lecture5 - HOHEM‘TUM FLU'X IN PUTEW/AL PLow'

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Unformatted text preview: HOHEM‘TUM FLU'X IN PUTEW/AL PLow' ______________________..____._——————— Aime) J m 3 - BY} = "“ 2 -~ OH: 9 db \Rt) efit) kt «4 13y wane OF THE + P <§§ V U» ‘13 2 'nzANS'PoET 'H-LEOQEM SCt) luvonwa EuLee’s ELQUATHS‘Nfi IN INVISCID Haw” FEE/4, (\7.V></:-LVP+ ‘ct we H 44 25051 THE RATE 0F cmwce OF THE HWEM’TW 1:»: HOMENTUM Fund In THE FoRH at CH=~f W [V(%+323+(V.V)U]Jv CH: \T'C-E) +§> Cg :70“ As 3 (t7 SO FAR ’V'CJC) 23 AN AKBH’EARY CLOSE-D TIME DEPENoan' VOLUME BOIUNOED BY 7745 THE DEPENDaJTSMZFACE 57%) . H525 we mean TO (MVOKE AM (HPO (17AM AND CorHLEx V1,:ch “Tm—L EE—CALL FROM THE PRooF oF BERMO‘ULLI'f EtQUG‘TION THATI mm : wwm-W >87) ' C By \IHZ‘TUE 0F GAu-ss's VECTGrz THH,’ [M ‘PO’TE-U'TIAL Fwd IT CAN BE SHowu THAT: ($3) 2(V.V>Y1C19 CEX’fivctacls S&) S&) 3 CS? Vh-C/CJS SL-t) P0200? LEFT As AM exacst .' TUST PEcDVE THAT I:on Vick—0', UPoN SUBSTITU’TION N) THE Mofuewum FLUX FoKHULA , w-E GEL/“24w : 21:4 = __P % ‘f {7 (Vial—Uh)? 9L3 CW, 3(6) FOLLowmc AN APPLICATION OF THE vac-rag THEGREM o-FCAUSS. IN THE EXPfZESS‘ImV ABOVE 75' 13 THE vow-r VECTon Favour/Na am- 0? ‘THE VOLWE (7)719?) szfi. v4: AND UV, [3 THE curt/6A1th Macawu. vewcrry OF THE SURFACE SIJC),__ THlS‘ FORMULAIIS OF CEUTRAL (H P012774 m5 HU PSTEUTIAL. FLOW HAEZVE HYDEOD‘IMAHMS BECAUSE-3 THE RATE OF CHAuaE OFTHE réwauz HGHE’U’TOH DEFINE—D ABDVE )s TUST iTHE FORCE ACTING ON T345 {32.12713 VOLWE . WHEN ITS ME‘kM \IflLUEVCAIV BE S‘HDwMTO VAMSH ) :MPOIz—rAM—r Fmg 5x92 esszams cm! Sour) 5w Mommas Foam) MD WILL BE; DERIVED (N w HAT Rummy... "6:99 Nara-Ex; S‘EPARA‘FEU/ TH E, TERM w THE, H9 ME WU H FLUX EX PRESS-(VON IN VOLWNC THE H 4 D ROSTATIC P QESSU s25 : =‘(3 #33?) amaze s=§F+si+§B 43; 4t 3 + 514, THE IWEQFAL ova-(L THE 800‘! SUQFALE , ASSUMma k FULLY SuBHEQQEO BODY 15‘ : BY vuzrue OF THE VECTOR Tm;er M 0F 6341193 “(Hrs 23 Nowe OTHER THAN THE Pewqu OF Aacmueoes .‘ we HAv THEQEFGRE CONSIDER THE SECOND VAN of: THE tau-(ea [EAL Invowwg WAVE ET'FEC'FS lUDEPENOE-NTL‘I AND w THE Arisavcs 0? "WE BODY , ASSUHEO Fuuxl S’UBHERGE‘D- In! 