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ME 106 Hwk 4 solutions

ME 106 Hwk 4 solutions - University of California Berkeley...

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University of California, Berkeley Mechanical Engineering Prof S. Morris ME106 Fluid Mechanics, Problem Set 4 1. The ¯gure shows the cross{section of a large{diameter pipe from which you are permitted to withdraw water through a hole of ¯xed area A i . You have attached a di®user with exit area A e . The °ow is incompress- ible and inviscid. Find (a) the volume °owrate Q as a function of the pipe pressure p o , atmospheric pressure p a , °uid density ½ and the exit area A e ; and (b) the pressure p i in the hole of area A i . Your analysis will predict that Q can be made arbitrarily large by making A e large enough, but what e®ect already discussed in this class would limit Q ? air p a free jet p o A i A e 2. The ¯gure shows a piston of cross{sectional area A p sinking under its own weight into an air{¯lled cylinder. The air thus expelled leaves the cylinder as a free jet of total cross{sectional area A j . The gap between the piston and cylinder is narrow, so that A j ¿ A p . (a) Sketch the stagnation streamline for a reference frame ¯xed in the cylinder wall, and hence sketch the streamlines. (b) Then ¯nd the pressure acting on the base on the piston. (Assume the °ow is quasi{steady, and that the gravitational potential energy of the air is negligible compared with the kinetic energy of the free jet.) (c) Hence show that the piston sinks at a speed V given approximately by V = q 2 mgA 2 j = ( ½A 3 p ). (A similar device is used as a hydraulic bu®er.) V m air Piston area A p Jet area A j g 3. Flow in a sink vortex can be approximated by V = K ^ Á=r ; here K is a constant, r and Á are plane polar coordinates, and ^ Á is a unit vector in the circumferential direction. (This simpli¯ed model does not include the downward °ow into the drain; that °ow is small compared with the swirling motion.)
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