CSE Winter School 2012

Transcription

1 0--4 CSE Wine School 0 FUNDAMENTALS OF FLUID MECHANICS Inodcion Some Chaaceisics of Flids Analsis of Flid Behaios Flid Popeies iscosi Chaaceisics of Flids Wha s a Flid? Wha s diffeence beeen a solid and a flid?

2 0--4 age idea Flid and Solid /3 ðflid is sof and easil defomed. ðsolid is had and no easil defomed. Molecla sce ðsolid has densel spaced molecles ih lage inemolecla cohesie foce alloed o mainain is shape Flids compise he liqid and gas (o apo) phase of he ph sical foms 3 Flid and Solid 3/3 ðliqid has fhe apa spaced molecles, he inemol ecla foces ae smalle han fo solids, and he molec les hae moe feedom of moemen. A nomal emp eae and pesse, he spacing is on he ode of 0-6 mm. The nmbe of molecles pe cbic millimee is on he ode of 0. ðgases hae een geae molecla spacing and feedo m of moion ih negligible cohesie inemolecla f oces and as a conseqence ae easil defomed. A no mal empeae and pesse, he spacing is on he od e of 0-7 mm. The nmbe of molecles pe cbic milli mee is on he ode of

3 0--4 Definiion of Flid A A flid is a sbsance ha defoms coninosl nde he applicaion of a shea sess no mae ho small he shea sess ma be. A A sheaing sess is ceaed henee a angenial fo ce acs on a sface. 5 Flid and Solid /3 When a consan shea foce is applied: ðsolid defoms o bends ðflid coninosl defoms. 6 3

4 0--4 Coninm Flid : aggegaion of molecles Mos engineeing poblems ae concened ih phsical dimensions mch lage han he limiing olme, d*, so ha densi is esseniall a poin fncion and flid popeies can be hogh of as aing coninosl in space sch a flid is called coninm. d* 0-9 mm 3 ( μm) 3 fo gases Ecepion : aefied gas 7 Coninm lim d d * dm d dm d d* d 8 4

5 0--4 Analsis of Flid Behaios / Analsis of an poblem in flid mechanics necessail incldes saemen of he basic las goening he f lid moion. The basic las, hich applicable o an f lid, ae: ðconseaion of mass ðneon s second la of moion ðthe pinciple of angla momenm ðthe fis la of hemodnamics ðthe second la of hemodnamics 9 Analsis of Flid Behaios / Flid saics : he flid is a es Flid dnamics : he flid is in moion Flid popeies ae closel elaed o flid behaio e.g.) gas : ligh and compessible liqid : hea and incompessible The flo of sp is sloe han ha of ae 0 5

6 0--4 Densi The densi of a flid, designaed b he Geek smb ol (ho), is defined as is mass pe ni olme. In n SI ssem, he nis ae kg/m 3. The ale of densi can a idel beeen diffee n flids, b fo liqids, aiaions in pesse and e mpeae geneall hae onl a small effec on he ale of densi. iscosi The popeies of densi and specific eigh ae mea ses of he heainess of a flid. I is clea, hoee, ha hese popeies ae no sffic ien o niqel chaaceie ho flids behae since o flids can hae appoimael he same ale of densi b behae qie diffeenl hen floing. 6

7 0--4 iscosi Definiion d m d Neonian flids The consan of popoionali is designaed b he Geek smbol m (m) and is called he absol e iscosi, i, dnamic iscosi, o simpl he iscosi of he flid. The iscosi depends on he pai cla flid, and fo a paicla fl id, he iscosi is also dependen on empeae. 3 Neonian and Non-Neonian Neonian Flid Flids fo hich he sheaing sess is lineal elaed o he ae of sheaing sain ae designaed as Neo nian flids afe I. Neon (64-77). 77). Mos common flids sch as ae, ai, and gasoline ae Neonian flid nde nomal condiions. Flids fo hich he sheaing sess is no lineal el aed o he ae of sheaing sain ae designaed as no n-neonian flids. 4 7

8 0--4 Non-Neonian Neonian Flids / Shea hinning flids: The i scosi deceases ih incea sing shea ae he hade he flid is sheaed, he less iscos i becomes. Man coll oidal sspensions and polm e solions ae shea hinnin g. Lae pain is eample. 5 Non-Neonian Neonian Flids / Shea hickening flids: The iscosi inceases ih i nceasing shea ae he hade he flid is sheaed, he moe iscos i becomes. Wae-con sach mi e ae-sand mie ae eamples. Bingham plasic: neihe a flid no a solid. Sch ma eial can ihsand a finie shea sess iho moio n, b once he ield sess is eceeded i flos like a flid. Toohpase and maonnaise ae common eamp les. 6 8

9 iscosi s. Tempeae /3 Fo flids, he iscosi deceases ih an incea se in empeae. Fo gases, an incease in empeae cases an in cease in iscosi. ðwhy? molecla sc e. 8 9

10 0--4 iscosi s. Tempeae /3 The liqid molecles ae closel spaced, ih song c ohesie foces beeen molecles, and he esisance o elaie moion beeen adjacen laes is elaed o hese inemolecla foce. As he empeae inceases, hese cohesie foce ae edced ih a coesponding edcion in esisance o moion. Since iscosi is an inde of his esisance, i follos ha iscosi is edced b an incease in empeae. The Andade s eqaion μ De B/T 9 iscosi s. Tempeae 3/3 In gases, he molecles ae idel spaced and inem olecla foce negligible. The esisance o elaie moion mainl aises de o he echange of momenm of gas molecles beeen adjacen laes. As he empeae inceases, he andom molecla acii inceases ih a coesponding incease in iscosi. The Sheland eqaion μ CT 3/ / (TS) 0 0