'rHE CASE OF A _S‘or2FACE Pzgectud BODY Amt) IN ‘THE Fu‘u—v NONLINEAR CASE MATrEvas ARE MORE CENPLEX’BUT TEA CTA BLE/ CoMSIDEQ THE APPLICA'FIDN OF THE Housman CONSERVATION “rt-160% W TI+E CASE oFA SOBHERQED 0R FwATtNG BZDOY IN $75510 ’ WAVES: HERE we CGNSIDEI? Tm; Two—DWENgIOWN, 0495 W caper? To PRES EMT Tue CONCEPTS. EXTENSIONS ‘To 'T'HQEE DtHENS‘IONS A25 THEN TQM/)AL. NOTE THAT uuume THE Enemy CauseszT/ow PEWCWLE , THE HDHEN‘TUM cowSEfivAT/ow THEOREM ova—Eweo ABOVE]? A warm: TDEM’H'TY WITHV‘A Homzoumt, AMP A Warm CAL Cayman/5101 THE mrrEGEAL OF THE H70 12037471.: “FE-12H ouee THE EEH AINING SURFACES LEADS TD: oligfi: __E 015‘ ‘-'-‘ ~€3VFLUID i: SF+S"+ 31 s” Tms lS‘ .S‘IHPU/ THE STATIC WEIGHT oF—‘T‘HE VOLUME OF FLUID Bovuoep 3., SF)9+} S f. AND Spa. 0mm no WAVES P42535NT,TH19 rs SIMPLY ‘THE szGH7 0? THE OCEAN 0.9147532 "COWN" gwoeo 31 S WHICH Dog-y NUT CONCEEN us HER-E- THIS‘ wEMHT D053 Mar CHANCE ltd PErNCIPLE WHEN wAVESAIZE PQESE‘UT AT LEAST DUNE—1U 3+)3'ARE FLA CED 5‘0 FFICIENTLV F—AIZ AMAY THAT THE w/MIE AMPIJTUDE H—AS D-EchaASEO TO “ZERO. So “ w Pewcmg“ THIS” TéRM BEtNCOF 14VDROSTATIC 0,2769N MAW BE \GworeED. HOwE'vE-IZ) 17(3 IN PEINC/Pu; 1‘ H Note fiAchALTO APPLY THE HOME-Wm CONSERVATION THEoQEH overz THE 0 II ' LINEAR! ZED Vourue “\J'LLH Comm (3 PERFECTLY P0 391 Bug correr THE FRAMEWORK DEBNED ABOVE. IN THIS CASE OHM IS EXACTLV EaQUAL To “mg with“; may “meta/A (mom PROBLEM. 63“ 3F 3” 4;:ng :o AMO HENCE ALL TERMS cow'er THE F1265 momma; INTEGRAL AND oven Spa [SE-AFLOoeJ CAN ' BE NE Q'LECTEQ I‘T Fouows THAT 3 NOTE THAT THE FREE summce {me—0mg ALSO VANIS H PM THE HcfiIZMUTAL Cat—(’PONENT SINCE THE HVQROSTATIC 'Fofacar Is ALLUAYS VEPW‘ML’ INTERACTIONS AND HA3 HANy 1MPO RTANT kPPLIcA'moNS ,9ouE oF Wt—HCH ARE “DISCUSSED BELOW. Q NGTE THA‘T THE MATHEMATICAL DEfi7VA-TIONS INVOLVED w ITS PROOF APPLY EdJuALLy wHEM'THE. VOLU‘HE \I‘ AND (TS ENCLOSED SURFACE ARE SELECTED TO BE AT mam LlNEAQV‘ZEQ Posmcnvs- IN SUCH A C1195 1T (5 ESSENTIAL To 587 Un :20 AND RECALL THATO'TU 2:67 /F#b AUG V»: #0 . LET THE MATH TALL: ovaa AND $044687 THE PRC) PEQ EXPRESSION FOR THE FORCE . IN Twig FULLY NONLINEAR CASE D Uni-7150 ' on 3.: AND 45:0 cm 31:: On A Soup BOUNDARY: Um = \fb AND E: 33107343 5‘15 d born! In POIMTWG IMSIDETHE BODY. we HM THEREFORE BECAS‘T THE Homemuu CE’NSERURTIQN THEOREM (N THE FORM : é£+>= ~ 3% fie SQ (r76? NMJHHCJS t Si WHER£ 3 i ARE FLUID BOUNDARIES AT SOME OtSTANCE Fem THE 86W. IF THE VOLUME 0F FLUID Suekouwoeo BY THE 800‘!) FREE SUEFACE AND THE SUfEFAC-E‘S 5*— Does NCSTG‘IZ‘OW IN TIME , THEM THE HO'IME-MTW OF THE EMCLDSED FLUID CAN/WW (3201» EITHHJSO “(£9 13 A STATION/42y PHYSICAL (QUAW'TY. l‘r IS A WELL muoqu RESULT THkT 7H2 HE—AM VALUE W ‘T’IMEoF THE‘TINEDEQ/VATIVE OP 4 STATION/\sz cgummz‘v 23 25%. so: -6: em _0 dt PKGOF ‘4: “r if t L 3 AFC?) CAT —E I T400 2‘T ._.T AT = 'W L [FCT)_PC—T)j ~19,” Z‘T‘ THIS (3 THE FUNDAMENTAL FonHULA Umoazu/wq, THE DEFINITION oF THE MEflN cums 92:97 FoECE§ AchG om FwATrwa BODIES. SUCH Peace: M25 V527 IMPoa‘MmFoiZ S’TATIBNARV FLEA‘TINC STEUCTUEZES AND Ck” BE EXPRESSED IN TEEMSGF (MTEQEDQLS op CUA'UE EéFECFS Oval CWEcDL So R’FACES I g *- OQHICH HA“! BE LGCATED AT INFINITY . THE EXTEN'S‘lON OF THE ABOUE FoiZMULA FOR it!