Fluid Flow through a Tube

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Veronica Stevens
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1 . Theory through Tube In this experiment we wi determine how we physic retionship (so ced w ), nmey Poiseue s eqution, ppies. In the suppementry reding mteri this eqution ws derived s p Q 8 where Q is the fow rte in units of voume per second; p is the pressure difference between the ends of the tube;, the tube rdius;, the ength of the tube, nd η the viscosity of the fuid. We wi mesure in the experiment Q, s function of these prmeters mentioned bove. One prmeter wi be vried whie the others re kept constnt, nd Q, s resut of vrition in this prmeter, potted on og-og pper. The reson for choosing og- og pper is tht it brings out power w dependence, such s Q nd Q much better thn iner grphs. Convince yoursef tht on og-og pper the exponent ( or - here) just determines the sope of stright ine, whie on iner pper, curved grphs with continuousy chnging sope woud resut. If we write the dependence of Q, on the vrious prmeters mentioned in terms of n exponenti power, we expect tht Q gets rger when, everything ese remining the sme, the pressure increses or the rdius increses; Q gets smer when the tube ength increses or the iquid gets more viscous. One woud not expect significnt effect from the mteri the tube w is mde from, uness it were very rough nd prevented smooth minr fow to tke pce. Temperture coud enter indirecty through the dependence of the viscosity on temperture. Coecting these terms resuts in formu of the kind: where K is proportionity constnt. Q Kp We cn determine the coefficients α, β, nd γ by coecting the fuid coming out of the tube during given mount of time s function of tht prticur prmeter. For exmpe, to find α, one woud mesure the mount of wter fowing through vrious tubes of different dimeters, everything ese (ength, pressure difference) remining the sme. One then writes the eqution: 30

Kp ogq og og This is stright ine of the form: ogq = B +α og, with s sope in ogq versus og pot. In simir wy one cn test for the other prmeters nd thus check the eqution.

The trnsition is determined by Reynod s number, Re. As mentioned in the suppement, 3 v p p Re or if one substitutes v, Re. 8 This trnsition occurs when Re equs 000.

2 Kp ogq og og This is stright ine of the form: ogq = B +α og, with s sope in ogq versus og pot. In simir wy one cn test for the other prmeters nd thus check the eqution. It shoud be kept in mind tht Poiseuie s eqution hods ony for minr fow. If the fuid moves too fst, smooth sher drg between yers disppers nd turbuence sets in. The trnsition is determined by Reynod s number, Re. As mentioned in the suppement, 3 v p p Re or if one substitutes v, Re. 8 This trnsition occurs when Re equs Procedure Wter Input Reservoir h Stopper Pipe Wter Output Fig. We wi use the pprtus shown in the figure. The reservoir hs sm openings in its side cosed by corks. By removing cork, the wter eve is set t the height, h, of the resuting opening bove the horizont pipe. The pressure difference between the two ends of the pipe, p, is the height of the wter coumn, thus given by p = ρgh. There is suppy of pipes of vrious rdii nd engths. The ony iquid to be used here is wter so tht the viscosity η, wi not be vried. Attch the desired pipe to the reservoir; set the 30

wter eve to such height tht the Reynod s number is ess thn 000. Adjust the wter fow from the fucet, so tht the wter eve in the reservoir remins stbe t the desired height.

Mesure the fow rte for different rdii tubes (keeping the ength nd pressure p constnt). Repet the procedure for different engths nd 3 different pressures.

3 wter eve to such height tht the Reynod s number is ess thn 000. Adjust the wter fow from the fucet, so tht the wter eve in the reservoir remins stbe t the desired height. Put the grduted beker under the opening of the tube nd mesure the time it tkes to coect 500 m. This wi give you the fow rte Q. Mesure the fow rte for different rdii tubes (keeping the ength nd pressure p constnt). Repet the procedure for different engths nd 3 different pressures. One coud pot og Q, versus the og of the rdius, the ength nd the height h. The sope of these ines woud give you the vrious exponents. Insted of ccuting the og ech time, it is much esier to pot the vue directy, but on og-og grph pper. Now choose your prmeters such tht Re >>000. Mesure the fow rte in this turbuent cse nd observe the devition from the ines found erier. There is short second experiment designed to test Bernoui s eqution in simpe cse. The pprtus is s shown in the figure. We use the sme reservoir s ws used erier. As wter fows through the horizont tube, the wter eve in the vertic tubes wi rise proportiontey to the wter pressure. Appying Bernoui s eqution to points nd gives: h h Fig. Thus; p v p v p p0 gh, p p0 gh gh v gh v 30 3

