Extra Examples for Chapter 1

Transcription

1 Exta Examples fo Chapte 1 Example 1: Conenti ylinde visomete is a devie used to measue the visosity of liquids. A liquid of unknown visosity is filling the small gap between two onenti ylindes, one is fixed and the othe is otating at a onstant angula speed. The toque neessay to otate the inne ylinde is measued and used to detemine the unknown visosity. = m T = Nm ω = 20 ad/s H = 0.2 m Fo the visomete shown on the ight, detemine the visosity of the fluid if a toque of Nm is equied to tun the inne = m ylinde at a onstant speed of 20 ad/s. Fluid is Newtonian. Assume linea veloity vaiation aoss the small gaps. = 0.05 m Solution: As the inne ylinde otates, it puts the fluid into motion and the fluid applies a shea foe (and toque) on the side and bottom sufaes of it. The given applied toque of Nm is equied to balane this esistive fitional toque applied by the fluid. We need to alulate the toque ating by the fluid on the side and bottom sufaes of the inne ylinde sepaately. Let s stat by alulating the toque on the side sufae. Toque ating by the fluid on the side sufae of the inne ylinde: Linea veloity vaiation aoss the small gap, V θ () ω θ, ω Fixed oute ylinde otating inne ylinde Flow Top view (Note that ) Veloity pofile inside the gap is given to be linea. Due to the no slip ondition, fluid speed at the side walls of the inne and oute ylindes will be ω and zeo, espetively. Aoding to Newton s law of visosity, shea stess at the side wall of the inne ylinde is τ side = μ dv θ d = μ ( ω = ) 1

2 whee dv θ /d is veloity gadient, i.e. the slope of the veloity vaiation in the small gap, whih is onstant and equal to ω/. Note that atually it is the negative of this (veloity deeases in the dietion), but we don t need to bothe with the minus sign hee, just like we did in the example solved in lass. Foe ating by the fluid on the side wall of the inne ylinde is in θ dietion, i.e. CW dietion when looked fom the top. Inne ylinde otates in CCW dietion and the fluid applies a visous esistane to it. τ side is onstant. Shea foe ating by the fluid on the side sufae of the inne ylinde is detemined by multiplying it with the aea of the side sufae. F side = τ side A side = (μ ω 2πμωH2 ) (2πH) = (in CW dietion) F side is also onstant. Toque ating by the fluid on the side sufae of the inne ylinde is detemined by multiplying it with the onstant moment am,. T side = F side = 2πμωH3 (in CW dietion) At this point it is a good idea to hek the units of this esult to see whethe it has the units of toque o not. [Pa s][s 1 ][m][m 3 ] = [Nm] (Coet units) [m] Toque ating by the fluid on the bottom sufae of the inne ylinde: This alulation is a bit moe ompliated beause the veloity hanges both in the and z dietions and the shea stess on the bottom sufae is not onstant. z otating bottom sufae of the inne ylinde ω ω Top view of the bottom sufae of the inne ylinde Fixed bottom sufae of the oute ylinde Veloity pofile V θ (, z) Veloity pofile of the fluid filling the bottom gap hanges linealy in both and z dietions. Sine the veloity is hanging with, shea foe ating on the bottom of the inne ylinde is also hanging with. Calulation of the shea foe equies an integation. Conside the foe ating on the ing shaped infinitesimal aea shown below 2

3 Top view of the bottom sufae of the inne ylinde ω d da = 2πd Shea stess ating on the infinitesimal aea is τ bottom = μ dv θ dz = μ ω (Not onstant, vaies with ) whee ω/ is the slope of the linea veloity vaiation in the z dietion at any abitay loation. We onside the slope in the z dietion beause Newton s law of visosity makes use of the veloity gadient nomal to the sufae. Fo the bottom sufae, the nomal dietion is z. Infinitesimal shea foe ating on the infinitesimal aea da is df bottom = τ bottom da bottom = (μ ω 2πμω2 ) (2πd) = d (in CW dietion) Simila to the foe ating on the side wall of the inne ylinde, the one ating on the bottom wall is also in the CW dietion. It is a esisting foe fo the CCW otation of the ylinde. Consideing the moment am to be, infinitesimal toque due to this infinitesimal shea foe is dt bottom = df bottom = 2πμω3 d (in CW dietion) Toque ating on the bottom sufae of the inne ylinde is obtained by integating this ove the bottom sufae T bottom = dt bottom =0 = 2πμω3 d =0 = πμω4 2 (in CW dietion) The unit of this esult should also be Nm. Chek it. Toque balane on the inne ylinde: To otate at onstant speed, net toque on the inne ylinde should be zeo, i.e. the Nm toque that is applied extenally need to be anelled out by the toque applied by the fluid. Toque = T side T bottom = πμωH3 πμω4 2 = 0 The only unknown in this equation is the visosity, whih an be alulated as μ = Pa s 3

