10. Euler's equation (differential momentum equation)

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1 3 Ele's eqaion (iffeenial momenm eqaion) Inisci flo: µ eslan of foces mass acceleaion Inisci flo: foces case b he esse an fiel of foce In iecion: ) ( a o a If (), ) ( a a () If cons he nknon aiables ae:,,, Benolli eqaion Inisci flo: µ V IV III II I a s s o s s a s a) Since f f f s a ineal II

2 4 b) If - au ineal IV (U U ) c) In case of sea flo ( ) ineal I ) ineal III, if - saic fli - o oenial flo - s lies in he lane eemine b an o ecos - s ineaion alon seamlines - s o ineaion alon oe lines e) If cons ineal V, if (), ineal V ( ) In case of inisci, sea flo of incomessible fli ( cons ), if - au an ineaion alon seamlines: U U The Benolli's sm cons alon seamlines Saic, namic an oal esse U U In sanaion oin, so namic esse saic esse oal, sanaion esse Benolli eqaion in case of inisci, sea flo of incomessible fli, iseain he fiel of foce: he oal esse is consan alon seamlines 3 Ele eqaion in seamise ("naal") co-oinae-ssem Sea flo of inisci (µ ) fli e cooinae is anen o he seamline, n is nomal o i an coss he cene of cae, b binomal cooinae eenicla o e an n

3 5 In e iecion Foce acin on iffeenial fli aicle of ee lenh b, n an e (mass: iecion: Fe b n e b n b n e e e, hee e he e comonen of he fiel of foce Since he flo is sea onl conecie acceleaion eiss, an bne e b n b n e e e e e e e n b a con e m bne ) in e In n iecion m cenieal foce is neee o moe m mass ih eloci alon a seamline of a ais of cae : ebn be n be ebn n n n In b iecion b b In nomal co-oinae iecion, iseain : n Conseqences: a) if he seamlines ae aallel saih lines ( ) he esse oesn' chane eenicla o he seamlines, b) if he seamlines ae ce he esse chanes eenicla o he seamlines: i inceases oas fom he cene of cae n

4 6 aino 4 oain ank Foce oe in absole ssem, [ / s] anla eloci,? 3 iffeen as of solion: a) co-oain co-oinae ssem: hosaics b) absole ssem Benolli eqaion; c) absole ssem, Ele eqaion in seamise ("naal") co-oinaessem a) ( ) U U s a s os s as b) I II III IV V Sea flo, ineal I, ineal II ( )/, ineal III since o, an no seamline connecs oins O an Since s ineal IV, cons ineal / V ( ) ( ) os an ( o) / / o, o an s ecos ae eenicla o each-ohe, an fhemoe an : ( ) c) n n, n (seamlines ae concenic cicles), n s,

5 7 5 Measemen of flo ae b sin Veni mee h [m] f(q )? an M ensi of ae an mec U U U-be manomee: h H) (m h) (H H m )h ( H h m H conini eqaion: / ( / ) h 4 H Flo ae: h K 4 q π 6 Unsea ischae of ae fom a ank

6 s I a s II s In oin s U, H s os III s IV V 8 a s, In oin, he eloci is () ' s hee / acceleaion eco / is inicae b a, / he same iecion a a s ' as al s, / an s oin a L H In case of sea flo ( ) s L s H s s s s L s L ah τ s L s h hee s H s τ 7 Floain of boies Bo olme: V, esse isibion is chaaceie b a, Pesse foce: F a V F V In aiaional fiel boan foce eih of he olme islacemen The boan foce eco cosses he cene of islace olme

7 9 The bo is floain if he aeae ensi is eqal o less han he ensi of fli Sabili of floain bo: sbmaines an shis If he cene of ai S is loe han he cene of islace olme K, a momen M aises, eceasin he anle of eflecion If S cene of ai is aboe he cene of islace olme K a momen is aisin o a ceain anle of eflecion eceasin he anle of eflecion eflecion he osiion, manie of eih an manie of boan foce oes no chane The line of alicaion of boan foce islaces s a conseqence of he eflecion a eeshae a of he bo () emees fom he ae an he B a of bo sinks So a cole of foces aise, islacin he boan foce eco The ne line of alicaion cosses he smme lane in oin M (meacene) If S is ne meacene M he shi is in sable eqilibim sae 8 aial-flo fan, Ele eqaion fo bines s: inle, k: scion nole, j: imelle, l: blaes, cs: casin, n: ole, : shaf, m: elecic moo, M: momen, : anla eloci Task: incease of oal esse of as:

8 i o i o aailable efomance: q P, hee q [k/m 3 ] is he flo ae Benolli eqaion in elaie cooinae-ssem (sea flo of incomessible an inisci fli) beeen oins an of he same seamline: V IV III II I a s s os s ) U a(u c Co c,, U U c s s, Since, if o o o Since o ( ) s s os Finall: ( ), ) ( i If i

9 9 Theoems fo oici: Thomson' an Helmhol' heoems Thomson' heoem (inisci fli) Ciclaion: Γ Γ s Temoal chane of ciclaion alon close fli line s? G au an cons o (), b sin Ele eqaion: G s In flo of incomessible an inisci fli in oenial fiel of foce no oici aises If licaions: Sain an soin oe (oe shein), makin eloci isibion nifom, flo in ae eseoi G Γ s o G (o ) (o ) Helmhol' I heoem µ o D ϑ ( ) ϑ D /< /<

Fli oe line: o s, fli oe sface: o Since s, a floin

ohe alon a oe line floin oe line, hich can be eae

of he same fli aicles Conseqence: The oe in smoke

smoke Helmhol' II heoem Floin oe be S s s s s S S S

be an i oes no chane emoall Conseqences: he oe be

10 Fli oe line: o s, fli oe sface: o Since s, a floin oe sface emains G oe sface To oe shees inesec each ohe alon a oe line floin oe line, hich can be eae as line of inesecion of o floin oe sfaces, consiss of he same fli aicles Conseqence: The oe in smoke in o in clo of smoke emein fom a chimne esees he smoke Helmhol' II heoem Floin oe be S s s s s S S S o s, S o o is consan oe all coss secions alon a oe be an i oes no chane emoall Conseqences: he oe be is eihe a close line (a in) o ens a he bona of he flo fiel o

11 3 Ince oe, i oe of finie aifoil Flih of il-ooses in V shae Voe in b afe oenin he sink Tonao Pesse measemens Sface ension F LC C [N/m] sface ension coefficien Fo ae ai combinaion C 5[N / m] ( α )ss Csα Cs C, In case of shees C /, an bbbles: 4C / If C 3 C C3 >, fli eans on he sface of fli (e oil on ae C 3 3 α s C s C cos s

12 4 cosα (C 3 C ) / C3 C 3 > C α < 9, α > 9 (mec) Ha C 3 > C C3, he fli eans oe he sface of soli bo eolem es o of oen bole Cailla ise C3 / C3 cos α / m C3 m cos α In case of mec cailla o Measemen of esse Manomees (fo measin esse iffeences) Micomanomees: U-be manomee - ( m - ) h, "inese" U-be manomee - ( - a ) h, incline be manomee LH/sinα, elaie eo: e s/l ( s/h) sinα, ben be manomee (e cons): Be-micomanomee Pesse as

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