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ME 363 - Fluid Mechanics Final Exam May 15, 2008 Spring Semester 2008 Problem 1a (5 points) A 6-mm diameter hole is punched near the bottom of a 32-oz drinking cup full of cold water (ρ = 1000 kg/m3, μ = 0.0018 kg/m-s). Estimate the velocity of the stream that issues through this hole. Problem 1b (5 points) A 6-mm-inside-diameter straw, 10 cm long, is held up to the hole so that the water flows through it. Assume the average velocity V in the straw is equal to your answer to problem 1. Do you expect the flow through this straw to be laminar or turbulent? (no matter what you find, assume it is laminar for all calculations below: this straw is short enough to prevent the flow from completely developing turbulence) Problem 1c (10 points) The setup is exactly as in problem 1b. Use the Moody chart or similar information to estimate the pressure gradient dP/dx [Pa/m] in the straw. You may ignore minor losses 2 and flow kinetic energy (the V terms). 2 Problem 1d (10 points) In fluid mechanics textbooks, one can read about extra losses existing at pipe entrances. For example, Figure 6.6 from White, Fluid Mechanics (6th edition) highlights the “entrance pressure drop”: Your answer to 1c corresponds to the slope of the curve at the label “1C” above. Below we will estimate the (steeper) pressure gradient at the point labeled “1E” above. To do this, we will consider only the first 1 cm of the straw: Here, boundary layers grow from the straw’s interior surface inward. If we imagine slicing the whole straw lengthwise and unrolling it to form a flat plate (18.8 mm by 10 cm), we can apply our understanding of flat plate boundary layers to approximately understand this boundary layer region. Using flat plate theory, how thick is the boundary layer 1 cm from the entrance to the straw [mm]? Problem 1e (20 points) Assume your answer to 1d is thin enough that (1) the flat plate analogy makes sense and (2) the inviscid core flow does not accelerate significantly in the first 1 cm. By considering the friction (drag or shear stress) in the first 1 cm, estimate the pressure gradient [Pa/m] in the first 1 cm of the straw (your answer corresponds to the slope at label “1E” above). Problem 2a (20 points) A large horizontal culvert with severe scale build-up on its interior can be considered as a 2-m diameter pipe with a relative roughness of 0.05. Initially, this culvert is full of water (ρ = 1000 kg/m3, μ = 0.001 kg/m-s) at rest. Then, as a valve is opened upstream, the water velocity V increases linearly in time, reaching 20 m/s after 6 seconds. Ignoring viscous friction, what is the horizontal reaction force required to hold a 1-m length of the culvert stationary during this valve-opening event? Problem 2b (20 points) Now we will reconsider the inviscid assumption from problem 2a. Including viscosity, at what average velocity V [m/s] will the total reaction force be double your answer in 2a? Problem 3 (10 points) A 1.24-mm-diameter rod, 4.4 mm long, is coated in glue. This rod is to be inserted completely into a cylindrical hole with a diameter of 1.26 mm. The maximum force F that can be used to insert the rod is 1 Newton. During insertion, the velocity of the rod must always exceed 0.2 mm/s. What is the maximum allowable viscosity of the glue [kg/m-s]? note untis to answer above should be Pa/m