
oseen samavaakyangal

draavaka chalanathmakathayil, 1910 ll kaal wilhem oseen roopappeduthiyathupole, cheria raynold nambarukalil viscos, compress cheyyaan kazhiyaatha draavakathinte ozhukkine oseen samavaakyangal (allengil oseen ozhukk) vivarikkunnu. stokku ozhukkinekkaal, eee ozhukkukalude mechappetta vivaranamaan oseen ozhukk, samvahana thvaranam (bhagika) ulppeduthunnu.
jii yude pareekshanangale adisthaanamaakkiyullathaanu oseente kruthi.jii. oru viscos draavakathiloode oru golam veezhunnathinekkurichu padicha stooks. stoxinte paradox ennariyappedunna prashnam pariharikkunnathin stoxinte kanakkukoottalukalil upayogikkunna ozhukk vegathaykkaayi jadatwa gadakangal ulppedunna oru thiruthal padam addeham vikasippichu. addehathinte ekadesha kanakk stoxinte kanakkukoottalukalil mechappeduthalilekku nayikkunnu.
samavaakyangal.
oru vasthu draavakathiloode sthiramaaya ozhukk vegathayil neengumbol-athu vasthuvil ninnu valare akaleyaan-vasthuvil gadippichirikkunna oru reference framilaanu oseen samavaakyangal
evide
oru karkkashamaaya vasthuvin chuttumulla oseen ozhukkinte athirthi vyavasthakal ivayaanu
r" ennathu vasthuvinte kendratthil ninnulla dooravum "p" φ ennathu vasthuvil ninnu akaleyulla thadassamillaatha marddavumaanu.
rekhamsha, traansversal tharamgangal.
oseen samavaakyathinte adisthaana savisheshatha pothuvaaya parihaarathe "rekhamsha", "traansversal" tharamgangalaayi vibhajikkaam ennathaanu.
oru parihara soothravaakyam _ 3 ennathu oru rekhamsha tharamgamaanu, vegatha itrotteshanal aanenkil viscos padam kurayunnu. samavaakyangal maarunnu
athinte anantharafalamaayi
pottensial sidhaanthathil ninnaan vegatha uruthirinjathu, marddam linearised bernauliyude samavakyangalil ninnaan.
mardha soothravaakyam _ 7 thulyamaayi poojyavum vegatha feeld solinoyidalumaanenkil oru parihara soothravaakyam _ 6 oru traansversal tharamgamaanu. samavaakyangal ivayaanu
appol poornnamaaya oseen pariharam nalkunnu
hooras lamb moolamulla oru vibhajana sidhaantham. ananthathayile vyavasthakal (soothravaakyam _ 10 ennu parayuka) vyakthamakkiyaal vibhajanam savisheshamaan.
chila oseen pravaahangalkk, traansversal tharamgathe erotational, rotational gadakangalaayi kooduthal vibhajikkunnath saadhyamaaya soothravaakyam _ 11 ayirikkatte formula _ 12 ennathu formula _ 13-neey thripthippeduthukayum ananthathayil aprathyakshamaakukayum cheiyunna scaler functionayirikkatte, neremarichu formula _ 14 nalkatte, angane formula _ 15, appol traansversal tharamgam
ivide formula _ 12 formula _ 18 ll ninnu nirnnayikkappedunnu, formula _ 19 yoonit vektaraanu. formula _ 20 allengil formula _ 21 randum transversal allah, ennaal formula _ 22 transversal aanu. athinaal,
formula _ 21 aanu eaka bhramana ghadakam.
adisthaana pariharangal.
oru oseen ozhukkile ekabinduvinte balam moolamundaakunna adisthaana pariharam oseenlettaanu. neutonian, micropolar draavakangalkkaayi ekapaksheeyamaaya samayathe aasrayikkunna vivarthana, bhramana chalanangalumayi bandhapetta saamaanyavalkkaricha asthiramaaya stokkukalkkum oseen ozhukalukalkkumulla adanja roopathilulla adisthaana pariharangal uruthirinjittundu.
