3.4 Energy Equation. Energy Equation

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1 HC Che 4/8/8.4 Eer Eqto Frst Lw of herocs Het Q Eer E S DE Q W Power W Both Q W re th fctos rocess eeet bt the et Q W to the sste s ot fcto E s totl fferetl theroc roert Eer Eqto Reols rsort heore fter l the cott eqto D MV F CV DF F e Coserto of Eer e DE D D e e MV CV q S Q f t S CS CV CV CS Het fl cross CS Het eerto Power roce b bo force Power roce b srfce force CVEN 68: Chter B

2 HC Che 4/8/8 Eer Eqto Iterl for DE D D LHS e e MV CV RHS q S Q f CS CV CV S CS CS CV CV CS q Q q S Q f S CV CV Dfferetl for D e D e q q CV Q f f Q f CV or Mechcl Eer Eqto Moet Eqto D f e er roct wth eloct ector Mechcl Eer Eqto D f D f D f CVEN 68: Chter B

3 HC Che 4/8/8 Eer Eqto Sbtrct echcl eer fro totl eer D e D q herl eer eqto De q q q Q f f Q Q Q 9 ters 7 ters totl echcl.5 Costtte Eqtos Cott Moet Alr Moet Eer sclr D t q ee to be etere fro costtte eqtos e f De q Q CVEN 68: Chter B

4 HC Che 4/8/8 CVEN 68: Chter B 4 Forer s Lw Het fl s roortol to teertre ret Het trsfer fro hh to low teertre Ooste to teertre ret q s ot fcto bt other s of het wor terctos be th fctos q q q Stress Rte-of-Str Rte-of-str tesor Rotto tesor Veloct ret ret eloct

5 HC Che 4/8/8 Rte-of-Str esor Strt wth Crtes coortes Norl Str Eleet Stretch or coresso Ler eforto t t t t t t t t CVEN 68: Chter B 5

6 HC Che 4/8/8 CVEN 68: Chter B 6 Cott Eqto wo-esol coressble flow Eso oe coorte recto st be ccoe b coresso other recto coserto of ole re t t Me Sher Str Eleet rotto or lr eforto t t t t t t t t

7 HC Che 4/8/8 CVEN 68: Chter B 7 Me Rotto Eleet rotto or lr eforto t t Sle Sher Flow Sle sher flow oles both lr eforto rotto Prllel flow = c bt ot rrottol! cott c c rottol flow z c

8 HC Che 4/8/8 Sle Sher Flow Sle sher flow = re sher + re rotto + Prllel streles bt rottol Irrottol Flow Free orte zero ortct Cre streles bt rrottol c r CVEN 68: Chter B 8

9 HC Che 4/8/8 CVEN 68: Chter B 9 Rte-of-Str Geerl Bss P Q R S Rte-of-Str P Q R S r P R P S P Q D : ot R : ot S: ot Q :

10 HC Che 4/8/8 CVEN 68: Chter B Rte-of-Deforto P Q r l Veloct Grets Veloct Gret Geerl Bss Recll tht Veloct ret s c seco-r tesor Note: two bses eee for c r recto ret bss cooet Also

11 HC Che 4/8/8 CVEN 68: Chter B Stress Rte-of-Str Sol Mechcs stress s. str Hoo s lw ler relto Fl Mechcs stress s. rte-of-str Newto fl ler relto Rte-of-str s the setrc rt of the eloct ret tesor Rte-of-str tesor Rotto tesor ortct Rte-of-Str esor Veloct ret Rte-of-Str tesor Dfferet cooets of the se rte-ofstr tesor

12 HC Che 4/8/8 Rte-of-Str esor Cotrrt cooets Cort che of ll three rectos Ioles ches both the te recto crler bss 9 cooets Detorc Stress Hrosttc ressre ε τ Detorc stresses or re cse b fl oto eloct ret I σ I I Fl sttcs Fl cs CVEN 68: Chter B

13 HC Che 4/8/8 Stress Rte-of-Str Sttc Fls t Pressre s orl to the srfce Mo Fls ; or ot ε ; Note: τ τ τ Ele: Newto Fl Stress s lerl roortol to rte-of-str D D Not tesor rt esor rt Crtes tesor Geerl tesor No-Newto fl oler relto C D E q q CVEN 68: Chter B

14 HC Che 4/8/8 Prcl Aes Prcl es At eer ot t eer stt there est three tll ereclr es of rteof-str tesor sch tht Crtes bss If we refer to the - - z-rectos tht re cle to the rcl es we st he ŷẑ z ẑ z ŷ zz z ẑŷ ẑẑ ẑ ŷ ŷŷ z ŷẑ Prcl Aes ŷ ẑ z ŷ z ẑ z z zz orl & sher str orl str ol z CVEN 68: Chter B 4

15 HC Che 4/8/8 ε ε Prcl Aes r r r I Crtes coortes ε z ε r r cost ε z ε z cost z z z c Qrtc Eq. zz 4 5 hs st be ellso or herbolo qrtc lebrc eqto It c be trsfore to cocl for lo chrcterstc rectos he srfce st he three tll ereclr es of setr Frtherore these es re orl to the srfce whe the eet t 6 z z Fl Isotro he fl shows o referece for rtclr recto he rcl es of stress tesor coce wth those of rte-of-str tesor he erto of stress rte-of-str relto c be slfe s rcl stresses cooets ol rcl rtes-of-str cooets CVEN 68: Chter B 5

