4.2.4 Entropy Functions is very general and can be readily extended to systems of equations SLIDE 39 function if it is positive, convex, al corresponding entropy fux such that F(u=U(u)f(u For smooth solutions au aF Proof: Multiply the original equation ut+f=0 by U' and the above results The function u(a, t)is the entropy satisfying solution of the governing equations if, for all convex entropy functions U(u)and corresponding entropy fluxes F(u) U(u af(a) is satisfied in the weak sense 10 If f(u) is convex we only need to check for one U(u) Note 10 Entropy Function This is an alternative approach to enforcing the entropy condition. In gas dyo namics,a physical quantity exists called the entropy that is known to te constant along particle paths in smooth flow, and to ump to a higher value as the gas crosses a shock. The entropy can never ump to a lower value ( econd aw of Thermodynamics)and this gives the extra condition which enat les us to select the physically correct solution The at ove inequality is derived t y multiplying the viscous equation ty U(u) Ut+ Fa U+F2=1{U since U is conVex, U=0 and so Ut+Fa< vx✫✛✬✮✭✯✬✰✫ ✱✒✲✴✳✶✵✸✷✺✹✼✻✾✽✛✿✛✲❁❀❂✳❄❃❅✷✥✲✛❆ ❇❉❈❂❊❋❊❍●❏■❑❈✸▲◆▼❑❖❅P◗●❙❘✡❚❯●✶❱✒❲❂▲✥❈❂❱❨❳◆●❙❘❩■❑❳✢❲✸❘❬❲❯■✶❱❭●✶❘❪❖✰❫❄❲✸❱❴❖❅❈✸▲❵❈❜❛❝❱✗❳✢●❏●✶▲◆❱❴❘❞❈❑❡✥❚❢■❑❈✸▲✥P❂❖❨❱❣❖❅❈❂▲✢▼✡❊✛❳❤❖❅■❑❳ ❖✮▼✁✐❥●✶❘❪❚❉❦❧●❙▲✴●❙❘❞❲❂♠❁❲❂▲✥P❏■❑❲❂▲♦♥❑●✁❘❞●❑❲◗P✸❖❨♠♣❚✝●❞q◗❱r●❙▲✴P❯●❑P❝❱r❈❝▼❪❚❥▼❪❱❭●✶s❉▼❉❈t❛✉●❬✈✶✇◆❲✸❱❴❖❅❈✸▲◆▼✶① ②✥③❧④❨⑤★⑥♦⑦✢⑧ ⑨❝⑩❅❶✴❷❹❸✮❺❩❻❂❼❵❽✲✴✳✶✵✸✷✺✹✼✻❿❾❭✿✛✲❁❀❂✳❄❃❅✷✥✲ ❸✰➀✒❸✰➁❏❸✮❺ ✹❍✷✥❆✪❃❅✳✪❃✗➂❽✥➃ ❀◗✷✺✲✴➂❽◗➄✙➃➅❻❯❼◆➆✾➁❞➇◆➈✶➉❬➈➊➈❙➋➌❸✔❺t➁❬❺✡❻ ➍❙➎➉❞➉❬➈✪❺❞➏ ➎❼◆➆➌❸✔❼✢➐✡❽✲✴✳✶✵✸✷✺✹✼✻➒➑➓✿➄➔❺❞→➍➇❏➁❬➇◆❻✸➁ ➣✁↔❴⑩❅❶✴❷➙↕➛⑨➜↔❴⑩✗❶✼❷t➝✴↔❴⑩✗❶✼❷ ➞➎➉➅❺t➟➎❤➎➁❬➇☞❺➎❯➠→➌➁❬❸➎❼◆❺ ➡ ➢⑨ ➢✺➤ ➥ ➢➣ ➢✥➦ ↕➨➧ ➩➫❘❞❈❄❈❜❛❙➭❏➯➊✇➌♠♣❱❴❖❡✥♠♣❚➲❱❨❳◆●❏❈✸❘❪❖➳❦❯❖❨▲✥❲❂♠➵●❑✈✶✇◆❲✸❱❴❖❅❈✸▲➸❶✺➺ ➥ ➝✸➻✝↕➼➧➒♥❙❚➽⑨↔ ❲✸▲✴P➊❱✗❳✢●☞❲❯♥❑❈✸✐✸●➽❘❞●❙▼❑✇➌♠➳❱❅▼ ❛❙❈✸♠❨♠✔❈✸❊➓▼✶① ②✥③❧④❨⑤★⑥➲➾◆➚ ➪➵➇✢➈➫➀✗→✢❼➍➁❞❸➎❼❝❶➓⑩➦★➶❞➤ ❷❁❸✔❺✙➁❞➇✢➈➹➈✪❼◗➁❬➉➎➏❤➘✉❺❞❻❂➁❞❸✮❺❜➀✗➘❤❸✔❼✢➐✁❺➎◗➠→➌➁❬❸➎❼ ➎➀◆➁❬➇✢➈➹➐➎✸➴➈✶➉❬❼✢❸✰❼◆➐➷➈✪➬❧→◆❻❂➁❞❸➎❼◆❺ ❸✰➀❜➃❂➀➎➉➮❻➠✔➠◆➍❙➎❼➴➈❙➋❉➈✪❼❧➁❞➉➎➏❤➘✁➀✗→✢❼➍➁❞❸➎❼◆❺➵⑨❝⑩✗❶✼❷➓❻❂❼✺➆ ➍✶➎➉❬➉❞➈❄❺t➏ ➎❼✺➆➌❸✰❼◆➐✒➈✪❼❧➁❞➉➎➏❤➘✁➱◆→➌➋➌➈❄❺➙➣✡⑩✗❶✼❷❪➃ ➁❞➇◆➈✁❸✰❼✢➈❄➬❧→◆❻➠ ❸➳➁❜➘ ➢⑨❝⑩✗❶✼❷ ➢✺➤ ➥ ➢➣✡⑩❅❶✴❷ ➢✥➦ ✃ ➧ ❸✮❺➹❺❞❻❂➁❞❸✮❺❜❐◆➈❄➆❏❸✔❼☞➁❞➇✢➈✁❒➫➈✪❻❯❮❏❺❞➈✶❼◆❺❞➈❯❰ Ï✉Ð✪➧ Ñ❭➀➙➝➓⑩✗❶✼❷❍❸✮❺ ❀◗✷✺✲✴➂❽◗➄Ò❒➫➈ ➎❼➠➘❩❼✢➈✪➈✪➆➽➁➎✡➍➇✢➈ ➍❮✡➀➎➉ ✷✥✲❽☞⑨❝⑩✗❶✼❷ Ó♦Ô✼Õ❪Ö➲×✺Ø ÙÛÚ✙Õ❞Ü❯Ô◗Ý✼Þ➨ß➮à✺Ú➙á❤ÕtârÔ➌Ú❁ã ➪➵➇✢❸✮❺➷❸✮❺✒❻❂❼➲❻➠➁❬➈✶➉❬❼◆❻✸➁❬❸➴➈❉❻❂➏✢➏◆➉➎❻➍➇☞➁➎ ➈✶❼➌➀➎➉➍❸✰❼✢➐❏➁❞➇✢➈ä➈✶❼❧➁❞➉➎➏❤➘ ➍❙➎❼◆➆➌❸✰➁❞❸➎❼✯❰✒Ñr❼❢➐◗❻◗❺➷➆➌➘❧å ❼◆❻❯➟❹❸ ➍❺✪➃❥❻➹➏✢➇❤➘➌❺t❸ ➍❻➠ ➬❧→◆❻❂❼❧➁❞❸✰➁❜➘✒➈❙➋➌❸✔❺t➁❬❺ ➍❻➠✰➠➈✪➆✒➁❬➇✢➈Û●✶▲◆❱❴❘❞❈❬❡✴❚➓➁❞➇◆❻❂➁✛❸✮❺✯❮❤❼➎❒➹❼✒➁➎➅æ ➈ ➍✶➎❼◆❺t➁❬❻❯❼❧➁ ❻➠✔➎❼◆➐❏➏◆❻❯➉t➁❬❸➍✶➠➈ä➏◆❻✸➁❬➇◆❺✒❸✰❼♦❺t➟➎❤➎➁❬➇❢➱➎❒✁➃✯❻❯❼◆➆➊➁ ➎✉ç →✢➟❝➏➸➁➎ ❻❏➇✢❸✔➐❯➇◆➈✶➉ ➴❻➠→◆➈❹❻◗❺➷➁❬➇✢➈❹➐◗❻❯❺ ➍➉➎❺❬❺❞➈✪❺➅❻❩❺❞➇➎➌➍❮✴❰➷➪➵➇✢➈ä➈✶❼❧➁❞➉➎➏❤➘ ➍❻❂❼➲❼✢➈➴➈✶➉ ç →◆➟❹➏➲➁➎ ❻ ➠✰➎❒➫➈✶➉ ➴❻➠→◆➈➽⑩❣è➌➈➍❙➎❼◆➆❢é✛❻❥❒ ➎➀ ➪➵➇✢➈✪➉❞➟➎ ➆➌➘❤❼◆❻❂➟❝❸ ➍❺❑❷❍❻❯❼◆➆❏➁❞➇✢❸✮❺➵➐❯❸➴➈❄❺❍➁❬➇✢➈➜➈❙➋❤➁❞➉❑❻ ➍❙➎❼✺➆➌❸➳➁❬❸➎❼❏❒➹➇✢❸ ➍➇☞➈✶❼◆❻æ✢➠➈❄❺➵→◆❺➮➁➎ ❺❞➈➠➈ ➍➁ ➁❞➇◆➈✁➏✢➇❤➘❤❺❞❸ ➍❻➠✔➠➘ ➍❙➎➉❬➉❬➈➍➁➅❺➎❯➠→➌➁❞❸➎❼★❰ ➪➵➇✢➈✁❻æ✥➎✸➴➈➜❸✔❼✢➈✪➬❧→◆❻➠ ❸✰➁❜➘❩❸✮❺➹➆➌➈✶➉❬❸➴➈✪➆ æ➘❩➟❉→➠➁❬❸✰➏➠➘❤❸✰❼◆➐❝➁❞➇✢➈ ➴❸✮❺ ➍✶➎→◆❺➵➈❄➬◗→✺❻✸➁❞❸➎❼ æ➘☞⑨↔ ⑩✗❶✼❷ ⑨✁↔➳❶✥➺ ➥ ⑨➜↔✮➝ ↕ ê❪⑨✁↔➳❶✴➻✪➻ ⑨➓➺ ➥ ➣✛➻ ↕ ê❪⑨✁↔➳❶✴➻✪➻ ⑨➓➺ ➥ ➣✛➻ ↕ ê❄⑩❭⑨✙➻❄➻➜ë➔⑨↔ ↔ ❶✴ì➻ ❷ ❺❞❸✰❼➍➈❉⑨✞❸✮❺ ➍❙➎❼➴➈❙➋✼➃◆⑨↔ ↔✼í ➧❝❻❂❼◆➆✝❺➎ ⑨➓➺ ➥ ➣✛➻ ✃ ê❙⑨✙➻❄➻ Ð❥î