Problem #1:
Solve Problem 3A.1 in Chapter 3 of BSL.
Answer:
Equation 3.631 describes the torque required to turn an outer cylinder at an angular velocity
Ω
o
.
This equation is the same for the torque required to turn an inner rotating shaft at an angular
velocity
Ω
o
, derivable in like manner from the corresponding velocity profile in Eq. 3.632.
The power required is equal to the product of the torque and the angular velocity.
The numerical answers are given immediately below the statement of Problem 3A.1 in the
textbook.
Problem #2
:
The annular space between two coaxial long circular pipes is filled with a viscous, Newtonian
fluid. The radii of the inner and outer wetted surfaces are
κ
R and R, respectively. The inner pipe
is rotating with an angular velocity of
Ω
. Determine the velocity profile of the fluid and the
torque required to keep the inner pipe rotating, using the equations of motion and continuity. For
simplicity it is assumed that there is no flow in the axial direction of the pipes.
Answer:
The derivation proceeds as in Example 3.63 to Eq. 3.626, which we choose to rewrite as
2
12
u
R
DD
rr
θ
⎛⎞
=+
⎜⎟
⎝⎠
Applying the boundary conditions,
@
[email protected]
i
uR
r
ur
R
R
κ
=
Ω=
==
Inserting these boundary conditions into the above equation for the angular velocity gives,
1
2
1
;0
i
Ω= +
= +
2
Solve these equations for D1 and D2
2
1
2
2
2
2
1
1
i
i
D
D
Ω
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 Fall '08
 dicarlo
 Fluid Dynamics, Angular velocity, Umax

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