Homework Set #4:
Due date: 10/06/06, by 11:00 am.
Problem #1:
In Example 4.21 in Chapter 4 of BSL, a fluid approaches a stationary sphere from below. Now
in this problem, a sphere of radius R is falling in creeping flow with a terminal velocity u
∞
through a quiescent fluid of viscosity μ. The velocity distribution of the fluid given as Eqs. (4.2
13) and (4.214) in BSL are valid for this case.
a.
What velocity components (i.e. u
θ
, u
r
, and u
Ø
) are zero on the zplane at z=0 (see Fig. 2.6
1 in Chapter 2 of BSL).
b.
Also on the zplane at z=0, the fluid velocity increases from zero at the sphere surface to
u
∞
at sufficiently far from the surface. At what horizontal distance from the sphere
surface does the velocity of the fluid fall to 1% of the terminal velocity of the sphere?
Answer:
On the zplane at z=0, u
θ
is the only nonzero velocity component. For a fixed sphere and a fluid
approaching the sphere at u
∞
3
3
1
1
4
4
R
R
u
u
r
r
θ
∞
⎛
⎞
⎛
⎞
⎛
⎞
= −
−
−
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
On the zplane at z = 0, 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
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 Spring '08
 Peters
 Fluid Dynamics, terminal velocity, uθ, Sphere Surface, fluid velocity increases

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