p0634 - across which fluid flows is the horizontal plane...

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6.34. CHAPTER 6, PROBLEM 34 573 6.34 Chapter 6, Problem 34 Since the geometry of this problem is continuously changing with time, the motion is inherently unsteady. Thus, we select the control volume whose upper boundary moves with the ball as shown in Figure 6.34. Figure 6.34: Spherical ball falling into a tank of square cross section. The mass-conservation principle tells us that d dt 888 V ρ dV + 8 s 8 S ρ u rel · n dS =0 Since the upper boundary of the control volume moves with the same velocity as the ball, in a time t the height of the control volume changes by V b t . Until the ball reaches the bottom of the tank, the change in height of the control volume is the same across the entire width of the tank. Hence, because the tank cross section is square with side h ,the volume change is given by V = h 2 V b t = dV dt = V b h 2 Since the fluid is incompressible, the unsteady term becomes d dt 888 V ρ dV = ρ dV dt = ρ V b h 2 Turning to the mass-flux integral, we observe that the only part of the control volume
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Unformatted text preview: across which fluid flows is the horizontal plane between the ball and the tank walls. On this part of the control-volume surface, the absolute velocity of the fluid and the control-volume velocity are u = 1 10 V b k and u cv = V b k 574 CHAPTER 6. CONTROL-VOLUME METHOD Also, the outer unit normal on this part of the control volume is n = k . Therefore, because the area between the tank and the sphere is the difference between the tank cross-sectional area, h 2 , and the area of the spheres projection on a horizontal plane, d 2 / 4 , we have 8 s 8 S u rel n dS = 88 A } 1 10 V b k ( V b k ) ] k dA = 11 10 V b w h 2 4 d 2 W Therefore, the mass-conservation equation simplifies to V b h 2 + 11 10 V b w h 2 4 d 2 W = 0 = 1 10 V b h 2 = 11 40 V b d 2 Solving for d , we find d = 2 h 11 = d = 0 . 34 h...
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This note was uploaded on 02/09/2012 for the course AME 309 taught by Professor Phares during the Spring '06 term at USC.

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p0634 - across which fluid flows is the horizontal plane...

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