1. Starting with the local version of the first law of thermodynamics q. derive the statement of the first law of thermodynamics for a control volume

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1 EN10: Contnuum Mechancs Homewok 5: Alcaton of contnuum mechancs to fluds Due 1:00 noon Fda Febua 4th chool of Engneeng Bown Unvest 1. tatng wth the local veson of the fst law of themodnamcs q jdj q t and usng the mass consevaton equaton xconst deve the statement of the fst law of themodnamcs fo a contol volume d 1 1 ( n σ vda b vd q nda qd d da dt v v v v v n B R B R R B tat wth the local veson of the fst law and ntegate ove a fxed satal volume q jdj q t xconst Note that t xconst t xconst t xconst ecallng mass consevaton v 0 t xconst we see that v 0 t xconst t xconst Next v v v v t xconst t const t const Then v ( jv j Djj j v j j j Use the momentum balance equaton j v v v k b j k t const whch gves v ( jv v v D jj j v vk b j j k t const

2 Next, obseve that 1 ( v v 1 ( v v v v v ( v v v v vkv t const t t v v v vkv t k k k k k k fom mass consevaton. ubsttutng back, we have that v ( jv 1vk vv 1( vv q v b q t t const j k const Integatng ths esult ove a fxed satal volume, and then alng the dvegence theoem elds the esult stated. [5 POINT]. Idealze the a above the eath s suface as an deal gas, wth temeatue dstbuton 0, whee s the heght above the eath s suface, 0 s the temeatue at the eath s suface, and s a constant. (a Assumng the a s at est, wte down the smlfed vesons of the momentum balance equaton and the consttutve equatons fo the a Assumng the decton s vetcal, the lnea momentum balance equaton educes to g 0 The consttutve equatons ae cv cv cv log Rlog s0 R ( 1 [ POINT] (b Comute the essue and denst dstbutons as a functon of heght above the suface, n tems of the essue 0 at the eath s suface. What haens n the lmt 0? ubsttutng fo the denst, the essue equaton educes to gr / g 0 1 R( The denst follows as 1 R 0 0 g/( R 1 [ POINT] The 0 lmt s not obvous b lookng at ths fomula but t can easl be comuted b solvng the dffeental equaton agan, gvng 0 ex( g / R0

3 [1 POINT]. how that the eneg equaton fo a comessble, nvscd flud flow can be exessed n the fom h q q t t xconst whee h / s the secfc enthal, and s the essue. x The eneg equaton s The stess s j and so t xconst Whee we have used mass consevaton. Theefoe q jdj q v jdj t q ( q t xconst t t t h q q t t [ POINT] 4. The ncomessble Nave-tokes equatons ae sometmes e-wtten n so-called Imetus-Gage fom. Ths s done b ntoducng an abta scala feld (called the gage and then defnng a vecto feld m (called the metus as m v Wth these defntons, show that the govenng equatons fo m (mass consevaton and the ncomessble Nave-tokes equaton can be exessed as m m q 1 m jk v j kq vkv k t const t const k k kk m whee : b / s the bod foce otental (The ont of dong ths s that snce s abta, t can be chosen to satsf an auxla equaton that smlfes the govenng equatons fo a atcula examle. Fo examle, one could choose 1 vv 0 t k k k k const

4 whch educes the govenng equaton fo m to a ve smle fom esecall fo deal fluds The ncomessble Nave-tokes equaton s 1 v dv v 1 b ( vkvk jk jvk jv j dt xconst t k const We have that m v ubsttutng nto the Nave-tokes eq gves 1 m b j j 1 m ( vkvk jkjq mq vk t const q k m m q vkv k m kj vk jq t const j j j j [5 POINT] 5. The fgue shows a essuzed soda can on a cat. The ntenal essue above the flud s. A hole wth coss-sectonal aea A s unched n the sde of the can. Calculate the nstantaneous acceleaton of the cat, n tems of the essue, the suoundng atmoshec essue a, and the combned mass m of the cat, can and flud. Gavt can be neglected. You can assume that the cat s at est f ou wsh, but the nstantaneous acceleaton s actuall ndeendent of the veloct. Consde the contol volume shown n the fgue. Lnea momentum balance gves d n σda bd vd ( v v nda dt B R R B 0 ma Av Alng Benoull along the steamlne shown n ed gves 1 a v. Theefoe a ( a A / m [ POINT]

5 6. The flow suoundng a gd shee (wth adus a that s at the ogn at tme t=0 and moves steadl wthout otaton wth veloct can be comuted fom the followng otental a ( t ( ( k kt k kt a. Calculate the veloct feld e a e 1 a a ( ( k k k t t v 4 [ POINT] b. ef that the veloct feld satsfes the coect bounda condtons on the suface of the shee. The bounda condton s v n n ( t n a k k k 4 a ( t a ( t ( t ( t ( t vn a c. Calculate the essue dstbuton (neglect gavt Usng the Benoull equaton: 1 0 t ( ( k k k k k k 6 4 [ POINT] 1 a a a ( ( ( ( k k kt t a a k k kt a n n nt a a t t 0 0 a a a ( t ( t 8 8 [ POINT] d. Hence, comute an exesson fo the dstbuton of tacton actng on the suface of the shee. 5 9 k ( k kt ( t t jn j 0 kk 8 8

6 e. Detemne the dag foce actng on the shee. [ POINT] The dag s zeo b smmet. Ths s not susng n an deal flud thee s no dssaton. The soluton s not ve ealstc, howeve, because t assumes that the flow does not seaate fom the suface of the shee. [ POINT] 7. Consde a sold object (e.g. the shee n the ecedng oblem that moves though an deal flud wth veloct ( t (not necessal constant. The moton of the sold nduces some veloct feld v n the flud, whch can be calculated fom a flow otental n the usual wa. how that the total knetc eneg of the flud can be comuted fom KE ( t nda whee s the flow otental, s the suface of the sold object, and n s the outwad nomal to the sold suface. You wll need to use the govenng equaton fo the flow otental and the dvegence theoem You wll also need to assume somethng about the behavo of the veloct feld at nfnt. The knetc eneg s KE vvd ubsttutng fo the veloct n tems of the flow otental v v v KE v d d d because the flow s ncomessble. Alng the dvegence theoem gves KE vm da vm da whee R denotes some bounda ve fa awa fom the sold, and m s the outwad nomal to the flud suface.. As long as the flow otental decas faste than 1/R at nfnt, the contbuton fom the second ntegal must vansh. In addton, m n Ths oves the statement. Hence, calculate the KE of the flud suoundng a shee movng wth nstantaneous veloct ( t. Fnd an exesson fo the acceleaton of a shee wth denst s mmesed n an deal flud (the buoanc foce can be teated wthout devatons. R [ POINT] Fo the shee, k k k k 4 a a KE da da 4

7 Ths s the same ntegal that came u n HW. It s nonzeo onl f k, and fo ths case the ntegal must be the same egadless of whethe k 1,, o. We can just look u the soluton 4 / We thus get a W KE 4g j Whee W s the weght of flud dslaced b the shee. [ POINT] The buoanc foce on the shee s FB mg W and the ate of wok done b ths foce must equal the ate of change of otental eneg of the sstem, whch s W m d W m a 4g dt g ( mg W a g W mg [ POINT]