11 0--4 Eample: Neonian Flid Shea Sess The eloci disibion fo he flo of a Neonian flid be een o sides, paallel plaes is gien b he eqaion 3 é ê - êë h ù ú úû Eample Solion hee is he mean eloci. The flid has a iscosi of N s/m. When 0.6 m/s and h5 mm. deemine: (a) he sheaing sess acing on he boom all, and (b) he sheaing sess acing on a plane paallel o he alls and passing hogh he ceneline (midplane). d 3 m -m d h boom all midplane d 3 m -m d h d 3 m -m d h 0 -h 0 70 N / m

12 0--4 Kinemaic iscosi Defining kinemaic iscosi ν μ/ [N] ðthe dimensions of kinemaic iscosi ae L /T. ðthe nis of kinemaic iscosi in SI ssem ae m /s. 3 Flid Saics FUNDAMENTALS OF FLUID MECHANICS Pesse a a Poin Basic Eqaion fo Pesse Field Pesse aiaion in a Flid a Res Boanc, Floaing, and Sabili 4

13 Pesse? Pesse a a Poin /4 ðindicaing he nomal foce pe ni aea a a gien po in acing on a gien plane ihin he flid mass of in ees. Ho he pesse a a poin aies ih he oienaio n of he plane passing hogh he poin? 6 3

14 0--4 Pesse a a Poin /4 Conside he fee-bod diagam ihin a flid mass. In hich hee ae no sheaing sess, he onl eena l foces acing on he edge ae de o he pesse an d he eigh. 7 fig_0_0 8 4

15 0--4 Pesse a a Poin 3/4 The eqaion of moion (Neon s second la, Fma) in he and diecion ae, δδδ å F Pδδ -PS δδssinθ ρ a δδδ å FZ P δδ -PS δδscosθ - γ d ds cosq d ds sinq δ PZ - PS ( ρaz γ) P - PS ρa δ0 δ0 δ0 Z δδδ ρ δ P P P S a 9 Pesse a a Poin 4/4 The pesse a a poin in a flid a es, o in moion, is independen of he diecion as long as hee ae no sheaing sesses pesen. The esl is knon as Pascal s la named in hono o f Blaise Pascal (63-66). 66). 30 5

16 0--4 fign_0_p039 3 Basic Eqaion fo Pesse Field To obain an basic eqaion fo pesse field in a sa ic flid. Appl Neon s second la o a diffeenial flid mas s å δf δma ðthee ae o pes of f oces acing on he mass of flid: sface foce an d bod foce. dm d 3 6

17 0--4 d - d F B Bod Foce on Elemen F B gdm gd k -gdddk Whee ρ is he densi. g is he local gaiaional al acceleaion. dm d 33 Sface Foces /4 No shea sesses, he onl sface foce is he pesse foce. 34 7

18 0--4 Sface Foces /4 The pesse a he lef face p p d p pl p ( L - ) p - p - The pesse a he igh fac e p p d p pr p ( R - ) p p d d The pesse foce in diec ion p d p d p δf p - δδ - p δδ - δδδ 35 Sface Foces 3/4 The pesse foce in diecion p d p d p δf p - δδ - p δδ - δδδ The pesse foce in diecion p d p d p δf p - δδ - p δδ - δδδ 36 8

19 0--4 Sface Foces 4/4 The ne sface foces acing on he elemen δfs δf i δf X Y p p j δfz k - i p p p gadp Ñp i j k p j k δδδ δfs -gadp(δδδ) -Ñpδδδ 37 Geneal Eqaion of Moion df dfs dfb (-Ñp g) ddd (-Ñp g)d - Ñ pddd - gdddk addd The geneal eqaion of moion fo a fli d in hich hee ae no sheaing sesses - Ñp - g k a 38 9

20 0--4 Pesse aiaion in a Flid a Res Fo a flid a es a0 - Ñp - gk a 0 p - g p - g p - g diecion diecion diecion dp d -g -g g g 0,g -g 0, P 0 P 0 P -g 39 Pesse-Heigh Relaion The basic pesse-heigh elaion of saic flid : dp d -g Resicion: -g ðsaic flid. ðgai is he onl bod foce. ðthe ais is eical and pad. Inegaed o deemine he pesse disibion in a saic flid ih appopiae bonda condiions. Ho he specific eigh aies ih? 40 0

21 0--4 Pesse in Incompessible Flid A flid ih consan densi is called an incompessible flid. dp d -g -g p ò dp -g p ò p - p γ(-)γh pγh p d This pe of pesse disibion is called a hdosaic disibion. h -,h is he deph of flid meased donad fom he loca ion of p. 4 Eample: Pesse-Deph Relaionship Becase of a leak in a bied gasoline soage ank, ae has s eeped in o he deph shon in Fige. If he specific gai of he gasoline is SG0.68. Deemine he pesse a he gasolin e-ae ineface and a he boom of he ank. Epess he pe sse in nis of N/m, N/mm, and as pesse head in mees of ae. Deemine he pesse a he gasoline- ae ineface and a he boom of he ank 4

22 0--4 Eample Solion The pesse a he ineface is p SGg 34.7 p H O 0 0 ( kn / m ) h p (0.68)(9800N / m 3 )(5.m) p 0 p o is he pesse a he fee sface of he gasoline. The pesse a he ank boom p g H O (9800N / m 43.5kN / m h H O 3 p )(0.9m) 34.7kN / m 43 BUOYANCY / Boanc: The ne eical foce acing on an bod hich is immesed in a liqid, o floaing on is s face de o liqid pesse. F B Conside a bod of abia s hape, haing a olme, ha is immesed in a flid, We enclose he bod in a paa llelepiped and da a fee-bo d diagam of paallelpiped ih bod emoed as shon in (b). 44

