UMS Number: 22.381 Undergraduate Fluid Mechanics: Fall 2013 Test 1 :Chapter 1 and 2 Instructor : Prof. D. Willis Please perform all calculations and show all work neatly on this test. If additional space is needed, please get a sheet of paper from Prof. Willis/the proctor of this exam, and neatly show where the additional work can be found. GOOD LUCK! 1 UMS Number: 1. (20 pts) Concept Questions: Please answer the following concept questions fully, but concisely. Explanations for any questions are recommended. (a) (3 pts) How do liquids and gasses behave similarly and/or differently in regard to compressibility? (b) (3 pts) How does the viscosity of a liquid and gas change with temperature? (c) (4 pts) Give an example of a shear thinning and a shear thickening fluid and indicate clearly how they behave differently? (d) (3 pts) Pressure and shear stress both have force per unit area as dimensions. Define these stresses and forces and clearly show the difference(s) between the shear stress and the pressure force. (e) (6 pts) Consider a soap bubble with a 1cm diameter. If the surface tension of soap is 2.5 × 10−2 N/m, answer the following: a) Is the pressure of the air inside the bubble higher or lower than the air surrounding the bubble? b) What is the numerical value of the pressure difference between the inside and the outside? 2 UMS Number: (f) (5 pts) Draw an example of (a) a wetting and (b) a non-wetting fluid. Clearly show the contact angle for each example. (c) Write down what affects contact angle? (g) (3 pts) Consider the diagram below of two dams. What is the correct statement, and why?: a) F1 > F2 b) F1 < F2 c) F1 = F2 d) I don’t know. Figure 1: The two dams. (h) (3 pts) The center of a square window with sides of 2m is located 100m below the surface of the water. The window is mounted at an angle of 45 degrees. What is the net resultant hydrostatic force on the window? Note: you do not need to determine the location of this force. 3 UMS Number: 2. (40 pts) The multiple plate problem Consider the diagram below showing multiple plates separated by two different fluids. The top plate is stationary, the middle plate moves with a velocity 1.5m/s and the bottom plate moves with a velocity 0.5m/s. Figure 2: The multiple plates separated by fluid problem. Additional information: • The velocity of the fluid varies linearly with height in both channels. • The density of both fluids is 1000kg/m3 . • The width of the plates is 1m • The middle plate can be considered to have small thickness and negligible weight. QUESTIONS (a) (4 pts) On the figure above, indicate clearly (using an arrow or circle) all locations where the no slip condition applies. (b) (8 pts) Draw a free body diagram of the middle plate and clearly show all stresses/pressures acting on the plate. (c) (4 pts) On the same free body diagram, draw the resultant forces due to the shear stress. (d) (6 pts) Assuming that the velocity varies linearly in the y direction, draw the velocity profile for both fluids (using the dashed line as V = 0) (e) (8 pts) Determine the magnitude and direction of the shear stresses acting on both sides of the middle plate. Clearly indicate whether the shear stress is positive or negative. (f) (10 pts) Determine the Force Fplate required to maintain the equilibrium (non-accelerating) conditions shown above. 4 UMS Number: cont’d 5 UMS Number: 3. (30 pts) Two pools Two fluids are separated by the gate shown in figure 3. The gate is hinged at point ’A’ to allow it to freely rotate about that point. In the image shown, the gate is in static equilibrium. For this problem, answer the questions below. Figure 3: The gate separates oil from water. The problem asks you to determine the weight, W of the gate. Useful information and assumptions • You may assume the gate has a width of 1m • The second moment of area of a rectangle is Ixc = 1 12 width × height3 Questions (a) (7 pts) CLEARLY draw and label the free body diagram you will use for solving this problem. Include all pressure distributions and their equivalent resultant forces. (b) (8 pts) Determine the resultant forces due to the pressure of the fluids on both sides of the gate. Also, determine the location of the resultant force (center of pressure) and clearly show that measurement on a diagram. (c) (15 pts) Determine the weight of the gate required to maintain the static equilibrium position shown. 6 UMS Number: cont’d 7
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