HomeworkSolutions_C5

HomeworkSolutions_C5 - Problem#1 A sphere of radius R is...

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Problem #1 : A sphere of radius R is falling in creeping flow with a terminal velocity u through a quiescent fluid of viscosity μ. At what horizontal distance from the sphere does the velocity of the fluid fall to 1% of the terminal velocity of the sphere? Answer: On the z-plane at z=0, u θ is the only non-zero velocity component. For a fixed sphere and a fluid approaching the sphere at u 3 31 1 44 RR uu rr θ ⎛⎞ ⎛⎞ ⎛⎞ =− ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ On the z-plane, the fluid velocity increases from zero at the sphere surface to u at sufficiently far from the surface. If an observer sitting in the fluid at a position sufficiently far from the sphere surface (R/r << 1), will see the sphere moving at u in an opposite direction. Therefore, 1% of the terminal velocity of the sphere (u ) in this reference frame is the same as 99% of the approach velocity of the fluid when the sphere is fixed. Or Since R/r << 1, ( 3 / ) R r is not important 3 0.99 0.99 1 −= 3 0.01 75 4 Rr rR = ⇔= Problem #2: Recall the example of flow around a solid sphere (see textbook, Example 4.2.1)

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HomeworkSolutions_C5 - Problem#1 A sphere of radius R is...

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