23 0--4 F F - F g ( h F B B F - F -W g ( h BUOYANCY / - h )A - h )A - g [( h - h )A - ] FB is he foce he bod is eeing on he fli d. W is he eigh of he shaded flid olme (p aallelepiped mins bod). A is he hoional aea of he ppe (o loe) sface of he paallelepiped. Fo a sbmeged bod,, he bo anc foce of he flid is eqal o he eigh of displaced flid A F B g 45 Achimedes Pinciple Fo a sbmeged bod, he boanc foce of he fli d is eqal o he eigh of displaced flid and is diec l eicall pad. F B g The elaion epoedl as sed b Achimedes in 0 B.C. o deemine he gold conen in he con of King Hieo II. 46 3

24 0--4 Eample Boan foce on a sbmeged objec The.8 N leas fish sinke is aached o a fishing line. The specific gai of he sinke is SG sinke.3. Deemine he diffeence beeen he ension in he line aboe and belo he sinke. 47 Eample Solion T A -T FB -W B 0 F F T B g W gsg B A -T sin ke W / SG B sin ke W[ - (/ SG sin ke )] 48 4

25 0--4 FUNDAMENTALS OF FLUID MECHANICS Flid Kinemaics The eloci Field The Acceleaion Field 49 Field Repesenaion of flo / A a gien insan in ime, an flid pope (s ch as densi, pesse, eloci, and accelea ion) can be descibed as a fncions of he fli d s locaion. This epesenaion of flid paamees as fnc ions of he spaial coodinaes is emed a field epesenaion of flo. 50 5

26 0--4 Field Repesenaion of flo / The specific field epesenaion ma be diffeen a d iffeen imes, so ha o descibe a flid flo e ms deemine he aios paamee no onl as fncion s of he spaial coodinaes b also as a fncion of ime. EXAMPLE: Tempeae field T T (,,, ) EXAMPLE: eloci field (,,, )i (,,,) j (,,, )k 5 eloci Field The eloci a an paicle in he flo field (he elo ci field) is gien b (,,,) (,,, )i (,,,) j (,,, )k The eloci of a paicle is he ime ae of change of he posiion eco fo ha p aicle. A d d A 5 6

27 0--4 Eample eloci Field Repesenaion A eloci field is gien b hee 0 and l 0 / l i - j ae consans. A ha locaion in he flo field is he speed eq al o 0? Make a skech of he eloci field in he fis qad an ( 0, 0) b daing aos epesening he flid elo ci a epesenaie locaions. ( )( ) 53 Eample Solion The,, and componens of he eloci ae gien b 0 /l, - 0 / l,, and 0 so ha he flid speed l / 0 ( ) ( The speed is 0 a an locaion on he cicle of adis l c eneed a he oigin [( ) / l] ] as shon in Fige E4. (a). ) / The diecion of he flid eloci elaie o he ais is gie n in ems of θ acan(/) as shon in Fige E4. (b) Fo his flo -0 / l - anθ 0 / l 54 7

28 0--4 eloci Field Mehod of Descipion Sead and Unsead Flos D, D, and 3D Flos Seamlines 55 Mehods of Descipion Lagangian mehod Ssem mehod Eleian mehod Conol olme mehod 56 8

29 0--4 Lagangian gian Mehod Folloing indiidal flid paicles as he moe. The flid paicles ae agged o idenified. Deemining ho he flid popeies associaed ih hese paicles change as a fncion of ime. Eample: one aaches he empeae-measing measing deice o a paicla flid paicle A and ecod ha paicle s empeae as i moes abo. T T A A () The se of ma sch measing deices moing ih aios flid paicles old poide he empeae of hese flid paicles as a fncion of ime. 57 Eleian Mehod Use he field concep. The flid moion is gien b compleel pescibing he necessa popeies as a fncions of space and i me. Obaining infomaion abo he flo in ems of ha happens a fied poins in space as he flid flos pa s hose poins. Eample: one aaches he empeae-measing measing deice o a paicla poin (,,) and ecod he empeae a ha poin as a fncion of ime. T T (,,, ) 58 9

30 0--4 D, D, and 3D Flos Depending on he nmbe of space coodinaes eqi ed o specif he flo field. Alhogh mos flo fields ae inheenl hee-dimen sional, analsis is based on fee dimensions is feq enl meaningfl. The complei of analsis inceases consideabl i h he nmbe of dimensions of he flo field. 59 Sead and Unsead Flos Sead flo: he popeies a ee poin in a flo fie ld do no change ih ime. h 0 hee η epesens an flid pope. Unsead flo:. Change ih ime. ðnonpeiodic flo, peiodic flo, and l andom flo. ðmoe difficl o anale

31 0--4 Seamlines / Seamline: Line dan in he flo field so ha a a gien insan he ae angen o he diecion of flo a ee p oin in he flo field. >>> No flo acoss a seamline. ØSeamline is eehee angen o he eloci field. ØIf he flo is sead, nohing a a fied poin changes ih ime, so he seamlines ae fied lines in space. ØFo nsead flos he seamlines ma change shape ih ime. ØSeamlines ae obained analicall b inegaing h e eqaions defining lines angen o he eloci field. 6 Seamlines / ØFo o dimensional flos he slope of he seamli ne, d/d, ms be eqal o he angen of he angle ha he eloci eco makes ih he ais d d If he eloci field is kno n as a fncion of and, his eqaion can be ine gaed o gie he eqaion of seamlines. 6 3

32 0--4 Eample: Seamlines fo a Gien eloci Field Deemine he seamlines fo he o-dimensional s ead flo discssed in Eample 4., 0 ( / l)( i - j)! Fige E4. 63 Eample Solion Since ( / l) and -( / ) 0 0 l The seamlines ae gien b solion of he eqaion d d ( 0 / l) - ( / l) - Inegaing. ò d -ò d 0 o ln -ln cons an The seamline is C, hee C is a consan 64 3

