Coco a Dffall Ha Naow Slo ja ala so: D. Co k Ca Us of zoa S/S
Coco a ffall a aow slo of fl ca sla a ff s of bao o os of fl a os of slo. basc cl s a fl a o wall wll s o s cas a a fl a cool wall wll fall. I a f slo as a fl acclas a o accla ss focs o a co back ow cool s. s ss a cclao as sow: a a b cosa o lal ca w. a as wll a b cosa o soal. o of cooa s ak o b c of slo a so s fl lo. s wll b w as -soal so o o as z. D coco s bo a a asf a a fl cacs obl a as sc w wll bo qaos fo fl oo a qao fo a asf. assos w a a. ossq Fl s aao ol affcs bo foc. Vscos al coc a a caac a cosa. Ds as w a as ρ ρ - β - f o qaos slf o fs qao s sl cosao of ass. s o c s s a ossq fl s o s c. l wo qaos a slf Na- Soks qaos o fo ac so. las s a qao. cos ω Τ O f f κ µ β ρ ρ µ β ρ ρ
Fo al a f slfcao w o-soalz s qaos l scal of slo a ffc a fl scos. solss qaos a / / / f υ ρ υ υ f s ak as aa a of wo walls wc ss a o. ll qaos s af wll b solss a sa wll b o. a a asof a al bs f as: κ υ υ β υ β W also f Ral b as oc of asof a al bs υκ β Ra fal oal of s sac s o sa flow a sabl of cas wc a a ozo. How ao oal s o o a colao of all o cass bo sa sa a as so sls a asl accssbl. lso as o ss oc ozo a obl o sc a a s bs o sa o w basc cas a sccssl bl o.
W all fos a qaos la o fs cas w sall a s sa sa solo of cosa a a cosa a walls. Fo a f slo f o w s asoabl la w ca aoa flow l os as aoal a as f slo was fl lo. I a cas w b b ass a aalll flow solo so ol flow s co. cas slo s fl lo a wall a s cosa w a ass a s ol a fco of. w V V O fo qaos a cosabl slf ow o W ca al sol fo. W o slo o a aabls f as a boa coos q a a fo Rlac back o 5 W kow a s ol a fco of so fo / ca ol b a cosa. a as sc w a s qao o s sso fo.
5 6 f a wc w a lf w a cbc oloal sso fo loc ofl w als o. o sl coo a wall fs o boa coos so a loc s zo a. 6 6 6 8 8 o ss w a alo co. ass s b cos a so s al sol b zo. / / 6 / / 8 / / 6 8 5 6 sl a ss a s zo s a c sl of a cos o a fc o b o a. a fall of fl s ab o boac focs ca b a cas a o ss a s cssa o fl. o caacsc of s sl s a s a o fco. s of obl s s wa o wol c. s o aso fo loc o fao o s o o. lso asof b s ol aa a affcs flow. oac os o o a. How w kow a laboao o a. lo walls s cas w sa o b cosa a bl of o fl a o a col fl a boo cas a sl
6 a a alo walls. W wll sol s cas a acall coa s cbc ofl w o colca ofls w wll lo. a w ass a aalll flow ol ffc s a ow a o lo js a fco of. I also as lal. w V V Rc o 7 8 lac 8 7 9 s ol a fco of fo a s of s qao s b a cosa. I fac w ca ak o b zo b kowl s o w kow a a cosa s b zo. ss sbo ca b fo as W ca al w ow solos fo a of cos ak al a - cos cos R cos cos R Ra D C D C Ra Ra
7 q a fcos b o cos a cos s o o. ol a wok s o f cosas. U al coos C C C C R R s solo aas o b q ff a a of fs cas. ca b cof a l as a s also os o zo cs o sa qaos. W ca a sas of bo a ol o. If s k cosa ol a of loc flow s ca b al b a. a cosa w look a flow as fo aos ol o Ra. fs o oc s a fo salls Ral b of obs al cbc loc ofl a la a sbo. s Ral b s cas a sbo bs o slowl slf. s acs loc a a sbo o ll o a l of flow a as Ral b cass f s ffc s ak o a aas o b al zo as s las s of as. fac ofls a oscllao w sall al s o. blk of flow a a aao a cof o os al a wall. lso a of s flows s c lss a cas w cosa a walls. ff ow of as sows s fo Ra /. a loc s abo 5 s a cas of ca wall a.
