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Problem 2.76
[Difficulty: 5]
Given:
Geometry of rotating bearing
Find:
Expression for shear stress; Maximum shear stress; Expression for total torque; Total torque
Solution:
Basic equation
τμ
du
dy
⋅
=
dT
r
τ
⋅
dA
⋅
=
Assumptions: Newtonian fluid, narrow clearance gap, laminar motion
From the figure
r
R sin
θ
()
⋅
=
u
ω
r
⋅
=
ω
R
⋅
sin
θ
⋅
=
du
dy
u0
−
h
=
u
h
=
h
a
R 1
cos
θ
−
⋅
+
=
dA
2
π
⋅
r
⋅
dr
⋅
=
2
π
⋅
R sin
θ
⋅
R
⋅
cos
θ
⋅
d
θ
⋅
=
Then
du
dy
⋅
=
μω
⋅
R
⋅
sin
θ
⋅
a
R 1
cos
θ
−
⋅
+
=
To find the maximum
τ
set
θ
⋅
R
⋅
sin
θ
⋅
a
R 1
cos
θ
−
⋅
+
⎡
⎣
⎤
⎦
d
d
0
=
so
R
μ
⋅
ω
⋅
R cos
θ
⋅
R
−
a cos
θ
⋅
+
⋅
Ra
+
R cos
θ
⋅
−
2
0
=
R cos
θ
⋅
R
−
a cos
θ
⋅
+
0
=
θ
acos
R
+
⎛
⎝
⎞
⎠
=
acos
75
75
0.5
+
⎛
⎝
⎞
⎠
=
θ
6.6 deg
⋅
=
τ
12.5 poise
⋅
0.1
×
kg
ms
⋅
poise
⋅
2
×
π
⋅
70
60
⋅
rad
s
⋅
0.075
×
m
⋅
sin 6.6 deg

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