33 0--4 Acceleaion Fo Lagangian mehod, he flid acceleaion is desc ibed as done in solid bod dnamics a a() Fo Eleian mehod, he flid acceleaion is descib ed as fncion of posiion and ime iho acall f olloing an paicles. a a(,,,) 65 Acceleaion Field /5 The acceleaion of a flid paicle fo se in Neon s second la is: a d / d The poblem is : Gien he eloci field find he acceleaion of a flid paicle, (,,, ) 66 33

34 0--4 Acceleaion Field /5 The eloci of a flid paicle A in space a ime : A A (,,,) The eloci of a flid paicle in space a ime d: A A ( d, d, d, d) d The change in he eloci of he paicle, in moing f om locaion d o, is gien b he chain le: d A A d A A d A A d A A d 67 a Acceleaion Field 3/5 A d d A d d d d d A d A d A A, A, A d d d d a A A Þ A A A d alid fo an paicle.. A a A A A A A A d d A A A A 68 34

35 0--4 Acceleaion Field 4/5 Scala componens a a a A shohand noaiona D D 69 D a D Acceleaion Field 5/5 ( ) ( ) ( ) ( ) ( ) D D Whee he opeao ( ) ( Ñ)( ) is emed he maeial deiaie o s bsanial deiaie

36 0--4 Phsical Significance- Unsead Effec Time deiaie: Local deiaie. I epesens effec of he nseadiness of he flo Local acceleaion Spaial deiaie: Conecie deiaie. I epesens he fac ha a flo pope associaed ih a flid paicle ma a becase of he moion of he paicle fom one poin in space o anohe poin. ( Ñ ) Conecie acceleaion 7 Fo aios Flid Paamees The maeial deiaie concep is e sefl in anal sis inoling aios paamee, no js he accelea ion. Fo eample, conside a empeae field TT(,,,) associaed ih a gien flo. We can appl he chain le o deemine he ae of change of empeae as dta TA TA da TA da TA da d d d d DT T T T T T Þ ÑT D 7 36

37 Eample: Eample: Acceleaion fom a Gien e Acceleaion fom a Gien e loci Field loci Field Conside he sead, o Conside he sead, o-dimensional flo field discssed in dimensional flo field discssed in p eios eios Eample Eample. Deemine he acceleaion field fo his flo. Deemine he acceleaion field fo his flo. ( ) ) j (i / 0 l - 74 Eample Eample Solion Solion ( )( ) D D a Ñ In geneal, he acceleaion is gien b In geneal, he acceleaion is gien b ( 0 / l ) and ) and -( ( 0 / l ) ) Fo sead, o Fo sead, o-dimensional flo dimensional flo j i a j () ()(0) i ()(0) () a l l l l l l a a l l

38 0--4 FUNDAMENTALS OF FLUID MECHANICS Analsis of Flid Flo Flid Elemen Kinemaics Conseaion of Mass Conseaion of Linea Momenm 75 Moion of a Flid Elemen Flid Tanslaion: The elemen moes fom one poin o anohe. Flid Roaion: The elemen oaes abo an o all of he,, a es. Flid Defomaion: _Angla Defomaion: The elemen s angles beeen he sides change. _Linea Defomaion:The elemen s sides sech o conac

39 Flid Tanslaion Flid Tanslaion eloci and acceleaion eloci and acceleaion The eloci of a flid The eloci of a flid paicle paicle can be epessed can be epessed The The oal acceleaion oal acceleaion of he paicle is gien b of he paicle is gien b k j i ),, (, D D a d d, d d, d d d d d d d d D D a Þ D D a is called he maeial, o sbsanial deiaie. is called he maeial, o sbsanial deiaie. Acceleaion field eloci field 78 Acceleaion Acceleaion Componen Componen a a a Recangla coodin Recangla coodin aes ssem aes ssem a a a q q - q q q q q q q q q q q Clindical coodin Clindical coodin aes ssem aes ssem

40 0--4 Linea Tanslaion All poins in he elemen ha e he same eloci (hich is onl e if hee ae no eloc i gadiens), hen he eleme n ill simpl anslae fom one posiion o anohe. 79 Linea Defomaion /3 The shape of he flid elemen, descibed b he angle s a is eices, emains nchanged, since all igh an gles conine o be igh angles. A A change in he dimension eqies a noneo ale of / A A A A / / 80 40

41 0--4 Linea Defomaion /3 The change in lengh of he sides ma podce chang e in olme of he elemen. The change in d d ( dd)( d) The ae a hich he d is changing pe ni olme de o gadien / d ( d) d d 8 Linea Defomaion 3/3 If / and / ae inoled ( d) d Ñ d d olmeic dilaaion ae 8 4

42 0--4 Angla Roaion /4 The angla eloci of line OA Fo small angles OA CCW OB CW ωoa lim δ 0 δα δ dd an da & da d d - fo CCW 83 Angla Roaion /4 The oaion of he elemen abo he -ais is defined as he aeage of he angla e lociies OA and OB of he o mall pependicla lines OA and OB. ( OA OB) In eco fom i j k 84 4

43 Angla Roaion Angla Roaion 3/4 3/4 ú ú û ù ê ê ë é k j i - k j i cl ú û ù ê ë é - ú û ù ê ë é - ú û ù ê ë é - Ñ Defining oici Defining oici Ñ Defining ioaion Defining ioaion 0 Ñ 86 Angla Roaion Angla Roaion 4/4 4/4 k j i k j i cl ú û ù ê ë é - ú û ù ê ë é - ú û ù ê ë é - Ñ

