Problem 2.61
[Difficulty: 3]
Given:
Data on the viscometer
Find:
Time for viscometer to lose 99% of speed
Solution:
The given data is
R5
0
m
m
⋅
=
H8
0
m
m
⋅
=
a
0.20 mm
⋅
=
I
0.0273 kg
⋅
m
2
⋅
=
μ
0.1
Ns
⋅
m
2
⋅
=
The equation of motion for the slowing viscometer is
I
α
⋅
Torque
=
τ
−
A
⋅
R
⋅
=
where
α
is the angular acceleration and
τ
is the viscous stress, and A is the surface area of the viscometer
The stress is given by
τμ
du
dy
⋅
=
μ
V0
−
a
⋅
=
μ
V
⋅
a
=
μ
R
⋅
ω
⋅
a
=
where
V
and
ω
are the instantaneous linear and angular velocities.
Hence
I
α
⋅
I
d
ω
dt
⋅
=
μ
R
⋅
ω
⋅
a
−
A
⋅
R
⋅
=
μ
R
2
⋅
A
⋅
a
ω
⋅
=
Separating variables
d
ω
ω
μ
R
2
⋅
A
⋅
aI
⋅
−
dt
⋅
=
Integrating and using IC
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 Fall '07
 Lear
 Fluid Mechanics

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