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9 W ow o cas of b ow w oc a oscllaos so a cos ω. a qao ca b ow asof b s a fco of a so o qao as a a soal. W a al o ss as ca b ak as zo. cos ω o o solo fo b W.Y. C & C.F. C 999 s N N 6 cosλ λ cos λ λ λ cos λ cosλ λ λ cos λ λ cos λ ω λ 8 / ω N ω ω N λ cos λ cosω λ cos λ cos λ λ λ cos Fo cas of w s sa sa w fo al. W coos ω a. ao o lo w also lo - a saal o s la ffc s as. sb loc as a a of abo 8.- wl a of oscllao s abo 7.- sow a ffc of a olao s o sa ooo as asof bs. s oscllao asof b aoacs cosa o a a as fqc of olao bcos sall aal coco as bco sabl. W wll o sabl of s flow la. λ
local s b o oc ozo o os ss. bfo w o a w wll a s flows as bao. W a qal o a a of walls a o loc flows w ol ow s ol o sa sa solos. a w b w sls cas. W ala kow sa sa solos so w o fo b a js w o qaos fo as a. a s sol w saao of aabls. lac back a sol fo U wll b o fco aso. solo fo a s obos b o ak alba a ll as w wll la aas as ls. No lo fo s. w s a as solo js oall cas o sa sa. lo s oabl a fo loc a acall wo as of ca. O s o al b lk fo a a o s o. U U U S S w V V C S C U C U C U S 6 Ra 6 Ra Ra Ra
Sol all o cass ols oc sa wa so l s o aa a j o os ffcl obl of bo a oscllao a o cosa walls a s wa w ca o. os aa o b c w ca o w s aal as. W ca asfo o of as. ˆ Ra ˆ ˆ ˆ ˆ ˆ Ra ˆ ˆ ˆ ˆ ˆ ˆ ˆ so w a lf w a s qao a oa ffal qao w oc coffcs. a cal cqs o wok fo b wll o b s. W wll s ao o l. W wll o s o of Cbs oloals as scb b Sa a W 99. Cbs oloals a a oooal s of oloals wc a q os ao a llcao aas ac o. a aclal wll s o sol sss w oc foc. Df Cbs oloals as f sf Cbs oloals as f as -. s ob oooal laos / 7 / w k k k w k U U U U U S S cos ω
also sasf follow laoss s s s s 8 If o f a ow co } } s} s } w s a cosa a f b abo ao laos o b: / / /8 /8 /6 / / /6 /8 O O O / / / / s also a llcao fola a sas k k k so fo wo fcos F f } } } } f f f } oc of F a s b F f } } } } } } w as a a cos of coffcs of : / / / 5 5
W s foao w ca sol fs o la ODE follow wa: f f f f } } } } } } } } } } } } } } Now s sl o sol fo coffcs of as: f } } I o solo ss. Noc a fola fo a ols coffcs a wll as - wc s s o aoao of. Ncall w wll fo ow sl o s s as zo. coffcs of a oba sl b oooal laos. w w o llsa ow s woks w sow sls of wo als o a. o cass ocall w b of s sa wa os fo al w a scal o. I calcla f a coffcs o olac fo aos was s o -6 a as sc s s w b of s sos ak a ffc. a so sacs wc s o fals o co o asw fo al w f 5 cos 5 5 5. lo f cos5 5 woks f. s of I a b abl o f a scfc ca a wll coc o o. os lkl aso s a fo ca f s osso of o s a fac cao b safl o. s ow ss o b co a a o a fo ow w wll co o s s o as s l ss s s all obl a s coc.
5 fo w ca s s cal o o sol o PDE s w s c o a ss of ODE s. W ca accols s alk os. W of o sos sll o follow U U U so ow w a wok o al -.
