465 A pliio of HEMIAL ENGINEERING TRANSATIONS OL 57 07 Gs Ediors: Sro Piri Jiří Jromír Klmš Lr Pi Srfim Bklis opyrigh 07 AIDI Srii Srl ISBN 978-88-95608-48-8; ISSN 83-96 Th Ili Assoiio of hmil Egirig Oli wwwidii/ DOI: 03303/ET75745 Th H d Mss Trsfr Modlig wih Tim Dly solod G Soroki Adry ymi* Alky I Zhro d yhsl Rik Adry D Polyi d Bm Mosow S Thil Uirsiy ory Bmsky l 5/ 05005 Mosow Rssi Dprm of hmil Egirig Mosow Polyhi Uirsiy Sry Bsmy /4 Mosow Rssi Siifi Rsrh Isi of Rr Idsry poslok NIIRP Srgi Posd Rssi d Ishliskii Isi for Prolms i Mhis RAS pr rdskogo 0 956 Mosow Rssi 958@lisr Nolir hyproli rio-diffsio qios wih dly i im r isigd All qios osidrd hr oi o rirry fio E solios r lso prsd for mor ompl olir qios i whih dly rirrily dpds o im E solios wih grlid sprio of rils r fod or spil ss w solios i h form of rlig ws r oid som of whih rprsd i rms of lmry fios All of hs solios oi fr rirry prmrs so h o s hm o sol modlig prolms of h d mss rsfr wih rlio phom Irodio Proli qio of h- d mss-rsfr hs physilly prdoil propry i ifii disr propgio r whih is o osrd i r Solig o-sdy-s h- d mssrsfr prolms i is ssry o k io o rlio phom ssoid wih h fiiss of h r of h d mss rsfr s for mpl Dmirl 007 Th hrml d diffsio rlio ims ry i rmly wid limis from millisods or lss o srl s of sods d shold k io o i solig my h d mss rsfr prolms Polyi d ymi 03 Th sod impor fr of oliory prosss ildig h- d mss-rsfr prosss wih hmil orsios is h i h grl s h r of riios i h dsird qiis i hmil iologil physiohmil hmil girig d ohr sysms dpds o oly o h s h gi im poi lso o h ir prios olio of h pross Jo l 00; Poks l 05 Ths sysms r lld hrdiry sysms I h prilr s whr h s of h sysm is oly drmid y prilr im poi i h ps rhr h h ir olio of h sysm h sysm is rfrrd o s dlyd fdk sysm Sysms wih dlyd fdk r frqly modld y rio diffsio qios i whih h kii fio h r of hmil rios dpds o oh h sogh orio fio = d h sm fio wih h dlyd rgm w = τ Th spil s of w = fw hs simpl physil irprio i h- d mss-rsfr prosss i mdi wih lol o-qilirim h iril propris i h sysm dos o r o io isosly h gi im poi s i h lssil lol qilirim s i rs y h dly im τ lr Solig o-sdy-s mss rsfr prolms i hmil girig i is ssry o k io o rlio phom ssoid oh wih h fiiss of h im of rsfr prosss τ d wih h fiiss of h ims τ of hmil orsios d/or h mirokii irio w diffr phss h form sigl rspor mromdim solios o h followig o-lir hyproli rio diffsio qios r drid d lyd i his sdy s lso Polyi l 05: w w Pls i his ril s: Soroki G ymi A Zhro AI Rik Polyi AD 07 Th h d mss rsfr modlig wih im dly hmil Egirig Trsios 57 465-470 DOI: 03303/ET75745
466 whr - diffsio offii - oordi I shold od h s prilr s τ = 0 osidrig qio ilds proli qios wih dly Mor ompl olir rio diffsio qios wih ril dly of h grl form τ = τ will lso osidrd I h dgr s τ = 0 i for h proli qio ri solios o Eq wr oid for mpls for rllig w y W d Zo 00;sig ompl grop lssifiio y Mlshko d Moyo 008; sig mhod of grlid d fiol sprio y Polyi d Zhro 04 or kii fio of wo grl forms w solios of qio will fod llow W mphsi h for h firs im solios r oid for h qio wih wo hrrisi dly ims whih h diffr physil mig d whih ppr i diffr rms of qio Ths rsls grli prios solios oid y ohr hors Mhods for fidig solios Th mril solig of rios olir pril diffril qios d sysms of qios wih dly d diffilis h ris i his s r dsrid y Jkiwi d Zik-Kowl 006 I y s h grl disdgs of mril mhods ild: s irsl ppliio wh hgig h gomri shp of h oj h yp of flid flow rio kiis d ippliiliy i h prs of siglr pois E solios o olir diffril qios promo h r drsdig of h qlii frs of h prosss dr