4 where h nd h refer to the height of the wter eve in the vertic tubes bove the center of the min tube. To obtin the veocities we note tht the fow rte is given by Q = Av,where A is the cross-section of the tube. Bernoui s eqution gives; Q Q gh gh A A Set the wter eve in the reservoir to bout -cm. Mesure h, h nd the fow rte Q. With these vues ccute the eft nd right hnd side of the bove eqution. How cosey re they equ? Bernoui s eqution is not ppicbe for turbuent fow. To see this quittivey, use tube with n brupt trnsition between nrrow nd wide dimeter tubes. This discontinuity introduces turbuence. Ccute now both sides of the eqution nd see how cose to equ they re. Questions: ) Wht is the most ikey cuse for differences between Poiseuie s theoretic formu nd your experiments? ) Wht is the most ikey cuse in your experiment for devitions from Bernoui s eqution? 3. Fuid Mechnics Suppement In Chpter of the book by Bueche, you find vrious equtions of hydrodynmics, notby the eqution of continuity nd Bernoui s eqution. We wnt here to discuss the fow of wter through cyindric tube, which is not discussed in the book. Aong the ws of the cyinder the fuid does not fow: it is sttionry. There is grdu increse of the fuid veocity, from zero t the w to mximum t the center. The friction force, F, tht resists the veocity, is given by F A dy where η = viscosity, A, the contct re nd the veocity grdient, dy tht is, the chnge of the veocity in going from the w (v = 0) towrds the center (v mx ). The center fuid drgs the next yer, which sips somewht behind etc., unti the fuid yer t the w, which does not move t. This type of fuid fow is ced minr fow. 30

5 This friction is the cuse of pressure drop ong cyindric tube. Poiseuie derived formu tht retes the quntity of iquid, Q, (the voume ) fowing per second through cyindric tube of rdius nd ength, with the pressure difference, p, between both ends. This formu is: p Q () 8 The constnt η is the viscosity defined erier. We wi derive this eqution. Let us consider cyinder with rdius r, where r <. At the entrnce surfce is force +πr p, nd t the exit surfce πr p. The force t the cyindric surfce, the friction force, is given by r ccording to the definition for η. Since the iquid does not dr experience cceertion but fows with constnt veocity, the sum of these three forces must equ zero. Since is negtive if we define r in the usu wy s the distnce from dr the center, we obtin: r p r p r 0 () dr p p This yieds rdr nd fter integrtion: p( r c) v (3) where c represents the integrtion coefficient stemming from rdr r c. We know tht v = 0 t the w, tht is, when r =. p p hs been written s p, for short. Appiction of the boundry condition gives us c nd resuts in: p( r ) v () This veocity profie is sketched in the figure. The tot voume of iquid, Q, fowing through cross section per second cn be obtined from: Q 0 p( r )rdr (5) This is nmey the sum of the fuid fowing through rings bounded by rdius r nd r + dr. One obtins, using: rdr 3 p nd r dr ; Q We wi now discuss few ppictions of this formu. The verge veocity of the iquid, v cn be obtined from: 30 5

6 Q p v 8 Reynod mde study of the ppicbiity of Poiseuie s eqution nd concuded tht bove certin v minr fow breks up nd turbuence sets in. This hppens when: vr = Reynod s number Re is > 000. In turbuent fow there vortex formtion, the fow is irregur nd noisy. Bood fow through the nrrow cpiries is minr s one verifies esiy. However, through the ort, with v = 0.6m/sec, r = 0.0 m, η= kg/m sec nd density ρ= grm/cm 3 = 000kg/m 3, one ccutes Re = 800. So bood fow through the ort is turbuent. There re sever other importnt medic impictions of eqution. If one requires rpid bood trnsfusion, incresing the height of the botte with bood by fctor of two increses the fow by tht sme fctor, but incresing the dimeter of the tube by two increses the fow by fctor of sixteen. In the cse of shock victim, the person s body temperture often drops. This resuts in significnt increse of the viscosity nd thus decrese in bood fow. To keep the body temperture up, covering of the victim with bnket nd if pproprite giving wrm drinks heps. Finy, it shoud be remrked tht our simpe mode coud ony pproximte bood fow. Veins re not soid tubes, but re estic, expnding somewht with incresed pressure. Aso bood is not s simpe s wter in its viscosity. For exmpe, in nrrow cpiries the bood ces ine up, reducing the viscosity. 30 6

Statistical Physics. Solutions Sheet 5.

Statistical Physics. Solutions Sheet 5. Sttistic Physics. Soutions Sheet 5. Exercise. HS 04 Prof. Mnfred Sigrist Ide fermionic quntum gs in hrmonic trp In this exercise we study the fermionic spiness ide gs confined in three-dimension hrmonic