“ THREE PIHENSUD’NS 13 TRIVIA!“ Smva REPLACE 3i 8? $09 ) A WHEEL §UEFACE AT IMF]N1TY.COMHO1\I CHO'ICi-ZY ARE; A VERTICAL CquDfiMAL BOUNDARY orz'rwc) VEQJTICAL PLANES PA KALLEL To THE AXIS 0F ForawAtao Nana“; 05:4 5H4) APHJCATIONS . HE‘AM HOEIZOU‘TAL HOHENTUM FLUX DUE TO A PLANE PROGRESSIVE cum/E 3+ §=A C§S(w":-kx) wl=3k+quch \\\'\'\\\ \ \\ \\\\\\\.\ EVALU ATE THE NOHE‘MTUM FLUX Ackass S + 2‘ IA cm 3 41:: ’ 3 (13+?“13 4% 'H 1‘): ~(3 g? -%f>(u7'+vz) —€j‘% :BngooLu AHx V ST[3+L+ Lbfi—u” +3%]JE A); 2+. 2 'H =P(S°+37> gwwawz 'H b . TAKHUC MEAN VALUES IN TIME AND ILEE'PM/d TERMS OF OCA”) CUE OSTAUU: ‘ 4c t t O Ar”: = emu-fit “as 3(v3uz><*z+%eaxz “H INUOKWG THE UMHRIZED DYUAHML FREE SoeFACE CONDITION: 2+: 2 — 3 bk i=0 (a yr Famoqu THAT: t at t 0 AW 2 t. (vium r=—%€%§Ctj+Z—§)~E 2’ o 1N DEEP W‘A'Tefi 71: at A no ’THE secmuo TERM IS tDEMthALw 25120 ) so ,. Jo t «Am 2 = - i ' 1.1%) «-1. ch; 2 ‘3 463A 0 IN (HATER a? le'rE DEPTHTHE wAUE PARTICLE TEA‘IECUMZIES AIZE ELLIPTICAL ._ 2 thH THE HEfiN HORIZWAL.V'EWCITIE:S 2, L/HLGHI THAN ‘THE MES-AN VERTICAL vacrnEs so; "‘2. H7, vH < LLH THE-4261:0125, w szTE DEPTH THE- MODULUS UFTHE MEAN MoHEMTU H FLUX Is HM‘HEKTHAN m DEEP wATEK F012 THE SAME A... So THE HE-ANHOQIZOMTAL HOHENTUM FLUX DUE To A PLANE PEoszSSWE wAvE [S AGAINST ITS DMZ-Scrum OF P£oFAGATIW AND ECDUAL To ~4L€3A3 . CQS‘N9iDEfZ A wAvEHA KER SHOWN IN THE FIGURE GENERATING A WAVE OF AMPLITUDE A AT INFINITY: WHAT !3 THE HE‘AN HQKIZOWAL FORCE. ON’ THE OuAuEHAtER Fnzch THE NOHEWUM Ccs-NSEQVATIGN THEOQEM " T wo (577452 (LLUS’FEAT/OUS‘ OF THE Momarflm FLUX PEWQIPLE ARE NaTc—Eooofa‘fHY: C0103! 0&2 A CuzcuLAQ manger [M The DIMEUS/opj EWATINC AT flw WCULAK FRScQuauo/ w UNDER A FREE SURFACE MD 001794 A SHALL AMPLMU DE ., w arr—(5.2 wow: "n+5 queue C—E‘NTEK REST/47125 AS. A FLUID PARTICLE nu A. REQULAQ WAV“ “Ago ‘ “IR/Ci)” b ‘- ' ~* w sz é": L=“Eg€3/AE,L IT cm 132 swam THAT Accoppwo To LHUEAR THEoey Tws \EGSTATIONAL. HGSTwN mean-es Rélcuwz comes PEDPACATINCZ To THLS RIGHT thH N0 wAuES PRIDPA GA'TYNG To THE LEFT/ wH AT )3 (ma Maw Hoez'zournp FoRCE Ac'mm 0N THE CIRCLE ? PM THE MoHeurug cowsEm/wrzow mwmpu; THE MEAN Fofics EAcrwa 011/ A THE Came MUST BE Ema 122 L >774”: (—‘ -2bL *A2 r ‘N . VDan (W533 HASSHoww THAT {Zeouma PLANE PEOGQESSIVE WMES (N (DEEP WATER WaPAcrAv-wc ova? Acmue FIXED UNDE’Q THE FREE svkfime M26 UMTHZELV TMmsmwep gz': A 00$ (W't—erf—Le w OTHER wows “I‘m; wAvE REFLECTlaw :3 25m)- “GTE “HHS 13THE ONLY Booy IN 2 mug/$10M: 00er “THIS P2005277, REFLECTIOM \3 MW~ZEEO w szrc—r DEPTH FOR ALL. WW” 1800183 . 