Fluid Mechanics Exercise sheet 1 – Fundamentals last edited April 22, 2015 These lecture notes are based on textbooks by White [4], Çengel & al.[6], and Munson & al.[8]. Except otherwise indicated, we assume that fluids are Newtonian, and that: ρwater = 1 000 kg m −3 ; p atm. = 1 bar; ρatm. = 1,225 kg m −3 ; µatm. = 1,5 · 10 −5 N s m −2 ; д = 9,81 m s −2 . Air is modeled as a perfect gas (R air = 287 J K −1 kg −1 ; γair = 1,4). 1.1 Speed of sound White [4] P1.87 Isaac Newton measured the speed of sound by timing the interval between observing smoke produced by a cannon blast and the perception of the detonation. If the cannon is shot 8,4 km away, what is the air temperature if the measured interval is 24,2 s? What is the temperature if the interval is 25,1 s? 1.2 Flow classifications 1. Can an inviscid flow also be steady? 2. Can a creeping flow also be inviscid? 3. [more difficult] Can a compressible flow also be isothermal? 4. Give an example of an isothermal flow, of an unsteady flow, of a compressible flow, and of an incompressible flow. 1.3 Flow in between two plates Munson & al. [8] Ex1.5 A fluid is forced to flow between two stationary plates. We observe that its velocity profile u (y) is linked to the average fluid velocity V by the relationship: " y 2# 3V u = 1− 2 h (1/15) in which y is measured from the middle of the gap, and h is half of the gap length (fig. 1.3). The fluid viscosity is 2 N s m−2 , the average speed is 0,6 m s−1 and the two plates are 10 mm apart. What is the shear effort generated on the lower plate? What is the shear effort at the center of the flow? 17 Figure 1.3: Velocity distribution for laminar flow in between two plates, also known as Couette flow. figure CC-0 o.c. 1.4 Viscometer Çengel & al. [6] 2-78 An instrument designed to measure the viscosity of fluids is made of two coaxial cylinders (fig. 1.4). The inner cylinder is immersed in a liquid, and it rotates within the stationary outer cylinder. Figure 1.4: Sketch of a cylinder viscometer. The width of the gap has been greatly exaggerated for clarity. figure CC-0 o.c. The two cylinders are 75 cm tall. The inner cylinder diameter is 15 cm and the spacing is 1 mm. When the inner cylinder is rotated at 300 rpm, a friction-generated moment of 0,8 N m is measured. What is the viscosity of the fluid? Would an non-Newtonian fluid induce a higher moment? 18 1.5 Boundary layer White [4] P1.56 A laminar fluid flow occurs along a wall (fig. 1.5). Close to the wall (y < δ ), we observe that viscous effects dominate the mechanics of the flow. This zone is designated boundary layer. The speed u (y) can then be modeled with the relation: u = U sin πy (1/16) 2δ in which U is the flow speed far away from the wall. Figure 1.5: Velocity profile across the boundary layer. image CC-0 o.c. The fluid is helium at 20 ◦C and 1 bar (9 · 10−6 N s m−2 ); measurements yield U = 10,8 m s−1 and δ = 3 cm. What is the shear effort on the wall? At which height y will the shear effort be half of this value? What would be the effort if helium was replaced with water (1 · 10−3 kg m−1 s−1 )? 1.6 Friction on a plate A plate the size of an A4 sheet of paper (210 mm × 297 mm) is moved horizontally at constant speed above a large flat surface (fig. 1.6). The fluid velocity profile in between the plate and the bottom surface is assumed to be strictly linear everywhere. 1. Express the shear force on the plate as a function of its velocity U , the gap height H , and the properties of the fluid in between the plate and the flat surface. 2. The plate speed is 1 m s−1 and the gap height is 5 mm. What is the shear force if the fluid is air (µatm. = 1,5 · 10−5 N s m−2 ), and if the fluid is honey (µ honey = 40 N s m−2 )? 19 Figure 1.6: A plate moved horizontally across a flat surface. figure CC-0 o.c. 1.7 Clutch Çengel & al. [6] 2-74 Two concentric metal axis are linked by two disks very close one to another, rotating in the same direction at similar (but not identical) speeds. The disk diameters are both 30 cm and they 2 mm one from the other; they are submerged in sae30w oil with viscosity 0,38 kg m−1 s−1 . Figure 1.7: Sketch of the two disks constituting the clutch. The gap width has been exaggerated for clarity. figure CC-0 o.c. The power shaft rotates at 1 450 rpm, while the powered shaft rotates at 1 398 rpm. We consider the simplest possible flow case in between the two disks. 1. What is the moment imparted by one disk to the other? 2. What is the transmitted power and the clutch efficiency? 3. What do you think the flow patterns in between the two plates would look like in the real case? 20 Answers 1.1 26,7 ◦C & 5,6 ◦C 1.2 1) yes, 2) no, 3) yes (in very specific cases such as high pressure changes combined with high heat transfer or high irreversibility, therefore generally no), 4) open the cap of a water bottle and turn it upside down: you have an isothermal, unsteady, incompressible flow. An example of compressible flow could be the expansion in a jet engine nozzle. 1.3 1) τy=−h = 720 N m−2 ; 1.4 µ = hM 2πωR 13 H 2) τy=0 = 0 N m−2 . = 1,281 · 10−2 N s m−2 . 1.5 −3 2 −1 for helium; 1) τwall = µ πU 2δ = 5,09 · 10 N m 3) τwall = 0,565 N m−2 for water. 2) y1 = 32 δ = 2 cm; 1.6 1) Fτ = L 1 L 2 µ UH = 1,87 · 10−4 N for air, and 500 N for honey(!). 1.7 µω 1) M = π2 h R 4 = 0,8228 N m; 2) W˙ 2 = ω2 M = 120 W; ˙ 2 3) ηclutch = W = 96,4 % (remember this is a very low-power, low-relative speed, W˙ 1 laminar-flow case). 21 22