6 f sal sccssfl al fcos w fall ca o coos faco - s a o s zo a boa a s s a sf Cbs oloals. Now w lac o qaos a oc. ow sa of ll b o sa al fco a a w ll b. w w j j j j s qs a w a so sso fo fs a sco as of Cbs oloals. fs a ca b as a a ls s folas ca b asl l s oloal llcao a so. Fo alk o w a ss of qaos. / R R
7 a R a acs fo ao: / 5 6 9 6 / 8 / 8 / / 8 / / 8 / 8 / / / 8 / 8 / / / O O O O O O O O O R W ca w all s as o l co qao of ls. R I I R Now s s a fo a ca b s b Cbs aso o ol al. Eac coffc s slf a o sf Cbs oloals as a fco of. W s fs o o sos. U s o a ac of qaos woks as follows: O F F F F F F F F F F F F } } } }..
8 E a s of sz a s a s a col cos of sz so w ow a a la alba obl of sz. I O O F F F F F F I Foal fo o qaos wll b lal sl o calcla. Sc ls a cosa - R s a I s a sl as I s ac l. a s of a wll also js coa a aoal sqc of s. W also a all s c fo qal o zo. s aa oba b ooooal laoss a ll sl co b -. of cos o al coos a wa w wa o b zo wok wll. ol awback s a o sf Cbs oloals wll ol wok fo a a of zo o o. s s accabl a f w wa a afl sls fo o calclaos w a o ak s wok fo as lo as w. os obos wa o o s o oc a s fal sl a as o al coo a aa a a as a s as cssa. s o s as o o c a all sf Cbs oloals a qal o o a o w al coos b acl wa o o fo coffcs fs lac. b a o s s sl a / as o al coo c o all ows as as. wa o alf of oloals a zo o alf a o - a aa w al coos a as o. I fac s os wok. obl s a lk a cal os al coos o a o fac al coos a o fo alo s solo. Cbs aso o o ol aoas solo o al also aoas al coo. Fo al Y -Y w Yo. aoa solo wll o sa a acl. b wll b off sll b a ao a ss as b of s cas. s s al ocabl a o sc a b al fo l qao sss b fo a la ss s o ca b q cl w o a ac solo sool o af o. Foal s o sl wa o o job. Isa of aoa as ak sl ajs o. Now s s al fo all fo zo o. col s ca ow.
9 oooal laoss ca: / w / / w / Slal c a / b / a b ao a wll b cas b a faco of. oc a wll a ca. Of cos as s la o s wll b o aa accac o al. s alo was w a sls w oo coa o aalc sls oba al. Fo cass w s cosa a a o c so solo a all bcos cosa bs aoac s o alo sal ss b a al coo o scb al. Eac ass wll b clos o fal solo a wll z o o lo sac. fo - a s o ocs o a bo al a fal os of lo a s s bs o sl sa w zo al coos a o a l ass. sls of sal calclaos a sow. aa Sac s s 5 sz 5 alf ss a Rla Eo 5.5- oa 5. a Hz
aa s sac : 5 alf ss. s 5 sac : 7 alf ss 6.
cos 5 s sac 8 No Dsac 8/5 ~6 os Eo: s w coa o aalcal solo o a o af al as bao a la o s < %. W a s w sac s a la o s < 5% a las s ~.5 os. s o oa was w - a a ozo aa. s o aalcal solo o coa o b w ca c a b oc a a sa sowa sla o ozo aa cass w kow. W ow alo oc so a o sls. Fo al o al coos w s o zo al als o w o clos o zo. Rcall w cos o o o coffcs a. I s o a s cao b o. o wok bfo cl s o s cass a of coc. f o back a ak s ajs oa l so o sfl look sls. Ra cos 5
aa 7 s 9 sac sz Dsac /5 ~ s sls a sla o cas of zo aa w solo s o b clos o sa solo w a soal a al. lso sla o sa sa solos s o b o ol wo aas. al s scal b a sa s b.
o s o cofc a s o s s asoabl sls fo solos o s qaos w co o a a b of cass all sos of ff aas. s aa s all aalabl fl foa. Wl ajo of cass look o b acca w so cass wc a as. W w a la al of olao o w o cosa a ol olao w c oo sls. Rcall cos ω. Df /. R a zo aa o os o a ow c s w cas os w o b a:. ~% Eo. ~.% Eo.5 ~8.7% Eo s s all o accabl. W qals f sls a q oo a a 9 sac s. s wll w obls w s o fo ca cofaos sc as: No a ol fo s b lo. Sc s fcos a o sac occ o a s o o lo o calcla o alf. fs a was w al coos s o zo sco was 8 s. I s o cocabl a as aas fs cas solo oscllas a a fqc a fqc of foc. as w wll sow la sol b o aso a al coos wol al fal bao.