4 Impotant Notes: Only non-zeo veloity omponent in the fluid is V θ. V and V z ae zeo. Atually this is not exatly the ase at the bottom one of the visomete, but one effets ae not onsideed hee. Fo the fluid filling the side gap, V θ is a funtion of only and in Newton s law of visosity we used the veloity gadient dv θ /d, i.e. the hange nomal to the wall. Fo the fluid filling the bottom gap, V θ is a funtion of both and z, and in Newton s law of visosity we used the veloity gadient dv θ /dz, i.e. the hange nomal to the wall. We always onside the hange of the veloity pofile nomal to the wall. Vaiation of fluid veloity in the gaps ae taken as linea, beause it is said so in the poblem statement. This is a valid assumption, onsideing the gap between the ylindes being small ( ). We ll lean how to detemine the exat pofile late in this ouse. Side sufae alulation equied no integation beause the shea stess and the shea foe is onstant on the side sufae. This is not the ase fo the bottom sufae, due to the veloity gadient (and theefoe shea stess) vaying in the dietion. Note that we did not bothe with the sign onvention of stess. Also we did not bothe with the signs of veloity gadients. Instead we detemined the oet dietions of the foes and the toques by onsideing the simple physis of the poblem; fluid should esist inne ylinde s otation, that s it. Let s say that you got onfused with the signs and detemined the side and bottom toques with opposite signs, one in CW and the othe in CCW dietion. So one is esisting inne ylinde s otation and the othe is helping it. This annot be tue. It s not logial. You need to do these kind of sanity heks at the end of you solutions to ath sign mistakes. Although we did not alulate (beause it is not needed), fluid also applies shea foe, and theefoe toque, on the side and bottom sufaes of the oute ylinde. Can you alulate it? Is it the same as the one we alulated fo the inne ylinde? Toque ating by the fluid on the side sufae is muh moe than that ating on the bottom sufae. If the toque due to the bottom sufae is negleted, visosity would be alulated only 6% highe. We alulated the visosity as Pa s. You an use the following web site to see what kind of a liquid this is. Note that we used the given values (numbes) of the paametes only at the vey last step of the solution. Until that point we woked with the paamete names (,, H, ω, μ, π). You should also follow this patie beause it has two advantages - When the equations ae witten in tems of paametes (not numbes) it is easy to hek thei units and eognize mistakes, if thee ae any. - It minimizes alulation mistakes, suh as enteing wong values into you alulato. 4

5 Example 2: (Fom Elge s book) A famous solution in fluid mehanis, alled Poiseuille flow, involves flow in a ound pipe. Conside Poiseuille flow with a veloity pofile in the pipe given by V() = V o (1 ( 2 ) ) o whee is adial position as measued fom the enteline, V o = 1 m/s is the veloity at the ente of the pipe, and o = 4 m is the pipe adius. Find the shea stess a) at the pipe wall, b) at the ente of the pipe, and ) at = 1 m. The fluid is wate at 15. Solution: Unlike the example we solved in lass and the fist example of this handout, now we have a nonlinea veloity pofile. To use Newton s law of visosity we need the veloity gadient, i.e. the hange of veloity in the dietion. dv d = 2V o 2 o We also need the visosity of wate at 15. Appendix of ou textbook gives it as Pa s a) At the pipe wall = 0 and the shea stess is τ wall = μ dv d = o = μ ( 2V o o ) = Pa b) At the ente of the pipe = 0 and the shea stess is ) At = 0.01 m the shea stess is τ enteline = μ dv d = 0 Pa =0 τ =1 m = μ dv d = Pa =0.01 Impotant Notes: The only non-zeo veloity omponent is the one along the pipe axis, and it hanges only with. Veloity pofile is not linea, i.e. its slope hanges with. Theefoe, shea stess also hanges with. Veloity hanges with the squae of and the shea stess hanges linealy with. It is possible to use Newton s law of visosity to alulate the shea stess not only on eal solid sufaes that ae in ontat with the fluid but also on imaginay sufaes that ae inside the fluid, as we did in pats (b) and (). In this example veloity pofile is given as a funtion of, and when we diffeentiate it we kept the minus sign. That minus sign also appeaed in the shea stess esults. Shea stess being negative in pat (a) does not mean that the shea foe ating by the fluid on the pipe wall is in x dietion (I am onsideing x to 5