44 oici oici Defining oici ζ hich Defining oici ζ hich is a measemen of he oaion is a measemen of he oaion of a flid elemen of a flid elemen as i moes in he flo field: as i moes in he flo field: In clindical coodinaes ssem In clindical coodinaes ssem: k j i Ñ ú ú û ù ê ê ë é cl Ñ q q Ñ q q q e e e 88 Angla Defomaion Angla Defomaion / / Angla defomaion of a paicle is gien b he sm of he Angla defomaion of a paicle is gien b he sm of he o angla defomaion o angla defomaion da db dg d d d d db da g d d 0 0 lim lim & Rae of sheaing sain o he ae of angla defomaion Rae of sheaing sain o he ae of angla defomaion d d d d db db d d d d da da & & an, an

45 an d d d d da da & Fo small angles 90 Angla Defomaion Angla Defomaion / / The ae of angla defomaion in plane The ae of angla defomaion in plane The ae of angla defomaion in plane

46 0--4 Eample: oici Fo a ceain o-dimensional flo field h eeloci is gien b 4 i ( - ) j Is his flo ioaional? 9 Eample Solion This flo is ioaional 9 46

47 0--4 Conseaion of Mass /5 Wih field epesenaion, he pope fields ae defin ed b coninos fncions of he space coodinaes a nd ime. To deie he diffeenial eqaion fo conseaion of mass in ecangla and in clindical coodinae ss em. The deiaion is caied o b appling conseaio n of mass o a diffeenial conol olme. The diffeenial fom of conini eqaion??? 93 Conseaion of Mass /5 The C chosen is an infiniesimal cbe ih sides of lengh d, d, and d. ò C p d ddd d ( ) d d - ( ) - d Ne ae of mass Oflo in -diecion 94 47

48 0--4 Conseaion of Mass 3/5 Ne ae of mass Oflo in -diecion é ê ë ( ) d ( ) d ( ) ù é ú d d - ê - û ë Ne ae of mass Oflo in -diecion Ne ae of mass Oflo in -diecion ù dd ú û ( ) ( ) ddd ddd ddd 95 Conseaion of Mass 4/5 Ne ae of mass Oflo ( ) ( ) ( ) é ê ë ù ú d dd û The diffeenial eqaion fo conseaion of mass ( ) ( ) ( ) Conini eqaion Ñ

49 0--4 Conseaion of Mass 5/5 Incompessible flid Ñ 0 Sead flo ( ) ( ) ( ) Ñ 0 97 Eample: Conini Eqaion The eloci componens fo a ceain incompessible, sead f lo field ae? Deemine he fom of he componen,, eqied o saisf he conini eqaion

50 0--4 Eample Solion The conini eqaion - - ( ) -3 - Þ -3 - f (, ) 0 99 Conseaion of Mass Clindical Coodinae Ssem /3 The C chosen is an infiniesimal cbe ih sides of l engh d, dθ,, and d. The ne ae of mass fl o hogh he conol sfa ce é ê ë q The ae of change of mass inside he c onol olme dqdd q ù ú ddqd û 00 50

51 0--4 Conseaion of Mass The conini eqaion B Del opeao Clindical Coodinae Ssem /3 ( ) ( q) ( ) q Ñ e eq k q The conini eqaion becomes Ñ Conseaion of Mass Clindical Coodinae Ssem 3/3 Incompessible flid ( ) ( ) ( ) q Ñ 0 q Sead flo ( ) ( ) ( ) Ñ 0 q q 0 5

52 0--4 Seam Fncion /6 Seamlines? Lines angen o he insananeos eloci ec os a ee poin. Seam fncion Ψ(,) [Psi]? Used o epesen he eloci componen (,,) and (,,) of a o-dimensional incompe ssible flo. Define a fncion Ψ(,), called he seam fncion, hich el aes he elociies shon b he fige in he magin as - 03 Seam Fncion /6 The seam fncion Ψ(,) saisfies he o-dimensional fo m of he incompessible conini eqaion 0 Þ - 0 Ψ(,)? Sill nknon fo a paicla poblem, b a leas e hae simplif he analsis b haing o deemine onl one nknon, Ψ(,), ahe han he o fncion (,) and (, ). 04 5

53 0--4 Seam Fncion 3/6 Anohe adanage of sing seam fncion is elaed o he fa c ha line along hich Ψ(,) consan ae seamlines. Ho o poe? Fom he definiion of he seamline ha he s lope a an poin along a seamline is gien b d d seamline eloci and eloci componen along a seamline 05 Seam Fncion 4/6 The change of Ψ(,) as e moe fom one poin (, ) o a neal poin (d,d) is gien b d d d -d d >> d 0 Along a line of consan Ψ d d seamline >> -d d 0 This is he definiion fo a seamline. Ths, if e kno he fncion Ψ(,) e c an plo lines of consan Ψo poide he famil of seamlines ha ae helpfl in isaliing he paen of flo. Thee ae an infinie nmbe of seamlines ha m ake p a paicla flo field, since fo each consan ale assigned o Ψa sea mline can be dan

54 0--4 Seam Fncion 5/6 The acal nmeical ale associaed ih a paicla seaml ine is no of paicla significance, b he change in he al e of Ψ is elaed o he olme ae of flo. dq d - d d d d q ò d - 07 Seam Fncion 6/6 Ths he olme flo ae beeen an o seamlines can be ien as he diffeence beeen he consan ales of Ψ defi ning o seamlines. The eloci ill be elaiel high heee he seamlines a e close ogehe, and elaiel lo heee he seamlines a e fa apa

55 0--4 fign_06_p74a 09 Seam Fncion Clindical Coodinae Ssem Fo a o-dimensional, incompessible flo in he θ plane, conseaion of mass can be ien as: q ( ) 0 q The eloci componens can be elaed o he seam fncion, Ψ(,θ) hogh he eqaion and q - q 0 55

