# Fluids & Bernoulli s Equation. Group Problems 9

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1 Goup Poblems 9 Fluids & Benoulli s Eqution Nme This is moe tutoil-like thn poblem nd leds you though conceptul development of Benoulli s eqution using the ides of Newton s 2 nd lw nd enegy. You e going to dw feebody digms, clculte Wok, nd conside kinetic nd potentil enegy. The lening gols fo this execise e to see Benoulli s eqution s sttement of enegy consevtion nd emove some of the mystey of the pessue dop in ventui. Pt I: The Ventui Conside non-viscous nd incompessible fluid flowing though hoizontl pipe tht chnges dimete so tht the coss-sectionl e is educed to ¼ its oiginl vlue s shown. The fluid flows fom left to ight nd smll cubic elements of fluid e dwn t thee loctions. 1) If the velocity t point A (v A ) is v, then wht is the velocity t C (v C )? Explin you esoning. A B C ) Wht is the velocity t point B (v B ) exctly hlf wy though the constiction whee the tube dimete ¾ the dimete t point A? Explin you esoning. Hint: Ae is popotionl to the dimete squed. 2) Dw fee-body digm of the fluid element t ech letteed loction. Include the diection of the cceletion nd the net foce if they e non-zeo. You my omit foces not in the left o ight diection if you wish. (All six fces of the cube hve pessue on them.) Point A Point B Point C ) Ae the foces pointing left nd ight the sme on ech fee-body digm? Explin biefly.

2 3) Wht gives ise to the foces esponsible fo the cceletion of the fluid? Explin you esoning. 4) On which side of the cube is the pessue highe duing the cceletion? Explin you esoning. 5) Do you gee o not gee with the following sttement. The foces mentioned in 3 do Wok on the fluid element nd tht Wok is equl to the element s chnge in kinetic enegy. Explin you esoning nd conside how you might compute this Wok fom the chnge in kinetic enegy of the fliud element nd elte tht to the chnge in pessue. 6) How does the pessue t point A compe to tht t point C? Hint: If the kinetic enegy inceses something hd to povide it. Tht thing must hve less enegy now.

3 Pt II: Potentil Enegy, Pessue, nd KE Conside non-viscous nd incompessible fluid flowing out hoizontl pipe t the bottom of lge tnk filled to height h s shown. Smll cubic elements of fluid e dwn t thee loctions. The two on the left epesent wte blloon dopped fom the sme height. 1) Compe the potentil enegies of the fluid elements t the vious loctions. Whee e they the sme, diffeent, lge, o smlle? 1 A h 2 B C 2) Compe the pessues on the fluid elements t the vious loctions. Whee e they the sme, diffeent, lge, o smlle? Explin you esoning. If you e stuck bout how to hndle i pessue, sk you TA. The tnk is vey lge so n element s velocity t A nd B e petty much zeo s is the velocity of the wte blloon t 1 tht is dopped fom est. 3) Use the ides in Pt I nd the chnge in PE to compe the velocities t 2 nd C. Explin you esoning.

4 2) A lge quntity of wte (density = 1.00 gm/ cm 3 o 10 3 kg/m3) is poued into n unusul setup in which thee is loudspeke mounted on the left hnd side of hoizontl pipe ttched to lge esevoi. The dimete of the pipes nd othe dimension e given in the figue. Fo pts () nd (b) the wte is ssumed to behve s Newton fluid. () With the speke off, wht is the pessue (in Pscls) in the hoizontl tube? (b) Now the powe to the speke is switched on nd the x motion of the speke is given by x(t)= A cos(ωt) whee the mplitude, A=0.020 m, nd the fequency, f, is set to 0.50 Hetz. The wte moves in sync with the speke. The height of the wte in the veticl column does not visibly chnge. Wht is the mximum pessue diffeence, p = p 1 p 2, between the pessue in the 2.0 m dimete section & nd 0.50 m section?

5 3) One lite of wte (1000 cm 3 with = 1.00 gm/ cm 3 o 10 3 kg/m 3 ) is poued into unifom U-shpe pipe with m 2 sque coss section boe nd shpe s shown in the figue t ight. The dwing is not to scle. () When the wte is t equilibium, wht is the pessue (in Pscls) t the bottom of the tube? You should ignoe the detils of the bend nd only conside the stight sections of the tube in you clcultion. (b) Now you gently blow into one side of the tube nd the wte sufce, fte coming to equilibium, sinks down m fom its stting height. Wht eltive pessue (i.e., chnge in pessue), in Pscls, did you pply? (c) You now double the pessue of pt (b) nd find tht the fluid sinks down twice s f. Afte this you elese the pessue nd the fluid sloshes bck nd foth. Assuming tht wte behves s pefect Newtonin fluid, t wht ngul fequency does it oscillte?

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