expessons obane he evese negal ansfoaon ay be coplcae a lo Theefoe fo analyss of he feaues of consee class of asks soe specfc physcal suaons ae consee such n pacula whch ae useful fo pacce of MDspeses Le fo splcy k= as befoe an conse a case of neglgbly sall nfluence of he suounng eu Boh assupons ae physcally coec because he speses ae ofen place n a gaseous aosphee o vacuu an equal wave nubes gve oe egula fl ecay no he ops of equal szes Then wh accoun of he above slaly yels W nexp gnz n 3 g 4k s k n exp g 3 n Z s g g exp g g 4k s k n exp g3 n Z s g g expg The Laplace-ansfo fo he fee fl suface s: s k s k 3 g g 4k g g 4k g g 4k (9) 3 3 g g 4k g g 4k k Z() s 4k F We By he equaon (3) can be splfe keepng he es of oe / whch follows o: 5 k s k 4 ( ) k k 5s s k k Z() s F We s k s k 5 Z s whee fo applyng he evese ansfoaon s: 5k 8 k 7k exp k k Jk J k F We (3) whee J k s he Bessel funcon of zeo oe ee a ffeenaon of negal wh a vaable op l he aonal nal conon s nouce: / Fo hee s vsble ha a he eal k nfluence of a fel on a fl peubaon s nsgnfcan especally a k as fa as he negal n he fs pa (3) s fas-oscllang an heeof he gh pa quckly aspes o zeo Naely hs pa causes a paaec pupng of peubaons of a fl (he lef pa efnes he Egen oscllaons) ISSN: Volue 6

9 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce Soluon of he negal-ffeenal equaon has been obane by he eho of aveagng of negal-ffeenal opeao by he fs schee [5] The aveage equaon s go n he fo 5k k J k (3) whee s k k J k F We 8 k 4 k k exp k 35 exp k Soluon of he aveage equaon has he fo: 5k Aexp k B 5k A( ) exp k B J k (3) 3 Paccal exaples To conse he soluon obane suppose onoonous nensy of an elecoagnec fel e k whee - a eal nube (by > nensy of a fel eceases on x an y an gows n e a <- on he conay) Then funcon has only he eal values: exp I exp We F whee I s he ofe Bessel funcon of zeo oe Chaace of gowng (eceasng) he suface wave s eene by sgn an value of he eal pa of / o by value hee The soluon (3) s coec fo lage enough ynols nubes wha s no always cae ou n pacce heefoe also a case of sho waves k s consee an an assessen of he es n equaon (9) s ae base on hs assupon Then he appoxae negalffeenal equaon s go as follows 6k k k (33) 4k exp k F We whee fo seen ha equaons (3) an (33) ae sla by sucue an paaec pupng of he enegy peubaons s coplcae wh pesence of he hghly-oscllang es by he exenal excng foce: by ncease of k he equaon (33) eflecs oe pecsely he nvesgae physcal pocess an fo he wave excaon hghe an hghe agnec fel s eque By coec fo any paccal asks he equaon (33) s solve usng he aveage negalffeenal opeao by he secon schee [5]: k a A A k 4k exp exp k (34) whee a - aan nube 64k k A 4 k F We 4k Soluon of he aveage equaon (34) s a k exp 4 exp k Ak D (35) whee D A k By he assupons ae appoxaely D Soluon (35) of aveage equaon (34) s close o he soluon of he negal-ffeenal equaon (33) on he neval c whee с=cons an egee of poxy of hese soluons ases wh gowh [5] Accounng n geneal case k k k fo he obane soluon fo vecal EM wave coes k k exp k x y k x y c (36) exp k x y c exp k x y whee c 4k / s phase spee of he wave speang along he sagh lne y=cons-х Expesson (36) shows ha n hs case he elecoagnec wave causes wo waves on a fl suface: sla o a wave of he evolng foce an ffeen fo speang wh velocy of unpeube fl flow The aplues ae: a exp k x y Ak D 4 exp k k (37) Thus he aheacal oels evelope an soluons obane on he base fo a se of physcal suaons show ha n a lnea sall-aplue ISSN: Volue 6

10 Inenaonal Jounal of Ccus an Eleconcs hp://wwwaasog/aas/ounals/ce appoach only he waves of wo kn ae avalable wh he aplues (37) By c k when fl flow velocy s zeo s go s an hee s only one wave Paaec esonance n a syse as shown by (37) akes place une fulfllen of he conon AD o ka an fuhe follows: k 8 5 k (38) Ga 4Be / k 4 / Oh Ga k 6 8 k whee Ga Be Oh- Galleo Bachelo an Ohnesoge nubes eene by physcal popees of ea an fl hckness (nepenen of flow ege) k / k - ynols nube efne by he lengh of coesponng esonance peubaon Coelaon (38) allows esablshng he elaon beween esonan values k an coesponng ynols nube k as well as he ohe efnng cea of he syse Then he elaon k k / can be esablshe as well The foula (38) can be subsanally splfe n any poan cases eg n MD-ganulaon he followng esaons ae val: k Ga Be Oh Ths yels: 8 k 8 k 8 k k Ga Ga 6 Oh The Egen oscllaons wh accoun of (34) ae 5 exp k (39) A whee Theefoe k 8 k A Conse Egen oscllaons fo eal values k an hen afe esaon of he es an accounng he assupons ae an he elaon acg s avalable o 6k 6k coe a 3k k F We 3 k 3k 6k 6 k (4) 4 6k base on conon e The conon (4) answes ncease of Egen flucuaons of syse n e (evelopen of fl nsably) A bg enough k conon (4) s no sasfe e sang fo soe value of k all subsequen flucuaons of salle lengh becoe fang n e A k s possble o ge he followng appoxae foula: e k Ga 8 k / Oh Ga 8 k / Oh 8 k Ga 8 k / Oh 4 (4) whee fo follows ha ecease of peubaons n a fl s possble only on conon ha s exeely selo n pacce Fl s unsable bu appaenly fo (4) he ncease ae of peubaons s sall a ha paccally eans uselessness of suppesson of Egen oes by paaec excaon of fl flow snegaon on he se wavelengh 4 Conclusons by he esuls obane The esuls obane ay be use fo analyss of paaec excaon an suppesson of fls ozonal pogessve elecoagnec wave exces on a fl suface he wave wh aplue ; paaec esonance n a fl flow by acon of consan agnec fel s possble n conas o he alenang fel; vscous foces ae poan: applcaon of nvsc appoach lea o naccuacy n calculaon of a wave aplue 3-4 ecal oes; paaec esonance s possble by eans of hozonal elecoagnec wave The peubaon aplue n case of k k / oes no epen on he wave nube an s eene only by fequency of a fel feences: [] Kolesnchenko AF Kazachkov IV Voyanyuk VO Lysak NV Capllay MD flows wh fee sufaces Kev: Nauk Duka p (In Russan ) [] Kazachkov IV Paaec wave excaon an suppesson on nefaces of connua/ DSc hess Kev In- Elecoyn p [3] Kazachkov IV Elecoagnec excaon of paaec oscllaons n a lqu eal fls speang ou n non-conucve ea- Pepn of Ins Elecoyn Uk Aca Sc- Kev N454-5pp (n Russan) [4] Tane CJ Inegal ansfos n aheacal physcs Lonon: Mehuen IX 95 [5] Flaov АN The ehos of aveagng fo ffeenal an negoffeenal equaons- Tashken: Fan pp (In Russan) ISSN: Volue 6