3H ow 8% HPPLYING F1745 MOMEMTOM CWSEQVA’TIO‘N prHVC-HOLE THAT Tm; MEAN HO EIZUNTAL F2120; [:26 a- 9 'PLA/Je Preoaaesszut; MAVES w GENE/ML DIRECTtowg. (A,w) \3 F1355 SUQFACE ELEVI‘ATIOM : 3‘: A gas ow“ lamp» 1:75!an 1A me {9 —~L‘{<’XcosF—~c‘lzysv'n{3 +:‘wi—} VELOCITY PJTENTSAL (N mes DEPTH d8; ng g cost, kéz—qLH) y w 003th 9 wAuES REFLECT‘IMG 0?? A VERTICAL emu. "(Hts IS owe @F THE FEW VERY )HPO {1an AMALY T'lCAL. SOLUTlcr‘NS OF REGULAR wAvE—S Imamc'rwa szH gouo 8mm DAR/ES SEEM lefiACTtCs, THE UFHESQ IS wa/EHA 14512 THEORV- Ag A1 A zwfik “Z, 7 / Z X BREAKduA—f‘EQ / 5 E Hff'llrr/rrrI/rrrlr. 25H LETfi‘FHE IMQOENrwAve ELEthTIC/N Be: —-Lk ‘w §::RQ{AIQ ¥++ ‘1} AND THE IZSFLECTED BE : g‘z? R2 gAR e-I—ikx +I'wi} wHaze THE 3mm OF THE e L1” TE‘EH H—AS BEEN yea/mac ‘70 951vde A (JO/“IVE PEM’AG‘A‘TING 707HE LGFT. AR CAN BE cmwusx m OKpER flquuma Fore PHASE 9 IFFEKENCESI ,— THE cotzczezspowowa vawcz'ry PWE/UT/ALS‘ ‘ ~ka-H'w6 Ci) 2 file; flzasfimaw) e )1 4‘- 60 Cost: 14H 00 609th 0'70 $<=o Wm». 23 CPR. vaAI-t-iKAZ :2. A2: 412A THE EES‘DLTIMG TUTAL. UELOCIW PWEWAL. (32 ¢:¢I+¢Rt Mira/4 Coshche-H) efwe W wstH ‘ 'wt". 22A COSL'KC&+H)0031<X€[ g N cosh kH THE RESULT/M5? WANDrma—w/WLS ELEVAT/ow 15: 3;: glfi-yg: 60st COSW‘Z). ._ CUEVIFv) o ExEQCIGEs _______..._—- APPLY THE HGHE‘LJ'TLIH Cm usEFruATraw F’EINCIPLETU'DETEEHIME THE HEAN HOEI'ZDMAL Far-ice 0w“ THE BEEAKwflTEfl. v YWFZ REPULT warm THE Mswfie u!th FfiwH Eula-.50"? Pat-35390915 {LITEGE'A Tram .. DE. ENE 15 THE 6-wva FLUX 7’. ES?! FRESTFGHS FUR THE: E—NEEGV y; EWEEGY FLLJX flue, HGHEWW .. DEENE. DE-NSIT FL.th UF Bl—CHEQJHA‘TIC. wAdES 9mm :1»:qu m THE: SAME. DerECTm \S: (q! Cng(th-Tk]r:) + gleasfw1£-k1x) HERE,de THE EYE ECUE: FOR mAues PKaPAG‘aTWC m “THE @PPosrn—f DmECTrami ‘ 3": fl! mjfwlt_ klx) + filmsfiwlt+tzfl A‘SSdHE DEEP HATER. bamlfig ‘THE “ST/moms ” cows 95:;- Ar 6R.an KHA‘TER. Font A wAuE. [Malpfiw h'T Au ANGLE [.‘f —|_.._ ...
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.24 taught by Professor Pauld.sclavounos during the Spring '02 term at MIT.

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lecture5 - HOHEM‘TUM FLU'X IN PUTEW/AL PLow'

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