dsripio oiqss spil loliio lowp rgims I shold mphsid h dly ssilly omplis h lysis of qios d is for h ld o h isiliy of h sysms ig modld Jord l 008 Th rm solios wih rsp o h olir dly of pril diffril qios r sd i h ss whr solio is prssd s follows: - Th solio prssd i rms of lmry fios or rprsd i h losd form h solio is prssd i rms of idfii or dfii igrls - Th solio prssd i rms of solios o ordiry diffril qios or dly ordiry diffril qios or sysms of hs qios - Th solio prssd i rms of solios o lir pril diffril qios - Th omiios of solios r lso llowl Solio mhods d rios ppliios of lir d olir ordiry diffril qios wih dly whih r ssilly simplr h olir pril diffril qios wih dly r dsrid for mpls y Bllm d ook 963; Kg 993 A mr of solios o ri olir pril diffril qios wih dly s wll s sysms of qios wih dly whih r diffr from rio diffsio qios r gi i ppr Thh 0 I his sdy o sk solios o olir hyproli rio diffsio qio sh s Eq w sd rios modifiios of h mhods of grlid d fiol sprio of rils s for iformio hdooks y Polyi d Mhiro 007; Glkioo d Sirshhskii 007; Polyi d Zis 0 d h fiol osris mhod for proli dly qios Polyi d Zhro 04; for proli qios wih ryig rsfr offiis Polyi d Zhro 04 rom his poi o irmdi llios r grlly omid for h sk of riy Eqio dos o modl y prilr hologil pross I grlis h diffsio qio; oid o h sis of h ik's lw i h qilirim for oqilirim prosss whih ks io o hir ow rs of prrio propgios i h mdim d is hmil rsformios Th oid rsls do o rqir y rifiio si hy r mhmilly r 3 E solios o Eq wih kii fio h dpds o h rio w/ L s osidr Eq i h followig form: w w whr is rirry fio 3 Eqio wih ril dly Eq yilds solio priodi wih rsp o os si 3
whr d λ r rirry oss d h fio ψ is dsrid y h followig ordiry diffril qio wih dly: Eq lso yilds solio of h form p p 4 whr h fio ψ is dsrid y h followig ordiry fiol-diffril qio: 3 Eqio wih os dly Now w osidr Eq wh τ = os I his s Eq yilds h sprl solio s h prod of h fios of diffr rgms s Eq 3 Th fio ψ i Eq 3 is dsrid y h followig ordiry diffril qio wih dly: 5 Eq 5 yilds h prilr solio ψ = A β whr A is rirry os d β is drmid from h lgri or rsdl qio 0 Eq lso yilds solio of h form Eq 4 whr h fio ψ is dsrid y h followig dly diffril qio: 6 Eq 6 yilds h prilr solio ψ = A β whr β is drmid from h lgri rsdl qio 0 3 Eq lso yilds h solio p 7 whr h fio θ is dsrid y h followig dly ordiry diffril qio: 0 / ' This qio yilds h prilr solio θ = A whr is drmid from h lgri rsdl qio 0 Solio i h form of Eq 7 is h olir sprposiio of wo diffr rlig ws 4 L h fio 8 y τ-priodi solio o h followig lir hyproli qio: 9 from his poi o for h sk of riy h dpd of Eqs 8 d 3 o h prmrs τ d whih ppr i Eqs 9 d 4 is o idid pliily I h s Eq yilds h grlid sprl solio 467
0 whr is rirry os I show h h grl solio o Eq 9 sj o h formiod odiio of τ-priodiiy wih rsp o im hs h followig form: si os p si os p 0 D B A whr A B d D r rirry oss whih sris i Eq r org h org srd g if w s A = B = = D = 0 > N whr N is y posii igr Th followig prilr ss disigishd: τ-priodi wih rsp o h im solios Eq 9 h dy r gi y Eqs d A 0 = B 0 = 0 = D = 0 d = ; τ-priodi wih rsp o h im solios odd r gi y Eqs d = D = 0 d = ; siory solio is gi y Eqs d A = B = = D = 0 d = 5 L h fio 3 τ-priodi solio o h followig lir hyproli qio: 4 I h s Eq yilds h grlid sprl solio 5 Th grl solio o Eq 4 hs h followig form: si os p si os p D B A 6 7 Solios τ-priodi o h im h dy r gi y Eqs 6 d 7 = D = 0 d = Eqs d 6 7 r ry similr Howr i h firs s h firs sm gis from = 0 d i h sod solio i gis from = ; h ls of β r lso diffr 4 E solios o Eq wih kii fio h dpds o h diffr w L s osidr Eq i h followig form: 468