56 0--4 Eample: Seam Fncion The eloci componen in a sead, incompessible, o dime nsional flo field ae 4 Deemine he coesponding seam fncion and sho on a sk ech seeal seamlines. Indicae he diecion of glo along he seamlines. Eample Solion Fom he definiion of he seam fncion - 4 f() - f() - Fo simplici, e se C0 C - Ψ0 ±, - / Ψ 0 56

57 0--4 Conseaion of Linea Momenm Appling Neon s second la o conol olme DP F Pssem dm d D òm(ssem) ò(ssem) SYS D( dm) df dm D D dm dma D Fo a infiniesimal ssem of mass dm,, ha s he T he diffeenial fom of linea momenm eqaion? 3 Foces Acing on Elemen / The foces acing on a flid elemen ma be classified as bod foces and sface foces; sface foces inclde nomal foce s and angenial (shea) foces. df dfs dfb dfs i dfs j dfs k df i df j df k b b b Sface foces acing on a flid elemen can be descibed in e ms of nomal and sheaing sess es. s n lim d 0 d F n da lim d 0 df da df lim d 0 da 4 57

58 0--4 fig_06_ 5 Doble Sbscip Noaion fo Sesses The diecion of he sess The diecion of h e nomal o he pla ne on hich he s ess acs 6 58

59 0--4 Foces Acing on Elemen / df s df s df s df df df b b b s ddd s ddd s ddd g ddd g ddd g ddd Eqaion of Moion 7 Eqaion of Moion df dma df dma df dma g g g s s s These ae he diffeenial eqaions of moion fo an flid saisfing he coninm assmpion. Ho o sole,,? 8 59

60 0--4 Iniscid Flo Sheaing sesses deelop in a moing flid becase of he isc osi of he flid. Fo some common flid, sch as ai and ae, he iscosi is small, and heefoe i seems easonable o assme ha nde s ome cicmsances e ma be able o simpl neglec he effec of iscosi. Flo fields in hich he sheaing sesses ae assmed o be ne gligible ae said o be iniscid, noniscos, o ficionless. Define he pesse, p, as he negaie of he nomal sess - p s s s 9 Ele s Eqaion of Moion Unde ficionless condiion, he eqaions of moion ae edced o Ele s Eqaion: g g g p - p - p - g - Ñp ( Ñ) 0 60

61 0--4 Benolli Eqaion /3 Ele s eqaion fo sead flo along a seamline is g - Ñp ( Ñ) Selecing he coodinae ssem ih he -ais eical so ha he acceleaion of gai eco can be epessed as g -gñ ( Ñ) Ñ( ) - ( Ñ ) - gñ - Ñp Ñ( ) - ( Ñ ) eco ideni. Ñ Ñ d s p Wih Benolli Eqaion /3 ( ) gñ ( Ñ ) ( Ñ ) ( ) ds gñ ds [ ( Ñ ) ] ds Ñp ds Ñ s ds di dj dk p p p Ñp ds d d d dp pependicla o pependicla o 6

62 0--4 fign_06_p80b 3 Benolli Eqaion 3/3 Ñ ds Ñ p p ( ) ds gñ ds 0 dp d Inegaing ò ( ) gd 0 dp g cons an g cons an Fo sead iniscid, incompessible flid ( commonl called ide al flids) along a seamline Benolli eqaion 4 6

63 0--4 Ioaional Flo /4 Ioaion? The ioaional condiion is Ñ 0 [In ecangla coodinaes ssem [In clindical coodinaes ssem q q - q - - q 0 5 Ioaional Flo /4 A A geneal flo field old no be ioaional flo. A A special nifom flo field is an eample of an io aion flo 6 63

64 0--4 Ioaional Flo 3/4 A geneal flo field ð A solid bod is placed in a nifom seam of flid. Fa aa f om he bod emain nifom, and in his fa egion he flo is ioaional. ð The flo aond he bod emains ioaional ecep e ne a he bonda. ð Nea he bonda he elo ci changes apidl fom eo a he bonda (no-sli p condiion) o some elai el lage ale in a sho di sance fom he bonda. Chape 9 7 Ioaional Flo 4/4 A geneal flo field ð Flo fom a lage eseoi enes a pipe hogh a seamlined enance hee he eloci disibion is esseniall nifom. Ths, a enance he flo is ioaional. (b) ð In he cenal coe of he pipe he flo emains ioaional fo some disance. ð The bonda lae ill deelop along he all and go in hi ckness nil i fills he pipe. iscos foces ae dominan Chape

65 0--4 Benolli Eqaion fo Ioaional Flo /3 The Benolli eqaion fo sead, incompessible, and inisc id flo is p g cons an The eqaion can be applied beeen an o poins on he s ame seamline. In geneal, he ale of he consan ill a fom seamline o seamline. Unde addiional ioaional condiion, he Benolli eqaio n? Saing ih Ele s eqaion in eco fom ( Ñ) - Ñp - gk Ñ ( ) - ( Ñ ) ZERO Regadless of he diecion of ds 9 Benolli Eqaion fo Ioaional Flo /3 Wih ioaional condiion Ñ 0 ( Ñ) - Ñp - gk Ñ( ) - ( Ñ ) Ñ( ) Ñ( ) - Ñp - gk d Ñ >> ( ) d d - Ñp d - gk d dp dp ( ) - - gd >> d( ) gd

66 0--4 Benolli Eqaion fo Ioaional Flo 3/3 Inegaing fo incompessible flo ò dp g con an p g cons an This eqaion is alid beeen an o poins in a sea d, incompessible, iniscid, and ioaional flo. p g p g g g 3 Saic, Sagnaion, Dnamic, and Toal Pesse /5 Each em in he Benolli eqaion can be inepee d as a fom of pesse. p g C Each em can be inepee d as a fom of pesse ðp is he acal hemodnamic pesse of he flid as i flos. To mease his pesse, one ms moe alo ng ih he flid, hs being saic elaie o he m oing flid. Hence, i is emed he saic pesse seen b he flid paicle as i moes. 3 66