lba of os all s cass aa fls s cl w s o. I l of s obls w a o ak so o ofcaos o Cbs aso alo cl a a o cl o s a accac ao o oba coffcs of asof aso ow s a ll ffc a w w abl o l o soc of obls js o. f alk o s w cos o s Cbs aso o sol o sl s of ffal qaos. s o ow s bs f w w back a js s a sl al al sol sc as R- o al w s sl ola qaos. So s s fac wa w a w fac c sls w o of obls a w a al. oo IVP sol wl sll slow calclao as c w a ol al w o s of oloals sac a a bo sac a as w w osl lo s c osl. Wa osl ook aw fo 5 s o os s ow b abo o o f s. ak a look a ow backwas El wol sol s w ca s a s o aso a al al wol affc lo bao. Rcall backwas El as f Fo o qaos s bcos R I I R Sol fo R I I R I I I cso s q aa a lc solo ca b oba as j j
5 s s aa o co f scal as of all s s lss a o. I s cas as os o f cobo of al coo saas a bao s oa ol b oc asof fco. W lo aa os wo fcos wc sow obls a s a w a b sls w backwas El. Sa fo zo al coos fco qckl acs s oc solo a os o sa fo. R a as wll a b s b qs a sall s sz o co. c s all aoac a sabl oc solo o wll a q al w a fs o sol. O o oblac cas s sow o a o obls a all w sa w zo al coos. f a bf al as sa wc js ao wll falls o ocal solo cl.
6 s wll a os fo lal sb o s oc solo cl. sccss of s la cal o ass o obos qso. W w o s fs lac? o was a w o lo os IVP sols wol q a lo of ss a wol ak fa oo lo. af a ss s s o. How l s o j o bawao js. W s b absoll s of o cal sc s accac a obsss bfo w ca co w la o a sabl a of s sac. W Cbs o o colss ss a w s o sa o s o bfo w ca o o. s s w c oss of s sac s w all sos of cass coa os a sc.
7 w a a a fls of fca s s ojc oa os a b w a bf sco of ac of s fls blow. b_co. backwas El o a a s s fl ca qs of bo co_. a _a. cbf. fco wc aks a a s a co coffcs fo. cb. Cbs aso alo. lo aas a s a l 7 a 7. s fco qw.. ll o s a o b oa. c. Cbs oloal alao. cb b. Rqs cbf.. ls. / ls. s a a R acs. a cal c ls a sz aabl s wl ls s. s allow fo o fo wok w oas a ala a o of os ls s as so ls. j. aalc solo fo zo aa a oscllao a as calcla b W.Y C a C.F. C. ll aas a s fl. s was w fo a -½ ½ a so asof b s b cas b a faco of 8 o l coss sls w -. losc. / losc. s wo fls a scs wc s Cbs coffcs a co b cas fco sc. a los o a sac of o. qw. asof fco s b cb. ll b w fco a Cbs oloal. kco. R o a a s s fl ca qs of co. sc. sf Cbs oloal alao. scb b. Rqs cbf.. l. /. Fcos call b _lo_f. o ala loc a a. _lo_f. Plos aalc solo s a fo ozo aa a cosa. ll aas a fl. -. Rqs l. a.. co. Is fco fo a s kco. co_. as a s b backwas El s of I- - a. as co s b backwas El
8 Ra Sa Daa Fls fl as w sa as: ssac s Yaa ω_al_sac sz_ a_u. back ssos a o cssal aa fla. Fo al 9Y 5 8 o_u. s 9Y 9 sac s aa. 5_ cos 5 _ _ sac lo was wo os / w 8_ 8 alf-ss cas a ol ss w lo: a o boo a l of oc ccl cas a so was lo w os o cas a a was o a a - o as a al los U cas a s fl ol coas als a a cssa o lo asw a o of a aabls b c fl sz fcal. s was o cl w los fl as alo oo ol sa sfl foao. o w s fls sl o a s sf coa. sfu fo loc o sf fo a. Sa fs I al als fs w sa sa of aa fls. fla sc s sla o wa was o abo alo s sall a o a V a b o wc aks obos as o w s f s lo a of loc. f zo ola.f s cl o osa ow oc woks.