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quiz1_sol

# quiz1_sol - Question 1 X51 θ X4C X52 X67 X79 X79 X54 X42...

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Unformatted text preview: Question 1: X51 θ X4C X52 X67 X79 X79 X54 X42 X78 Figure 1: Schematic Consider a channel of height 2 R with rectangular cross-section as shown in the sketch. A hinged plank of a length L < R and at an angle θ is located at the center of the channel. (You may assume that L is small compared to the length perpendicular to the sketch.) The plank has a mass per unit width, m . A constant volume ﬂux per unit width, Q , of an inviscid, incompressible ﬂuid is applied at the inlet of the channel. Find all equilibrium values for θ (include BOTH stable and unstable equilibria!) To simplify your calculation, you may assume that we have chosen the mass, m , and volume ﬂux, Q , such that for at least one equilibrium state, θ is small. Solution: (4 points total, 1 for each equilibrium and 1 each for stability.) Before we do any calculations we can see by inspection that for inviscid ﬂow, there are two trivial equilibrium positions. If the plank is vertically aligned, the pressures on the front and back balance (since the ﬂow is symmetric) and the weight of the plank is supported by the hinge: 1 Streamlines FBD X6DX67 X46 X68X69X6EX67X65 X46 X70 X46 X70 X6DX67 X46 X68X69X6EX67X65 X46 X70 X46 X70 Thus there is an equilibrium at θ = 2 π/ and at θ = 3 π/ 2 . Note, neither of these are physically realizable since, in a real ﬂuid that has some viscosity, separation at the plank ends will break the front/back symmetry. To find the stability of there two solutions, we perturb away from equilibrium and inspect the resulting torques. For the θ = 3 π/ 2 solution: Streamlines FBD X6DX67 X46 X68X69X6EX67X65 X46 X70 X46 X70 Considering the torque about the hinge, both gravity and pressure act to restore the plank to its equilibrium position so θ = 3 π/ 2 is STABLE . Drawing a similar picture for π/ 2, we find that the pressure is stabilizing and gravity is destabilizing so we need to do a calculation to figure out which one wins (we will come back to this after the next section). (15 points total: 3 points for mass conservation, 3 points for conservation of momentum (Bernoulli), 6 points for force and torque balance on the plate, 2 points to put it all together and find θ eq (i.e. integrate!) and 1 point for stability.) Solving this problem exactly is nontrivial since technically, we would need to 2 solve 2 ψ = with no ﬂux boundary conditions at the plank and at the top and bottom channel walls. To approximate the equilibrium conditions, we need to estimate the torque on the plank. Since there are several reasonable assumptions you can make at this point, credit was given if you (1) made a reasonable set of assumptions (2) applied conservation of mass, conservation of momentum (in this case...
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quiz1_sol - Question 1 X51 θ X4C X52 X67 X79 X79 X54 X42...

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