ooseen samavaakyam upayogichu, 1911-ll oru golatthinu chuttumulla viscos floyk mechappetta padaprayogangal nedaan hooras lambinu kazhinju, ithu stokku niyamathil kurachu uyarnna reynolds nambarukalilekku mechappeduthi. koodaathe, aadyamaayi uruthirinja kunjadum-oru vrithaakrithiyilulla silindarinu chuttumulla viscos ozhukkinte oru pariharam.
baahya athirthikalonnum illaathirikkumbol oru ekabalam fozhsu formula _ 25-nte prathikaranatthinulla pariharam ingane ezhuthanam
formula _ 27, ivide formula _ 28 ennathu pooyint formula _ 29 ll kendreekarichirikkunna ekavachana shakthiyum formula _ 30 oru ekapaksheeyamaaya binduvum formula _ 31 nalkiyirikkunna vektarumaanenkil, athu ekavachana shakthiyude disha nalkunnu, appol athirthikalude abhaavathil, vegathayum mardhavum adisthaana tensor formula _ 32 ll ninnum adisthaana vector formula _ 33 ll ninnum uruthirinjathaanu.
ippol formula _ 25 ennathu sthalathinte ekapaksheeyamaaya pravartthanamaanenkil, paridhiyillaattha oru domaininte pariharam ithaanu
ivide formula _ 37 ennathu pooyint formula _ 29 nuu chuttumulla ananthamaaya vooliyam/eriya gadakamaanu.
dvimaana.
uthbhavathilum soothravaakyathilum edutha generality formula _ 39 nashtappedaathe. appol adisthaana tenzer vektarum
evide
ivide formula _ 43 ennathu randaam tharam poojyatthinte parishkkaricha bezel functionaanu.
trimana.
uthbhavathilum soothravaakyathilum edutha generality formula _ 44 nashtappedaathe. appol adisthaana tenzer vektarum
evide
kanakkukoottalukal.
golathe nischalamaayum draavakam golatthil ninnu ananthamaaya akalatthil ozhukunna vegathayodeyum (soothravaakyam _ 48) ozhukunnathaayi ooseen kanakkakki. stokkukalude kanakkukoottalukalil jadatwa padangal avaganikkapettu. reynolds nambar poojyathilekku neengumbol ithu oru parimithappedutthunna parihaaramaan. 0. 1 polulla reynolds nambar cheruthum parimithavumaayirikkumbol, inertial padatthinu thiruthal aavasyamaanu. inipparayunna phloo velosity moolyangal navier-stokks samavaakyangalilekku maatti.
iva navier-stokku samavakyangalil ulppeduthunnathum primed alavukalile quadratic padangal avaganikkunnathum oseeninte ekadeshathinte vyulppathiyilekku nayikkunnu
formula _ 51 aaksisumaayi bandhappett chalanam samamithiyayathinaal vorttitti vekttarinte vyathichalanathin ellaayppozhum poojyamaayirikkum
formula _ 51-ll anuyojyamaaya oru functionilekku cherkkunnathiloode function formula _ 53 illaathaakkaam, ithu vorttitti functionaanu, munbathe function ingane ezhuthaam
chila samyojanathiloode formula _ 12-nulla pariharam ithaanu
athinaal formula _ 51 athu uthppaadippikkunna "prathyeka dishayaayi" anuvadikkunnathiloode
appol namukk labhikkuna moonnu athirthi vyavasthakal prayogikkunnathiloode
puthiya mechappetta dragg coffificient ippol ingane maarunnu
oduvil, oseeninte ekadesha kanakk adisthaanamaakki stoxinte pariharam pariharikkappettappol, athinte falamaayundaakunna valichidal shakti nalkunnathu
evide
oseen samavaakyathil ninnulla balam stokkukalil ninnu oru ghadakam kondu vyathyasthamaan
stokks parihaaratthil pishak.