16 HC Che 4/8/8 Newto Fl Ler relto betwee the rcl stresses rcl rte-of-str b c b c b c Iterche coorte rt b c b c b = c b c Iterche e the se reslts Rotte cclc orer el the se reslt Newto Fl Isotroc o rectol referece b b b b b b = ere ressre howeer hrosttc ressre ol f he roortol costts b re fl roertes.e. fctos of sce te ol c = b CVEN 68: Chter B 6

17 HC Che 4/8/8 Newto Fl Let b = b = the = ere ressre = scost = seco scost I Newto Fl Coressble flow Crtes sste w z w z w w z z Aere ressre s etcl to the hrosttc ressre ol whe rte-of-str = fl sttcs w z If coressble or el s = / CVEN 68: Chter B 7

18 HC Che 4/8/8 Isotroc esor Ele for D cse wth setrc Rotte b 8 o or or Isotroc esor Iterche coortes Reces to coeffcets Frther slfctos? 8 coeffcets D cses!! CVEN 68: Chter B 8

19 HC Che 4/8/8 Newto Fl Stress s lerl roortol to rte-of-str Veloct rets Rte-of-str tesor; setrc Rotto tesor; t-setrc hs o cotrbto to the stress/rte-of-str relto becse both re setrc l e l J bss Newto Fl Stress s lerl roortol to rte-of-str D D s 4th-orer sotroc tesor 8 cooets b c etc. b c b c 4 b4 c 4 How o we etere the coeffcets eerl crler coortes? Fls hs o recto of referece Isotroc esors - her les re efe the se w ll coortes D b5 b6 b7 b8 b9 c c c c c CVEN 68: Chter B 9

20 HC Che 4/8/8 Isotroc esors here re ol fte ber of sotoc tesors for ech bss Seco-orer sotroc tesor Iett tesor bss bss or bss hr-orer sotroc tesor ertto tesor bss bss ol oe ech bss Isotroc esors Forth-orer sotroc tesor: cobto of ett tesors δ δ δ δ δ Ol three ossble cobtos e to setr: coorte rt er rotto/terche δ δ δ bss bss bss CVEN 68: Chter B

21 HC Che 4/8/8 Assocte esors How to che fro to ssocte tesors? b c Ler cobto D b b c b c ol ossble cobtos D c Newto Fl How to che fro to? or b c ol ossble cobtos! b c b c b CVEN 68: Chter B

22 HC Che 4/8/8 Alterte Dertos here re ol three eeet sotroc cotrrt tesors of orer for D D D D he lst ter hs o cotrbto to setrc therefore Setrc bt ot D Newto Fl [ herefore ] I CVEN 68: Chter B

23 HC Che 4/8/8 CVEN 68: Chter B Sr: Newto Fl D C ; to s lerl roortol costt ters ler ters seco-r forth-r sotroc tesors Cort Derte of Stresses esor for Vector for ] [ } {

24 HC Che 4/8/8 CVEN 68: Chter B 4.6 Ner-Stoes Eqtos Ste or Uste Lr or rblet Coressble or Icoressble Isotroc o rectol referece Bt ot hooeeos eeet of osto Newto D C f D Ner-Stoes Eqtos esor for coorte rt Vector for coorte rt f t D ] [ f t D

25 HC Che 4/8/8 CVEN 68: Chter B 5 Ner-Stoes Eqtos For el otoc s Icoressble flow Isotroc bt ot hooeeos f D costt f D t eerl Eer Eqto Forer s Lw Eer Eqto Asse costt scost Dssto fcto ete = / q q : Q q De ] [ ters 9 ters 8 ters

26 HC Che 4/8/8 Eer Eqto For rbtrr bt fe cotrol ole De De q Q Q srfce het coresso ssto het fl eerto wor fcto De Q : Eer Eqto Ethl Eqto h = e + / D D D D Dh D D e Q eertre Eqto e = C h = C C C D D Q Q D CVEN 68: Chter B 6

27 HC Che 4/8/8 CVEN 68: Chter B 7 Sr Coressble flow costt Icoressble flow Q t C De f t D t. Q C C t f t Prtl s. Colete rsfortos Colete rsforto Use cotrrt cooets of eloct rte-of-str or stress tesors Prtl rsforto Iole cort fferetto of sclrs ol ll ectors tesors re eresse reselecte tcll orthool coorte sste w w w ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ rtl colete

( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) + ( ) ( ) = ( ( )) ( ) + ( ( )) ( ) Review. Second Derivatives for f : y R. Let A be an m n matrix.

( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) + ( ) ( ) = ( ( )) ( ) + ( ( )) ( ) Review. Second Derivatives for f : y R. Let A be an m n matrix. Revew + v, + y = v, + v, + y, + y, Cato! v, + y, + v, + y geeral Let A be a atr Let f,g : Ω R ( ) ( ) R y R Ω R h( ) f ( ) g ( ) ( ) ( ) ( ( )) ( ) dh = f dg + g df A, y y A Ay = = r= c= =, : Ω R he Proof