6 be the flow dietion, i.e. towads ight). Fluid is flowing in the x dietion and by intuition we an say that it applies a foe on the pipe wall in the x dietion. Minus sign of the shea stess is atually onsistent with the sign onvention of stess. Shea foe on the pipe wall is in x dietion, sufae nomal of the pipe wall is dietion (the nomal that goes fom the pipe wall into the flow), and this plus/minus ombination gives negative shea stess at the pipe wall. Nothing is wong. But again, negative shea stess does not mean that the shea foe is in x dietion. Be aeful. Veloity pofile goes though a maximum at the pipe enteline, i.e. its slope is zeo. Aodingly, shea stess at the enteline is also zeo. If not given, how ould we know that the veloity pofile is not linea in this poblem? With the limited fluid mehanis knowledge you have at this point, you annot know that. That s why it is povided to you. Late in this ouse we ll lean how to detemine it. As a fluid flows inside a pipe, it applies shea foe on its wall. In tun, pipe also applies a shea foe on the fluid in the opposite dietion. To establish a etain flow ate (amount of fluid flowing pe unit time) in a pipe, we typially need to use a pump (fo liquid flows) o a ompesso (fo gas flows) to oveome this fitional foe at the pipe wall. As the flow ate ineases, i.e. maximum value of the veloity pofile ineases, the veloity gadient at the pipe wall ineases and theefoe shea stess at the pipe wall ineases. This means a moe poweful pump o ompesso is neessay to establish a highe flow ate. The fluid applies not only a shea foe, but also a nomal foe on the pipe wall. The nomal foe depends on the pessue of the fluid inside the pipe. We ae not inteested in that in this poblem. Example 3: (Fom Elge s book) When you plae a sewing needle gently on the sufae of a glass of wate it an float. This effet is due to sufae tension suppoting the needle. Detemine the lagest diamete of sewing needle that an be suppoted by wate. Assume that the needle mateial is stainless steel with a speifi gavity of 7.7. Solution: Befoe solving this poblem you need to fist wok on the eading assignment 4 of the fist study set. This is a stati foe balane poblem and we need to daw the fee body diagam of the needle. Fee body diagam involves two foes, weight of the needle ating downwads and the sufae tension foe ating upwads. Vetial foe balane is 6

7 W = 2F σ os(θ) ρ steel g π 2 L = 2σL os(θ) whee is the adius of the needle, L is the length of the needle and σ is the sufae tension of wate. In the appendix of ou textbook sufae tension of wate in ontat with ai at 20 is given as N/m. Using the given speifi gavity of stainless steel, its density is alulated as ρ steel = 7700 kg/m 3. adius of the needle is 2σ os(θ) = πρ steel g This is maximized when os(θ) = 1. Theefoe, the maximum adius is max = 2σ = 0.78 mm D πρ max = 2 max = 1.56 mm steel Impotant Notes: Sufae tension, σ, is a foe pe length. To alulate the sufae tension foe we multiply it by the length of the ontat line on whih it ats, whih is 2L. Speifi gavity is the density divided by the density of wate at 4, whih is 1000 kg/m 3. We did not onside the buoyany foe ating on the floating needle. Is thee suh a foe? If yes, how would that affet the esult? 7