navier stooks samavaakyangal ingane
ennaal vegatha feeld ingane aayirikkumbol
vidoora feeld formula _ 71 ′1, viscos stress avasana tame aadhipathyam pulartthunnu. athu ithaanu
jadatwa padatthil aadhipathyam pularthunnathu eee padamaanu
pishak appol eee anupatham nalkunnu
ithu soothravaakyathinu paridhiyillaathathaayi maarunnu, athinaal vidoora feeldil jadathvam avaganikkanaavilla. kurl edukkunnathiloode, stooks samavaakyam formula 76 nalkunnu, kaaranam body vortixiyude uravidamaanu, formula 77 valia formula 78 nuu paridhiyillaattha logaritmikkalaayi maarum, ithu theerchayaayum bhouthikamallaathathum stoxinte paradox ennariyappedunnathumaanu.
amarthaanaavaatha draavakathil chalikkunna golatthinulla pariharam.
sthiramaaya vegathayil nischalamaaya draavakathil neengunna oru kharagolam pariganikkuka.
draavakam amarthaanaavaatha draavakamaayi maathrkayaakkappedunnu (i.iii. sthiramaaya saandrathayode), nischalamaayirikkuka ennaal golatthil ninnulla dooram ananthathaye sameepikkumbol athinte vegatha poojyathilekku neengunnu ennaanu ardhamaakkunnathu.
oru yadhaartha bodic athinte chalanam aarambhikkumbol athinte thvaranam kaaranam oru shanikamaaya prabhavam undaakum; ennirunnaalum mathiyaya samayathinu sesham athu poojyathilekku neengum, angane ellaayidathum draavaka vegatha shareeram ithinakam ananthamaaya samayathekku neengunna saangalppika kesil labhichathine sameepikkum.
angane ananthathayil vishramikkunna amarthaanaavaatha draavakathil sthiramaaya vegatha soothravaakyathil _ 79-ll neengunna "a" vyaasamulla oru golam nammal anumaanikkunnu. golatthinte kendratthil sthithicheyyunna coordinate centarinoppam golathinoppam neengunna coordinate formula _ 80 ll nammal pravartthikkum. namukk undu
eee athirthi vyavasthakalum chalanangalude samavaakyavum samaya maattamillaathavayaayathil (i.iii. samaya soothravaakyam _ 82 maattunnathiloode avaa maattamillathe thudarunnu) soothravaakyam _ 80 kordinettukalil prakadippikkumbol, pariharam eee kordinettukaliloodeyulla samayathe maathram aasrayichirikkunnu.
chalana samavaakyangal resting frame coordinate formula _ 84 ll nirvachichirikkunna navier-stokku samavaakyangalaan. rund coordinate sisttangalilum speshyal derivateevukal thulyamaanenkilum, samavakyangalil drishyamaakunna time derivative thripthippeduthunnu
86 enna derivative formula chalikkunna coordinate formula 80-maayi bandhappettirikkunnavel, inii muthal nammal "m" subscript ozhivaakkunnu.
ooseente ekadesha kanakk _ 88 enna soothravaakyathile noonn-leanier enna padam avaganikkunnathilekku samgrahikkunnu. angane amarthaanaavaatha navier-stokku samavaakyangal ingane maarunnu
saandrata ρ, kainamattik viscosity ν = μ/ρ (dynamic viscosity aayirikkunnathu) ennivayulla oru draavakathinu. "pii" ennathu sammarddamaanu.
compress cheyyaanaavaatha draavaka soothravaakyam _ 90-nte thudarcha samavaakyam kaaranam, oru vekttar pottensial soothravaakyam _ 91 upayogichu pariharam prakadippikkaan kazhiyum. ithu soothravaakyathilekku nayikkappedunnu _ 92 dishayil athinte vyaapthi dvimaana prasnangalil upayogikkunna stream functionu thulyamaanu. ithu ingane maarunnu
ivide formula _ 94 golathinaduthulla ozhukkinte reynolds nambaraanu.
chila notteshan formulakalil formula 95-neey formula 96 upayogichu maattisthaapikkunnu, athinaal formula 98-ll ninnulla formula 88-nte derivation dvimaana kesil (dhruva kordinettukalil) stream functionil ninnulla derivationode kooduthal saamyamullathaan.
vishadamaakkal.