67 0--4 Saic, Sagnaion, Dnamic, and Toal Pesse /5 ðthe saic pesse is meased in a floing flid sin g a all pesse ap, o a saic pesse pobe. The saic pesse p gh3- p3 gh3- gh4-3 ðg is emed he hdosaic p esse. I is no acall a p esse b does epesen he change in pesse possible d e o poenial eneg aiai ons of he flid as a esl of eleaion changes. g h 33 Saic, Sagnaion, Dnamic, and Toal Pesse 3/5 ð / is emed he dnamic pesse. I can be inep eed as he pesse a he end of a small be insee d ino he flo and poining pseam. Afe he iniial ansien moion has died o, he liqid ill fill he be o a heigh of H. The flid in he be, inclding ha a is ip (), ill be saiona. Tha is, 0, o poin () is a sagnaio n poin. Sagnaion pesse p p Saic pesse Dnamic pesse 34 67

68 0--4 Saic, Sagnaion, Dnamic, and Toal Pesse 4/5 Thee is a sagnaion poin on an saiona bod ha is placed ino a floing flid. Some of he flid flo s oe and some nde he objec. The diiding line is emed he sagnaion seamlin e and eminaes a he sagnaion poin on he bod. Neglecing he eleaion ef fecs, he sagnaion pess e is he lages pesse obainable along a gien s eamline. sagnaion poin 35 Saic, Sagnaion, Dnamic, and Toal Pesse 5/5 The sm of he saic pesse, dnamic pesse, and hdosaic pesse is emed he oal pess e. The Benolli eqaion is a saemen ha he oal p esse emains consan along a seamline. p g pt cons an Consan along a seamline 36 68

69 0--4 The Pio-saic Tbe /5 Knoledge of he ales of he saic and s agnaion pesse in a flid implies ha h e flid speed can be calclaed. This is he pinciple on hich he Pio-s aic be is based. p p p / Sagnaion pesse p p 4 3» 4 p - p 3 4 >> p (p Saic pesse 3 / - p 4 ) / 37 fig_03_07b 38 69

70 0--4 fig_03_e06a fig_03_07a 70

71 0--4 Applicaion of Benolli Eqaion / The Benolli eqaion can be applied beeen an o poins on a seamline poided ha he ohe h ee esicions ae saisfied. The esl is p g p g Resicions : Sead flo. Incompessible flo. Ficionless flo. Flo along a seamline. 4 Floae Measemen in in pipes /5 aios flo mees ae goened b he Beno lli and conini eqai ons. p p Q A A The heoeical floae Q A ( p - p ) [ - ( A / A ) ] Tpical deices fo measing floae in pip 4 7

72 0--4 Eample: eni Mee Keosene (SG 0.85) flos hogh he eni mee shon i n Fige E3. ih floaes beeen and m 3 /s. Deemine he ange in pesse diffeence, p p, needed o mease hese floaes. Knon Q, Deemine p -p 43 Eample Solion / Fo sead, iniscid, and incompessible flo, he elaionship be een floae and pesse p - p Q [ - (A A / A The densi of he floing flid 3 SG 0.85(000kg/m ) H O The aea aio ) ] 850kg/m ( p - p ) Q A Eq. 3.0 [ - (A / A ) ] A /A (D / D ) (0.006m / 0.0m)

73 0--4 Eample Solion / The pesse diffeence fo he smalles floae is 3 3 ( ) p - p (0.005m /s) (850kg/m ) [( p / 4)(0.06m) ] 60 N/m.6kPa The pesse diffeence fo he lages floae is p - p ( ) (0.05 )(850) [( p / 4)(0.06m) ] N/m 6kPa.6kPa p -p 6kPa 45 iscos Flo To incopoae iscos effecs ino he diffeenial ana lsis of flid moion Geneal eqaion of moio n s g s g s g XSess-Defomaion Relaionship aaa 46 73

74 0--4 Sess-Defomaion Relaionship / The sesses ms be e pessed in ems of h e eloci and pesse field. Caesian coodinaes s s s -p - mñ m 3 -p - mñ m 3 -p - mñ m 3 m m m 47 Sess-Defomaion Relaionship / s s s qq q q -p m q -p m q -p m q q m m q m q q q Clindical pola coodinaes Inodced ino he diffeenial eqaion of moion

75 The Naie The Naie-Sokes Eqaions Sokes Eqaions /5 /5 These obained eqaions of moion ae called he Na These obained eqaions of moion ae called he Na ie ie-sokes Eqaions. Sokes Eqaions. ú û ù ê ë é Ñ - m ú ú û ù ê ê ë é m ú û ù ê ë é m - ú ú û ù ê ê ë é m ú ú û ù ê ê ë é Ñ - m ú ú û ù ê ê ë é m - ú û ù ê ë é m ú ú û ù ê ê ë é m ú û ù ê ë é Ñ - m - 3 p g D D 3 p g D D 3 p g D D Caesian coodinaes Caesian coodinaes 50 ú ú û ù ê ê ë é q m - q ú ú û ù ê ê ë é q q - m q - q ú ú û ù ê ê ë é q - q - m - - q q q q q q q q q q q q q q q q g P g p g p The Naie The Naie-Sokes Eqaions Sokes Eqaions /5 /5 Clindical pola coodinaes Clindical pola coodinaes