w w 8 whr is rirry fio 4 Eqio wih ril dly Eq 8 yilds solio of h form 9 whr 0 p p 0; si os 0 d h fio ψ is dsrid y h followig dly diffril qio: 4 Eqio wih os dly Now w osidr Eq 8 wh τ = os I his s Eq 8 yilds h sprl solio s h sm of h fios of diffr rgms of h form of Eqs 9-0 d h fio ψ is dsrid y h followig dly diffril qio: A = 0 Eq 8 yilds h sprl solio h is qdri wih rsp o : whr h fio ψ is dsrid y h followig dly diffril qio: 3 Th solio o Eq 8 h grlis solio of h form Eq 9 hs h form whr φ is drmid y Eq 0 d θ is dsrid y h dly ordiry diffril qio: 0 ' A > 0 Eq dsris h olir irio w priodi sdig w d rlig w 4 A = 0 h solio of Eq 8 h grlis Eq hs h form whr h fio θ is dsrid y h followig dly ordiry diffril qio: 0 ' 5 Eq 8 yilds h dgr grlid sprl solio whr φ is drmid y Eqs 0 d ψ is dsrid y h lir ordiry diffril qio: Mor ompl solios o Eq 8 drid sig h followig propry L 0 solio o olir Eq 8 d = ; y τ-priodi solio o lir Eq9 I h s h sm 0 3 469
470 is h solio o Eq 8 Th form of h fio ; is drmid y Eqs or mpl h rlig w solio 0 = 0α + β sd i Eq 3 s h solio o olir Eq 8 5 olsios or hyproli diffsio-rio qios wih im dly solios r oid i poil form wih irm ompd from rsdl qio of spil yp hrogh dly ims Nw solios of his qio r fod i h form of olir sprposiio of wo diffr rlig ws Th form of solios sisfyig iiil-odry prolms is slishd ri solios r dsrid for mor ompl olir rio diffsio qios sh s hos wih ril dly of h grl form τ = τ Th drid solios oi fr prmrs i som ss hr y mr of hs prmrs d sd o sol ri modl prolms d s pproim lyil d mril mhods for solig similr or mor ompl olir dly pril diffril qios Akowldgms Th work is sppord y h Rssi odio for Bsi Rsrh proj No 6-08-05 Rfr Bllm R ook KL 963 Diffril-Diffr Eqios Admi Prss Nw York/Lodo Egld Dmirl Y 007 Noqilirim Thrmodymis: Trspor d R Prosss i Physil hmil d Biologil Sysms Elsir Amsrdm Nhrlds Glkioo A Sirshhskii SR 007 E Solios d Iri Ssps of Nolir Pril Diffril Eqios i Mhis d Physis hpm & Hll/R Bo Ro USA Jkiwi Z Zik-Kowl B 006 Sprl olloio d wform rlio mhods for olir dly pril diffril qios Applid Nmril Mhmis 56 433-443 Jord PM Di W Miks RE 008 A o o h dlyd h qio: isiliy wih rsp o iiil d Mhis Rsrh ommiios 35 44-40 Jo D ss-q J Lo G 00 Edd Irrrsil Thrmodymis Sprigr Nw York USA Kg Y 993 Dly Diffril Eqios wih Appliios i Poplio Dymis Admi Prss Boso USA Mlshko S Moyo S 008 O h ompl grop lssifiio of h rio diffsio qio wih dly Jorl of Mhmil Alysis d Appliios 338 448 466 Poks BG Krlo SP ymi A Nkrso DA 05 Diffsio phom i gls hmil Egirig Trsios 43 68-686 Polyi AD Zis 0 Hdook of Nolir Pril Diffril Eqios hpm & Hll/R Bo Ro USA Polyi AD Zhro AI 04 Nw grlid d fiol sprl solios o o-lir dly rio diffsio qios Iriol Jorl of No-Lir Mhis 59 6 Polyi AD Zhro AI 04 iol osris mhod for osrig solios o dly rio diffsio qios d mor ompl olir qios ommiios i Nolir Si d Nmril Simlio 9 47 430 Polyi AD Zhro AI 04 Th fiol osris mhod: ppliio o o-lir dly rio diffsio qios wih ryig rsfr offiis Iriol Jorl of No-Lir Mhis 67 67 77 Polyi AD Mhiro A 007 Hdook of Mhmis for Egirs d Siiss hpm & Hll/R Bo Ro USA Polyi AD Soroki G ymi A 05 E solios d qlii frs of olir hyproli rio diffsio qios wih dly Thoril odios of hmil Egirig 49 6-635 Polyi AD ymi A 03 Diffril-diffr h-odio d diffsio modls d qios wih fii rlio im Thoril odios of hmil Egirig 47 7-4 Thh J 0 Symmry lysis of h ohomogos iisid Brgrs qio wih dly ommiios i Nolir Si d Nmril Simlio 7 4978 4987 W J Zo X 00 Trllig w fros of rio diffsio sysms wih dly Jorl of Dymis d Diffril Eqios 3 65 687