formula _ 95 thaazhepparayunna reethiyil prakadippikkaam
evide
_ 104 enna soothravaakyathinte oru vektarinte vector laplasian ingane vaayikkunnu
angane ithu kanakkakkaam
athinaal
athinaal vorttitti ithaanu
vekttar laplatian eratta kerlineyum bandhappedutthaan nammal soothravaakyathinte vyathichalanathinte aprathyakshatha upayogichu.
chalanatthinte idathu vashathinte samavaakyam inipparayunnavayude churulaanna
oro padatthinum nammal derivative formula _ 95 ll vevvere kanakkakkunnu.
shradhikkuka
koodaathe
athinaal namukk undu
njangalude ellah nibandhanakalum samyojippikkuka
kell edukkumbol, thaazhepparayunna functionte gradiantinte 116 madangu formulakku thulyamaaya oru padaprayogam nammal kandethunnu, athu mardhamaanu
118 enna soothravaakyam ananthathayile mardhamaanengil, formula_119.is dhruvakonam munvashathe stambhana binduvinte ethirvashathu ninnaan uthbhavichath (soothravaakyam 120 munvashathe stambhana binduvaaya ivide).
koodaathe, _ 91 enna soothravaakyathinte churul edukkunnathiloodeyaanu vegatha uruthirinjathu
eee "p", "u" enniva chalana samavaakyathe thripthippeduthukayum angane ooseente ekadesharekhaykkulla parihaaramaayi maarukayum cheiyunnu.
oseente ekadesha kanakkilekkulla mattangal.
ennirunnaalum, thiruthal padam yaadruchikamaayi thiranjedutthathaano ennu oraalkk chodyam cheyyaam, kaaranam golavumaayi neengunna oru reference framil, golathinaduthulla draavakam eathaandu vishramikkunnu, aaa pradeshathu jadatwa shakti valare kuravaanu, stoxinte samavaakyam nannayi nyaayeekarikkappedunnu. golatthil ninnu valare akale, ozhukk vegatha "yuu" yilek adukkukayum oseeninte ekadesha kanakk kooduthal krithyatha pulartthukayum cheiyunnu. ennaal muzhuvan phloo feeldinum samavaakyam prayogichaanu oseente samavaakyam labhichathu. 1957ll eee chodyathinu marupadi nalkiyathu, navier-stokks samavaakyangal pariharikkukayum golatthinte sameepat mechappetta stokks pariharavum ananthathayil mechappetta ooseens pariharavum nalkukayum rund parihaarangalum avayude saadhuthayude pothuvaaya pradeshathu poruthappedutthukayum cheythu. avarkk labhichathu
aaplikkeshanukal.
valare kuranja rhinold nambaril ozhukk vishakalanam cheiyunnathinulla reethiyum roopeekaranavum pradhaanamaan. oru draavakathile cheria kanangalude mandagathiyilulla chalanam bio enchineeyaringil saadhaaranamaanu. kanikakal adangiyirikkuka, kanangalude sedimenteshan, centrifugation allengil suspensionukal, colloydukal, tumourukalum aantijanukalum verthirichedukkunnathile raktham ennivayude ultra centrifugation thudangiya vividha prathyeka saahacharyangalil draavakangalude ozhukkumaayi bandhappett ooseen dragg formulation upayogikkaam. draavakam draavakamaayirikkanamennilla, kanikakal kharamayirikkanamennumilla. pukamanju roopappedal, draavakangalude aattomaiseshan thudangi niravadhi prayogangalil ithu upayogikkaam.
kaappilarikal polulla cheria naalangalile rakthapravaahathinte savisheshatha cheria rhinoldukalum vumezhsli nambarukalum aanu. rakthathinte ozhukk, viscosity, saandrata, hridayamidippu ennivayulla oru paathratthinu 0.005 enna renauld number 0.0126 enna vumezhsli number undaakum. eee cheria reynoldukalilum vumezhsli nambarukalilum draavakathinte viscos efakttukal prabalamaakunnu. eee kanangalude chalanam manasilaakkunnath marunnu vitharanam cheiyunnathinum kaansarukalude metaastaasis chalanagal padikkunnathinum athyaavasyamaanu.