76 The Naie The Naie-Sokes Eqaions Sokes Eqaions 3/5 3/5 Unde Unde incompessible flo ih consan iscosi incompessible flo ih consan iscosi condiions condiions, he Naie he Naie-Sokes eqaions ae edced Sokes eqaions ae edced o: o: m - m - m - g p g p g p 5 The Naie The Naie-Sokes Eqaions Sokes Eqaions 4/5 4/5 Unde Unde ficionless condiion ficionless condiion, he eqaions of moion he eqaions of moion ae edced o ae edced o Ele s Eqaion Ele s Eqaion: g p g p g p p g D D - Ñ

77 0--4 The Naie-Sokes Eqaions 5/5 The Naie-Sokes eqaions appl o boh lamina an d blen flo, b fo blen flo each eloci componen flcaes andoml ih espec o ime a nd his added complicaion makes an analical soli on inacable. The eac solions efeed o ae fo lamina flos i n hich he eloci is eihe independen of ime (se ad flo) o dependen on ime (nsead flo) in a ell-defined manne. 53 Some Simple Solions fo iscos Incompessible Flids A pincipal difficl in soling he Naie-Sokes eqaions is becase of hei nonlineai aising fom he conecie accel eaion ems. Thee ae no geneal analical schemes fo soling nonlinea paial diffeenial eqaions. Thee ae a fe special cases fo hich he conecie accelea ion anishes. In hese cases eac solion ae ofen possible

78 0--4 Sead, Lamina Flo beeen Fied P aallel Plaes /5 The Naie-Sokes eqaions edce o p 0 - m p p g 0-55 Sead, Lamina Flo beeen Fied P aallel Plaes /5 p g p 0 - p 0 - m Inegaing Inegaing p -g f ( ) p c c c? c? m 56 78

79 0--4 Sead, Lamina Flo beeen Fied P aallel Plaes 3/5 Wih he bonda condiions 0 a -h h p c 0,c - h m eloci disibion 0 a ( ) h p - m 57 Sead, Lamina Flo beeen Fied P aallel Plaes 4/5 Shea sess disibion p olme flo ae h h 3 - h ò- h -h q d ò p ( m h p )d - 3m p p cons an 3 h Dp >> q 3ml - p l Dp - l 58 79

80 0--4 Sead, Lamina Flo beeen Fied P aallel Plaes 5/5 Aeage eloci q h Dp aeage h 3ml Poin of maimm eloci d 0 a 0 d h p ma U - m 3 aeage 59 Coee Flo /3 Since onl he bonda condiions hae changed, hee is no need o epea he enie analsis of he boh plaes saiona case. p c c c? c? m 60 80

81 Coee Flo Coee Flo /3 /3 The bonda condiions fo he moing plae case a The bonda condiions fo he moing plae case a e 0 a 0 0 a 0 U a b U a b 0 c b p b U c m - Þ eloci disibion eloci disibion ú ú û ù ê ê ë é - m - m - m b b p U b b U b p p b U m p U b P 6 Coee Flo Coee Flo 3/3 3/3 Simples pe of Coee fl o b U 0 p Þ This flo can be appoimaed b he flo beeen closel spa ced concenic clinde is fied and he ohe clinde oaes i h a consan angla eloci. Flo in he nao ga p of a jonal beaing. ) /( b U o i i o i - m -

82 0--4 Plane Coee Flo A ide moing bel passes hogh a conaine of a iscos liqid. The bel moes eicall pad ih a cons an eloci, 0, as illsaed in Fig e E6.9(a). Becase of iscos foces he bel picks p a film of flid of hic kness h. Gai ends o make he fl id dain don he bel. Use he Naie Sokes eqaions o deemine an ep ession fo he aeage eloci of he flid film as i is dagged p he bel. Assme ha he flo is lamina, sead, and fll deeloped p 0 Eample Solion / Since he flo is assmed o be fll deeloped, he onl eloci componen is in he diecion s o ha 0. Fom he conini eqaion, and fo sead flo, so ha () p 0 d -g Inegaing 0 m d d d g m c d m d 64 8

83 0--4 Eample Solion / 0 a h c gh - m 0 a 0 Inegaing g gh - c m m g gh - m m 0 65 Sead, Lamina Flo in Cicla Tbes /5 Conside he flo hogh a hoional cicla be o f adis R. 0, q 0 Þ 0 ( ) 66 83

84 0--4 Sead, Lamina Flo in Cicla Tbes /5 Naie Sokes eqaion edced o p 0 -gsin q - p 0 -gsin q - q p 0 - m Inegaing Inegaing ( sin q) f ( ) p -g p -g f ( ) p c ln c c? c? 4m 67 Sead, Lamina Flo in Cicla Tbes 3/5 A 0, he eloci is finie. A R, he eloci is eo. c p 4m 0,c - R eloci disibion ( ) R p - 4m 68 84

85 0--4 Sead, Lamina Flo in Cicla Tbes 4/5 The shea sess disibion m d d p olme flo ae R p ò R p Q pd m p p cons an 4 - p l 4 -Dp / l pr p pr Dp >> Q - 8m 8m l 4 4 pd pd 8ml 69 Sead, Lamina Flo in Cicla Tbes 5/5 Aeage eloci aeage Q A Q pr R Dp 8ml Poin of maimm eloci d 0 d a 0 ma R Dp - 4ml aeage Þ ma - R 70 85

86 0--4 Sead, Aial, Lamina Flo in an Annls / Fo sead, lamina flo in cicla bes p c ln c c? c? 4m Bonda condiions 0, a o 0, a i 7 Sead, Aial, Lamina Flo in an Annls / The eloci disibion é p i - o ê - o ln 4m êë ln( o / i ) The olme ae of flo p é - ù ò o p 4 4 (o i ) Q (p)d - êo - i - ú 8m êë ln( o / i ) i úû The maimm eloci occs a m o ù ú úû 0 m é o - ù i ê ú êë ln( o / i ) úû / 7 86