Download Homework #9 Extra credit

Homework # 11 Extra credit
Due Wednesday, April 16, 2008
ME 363 - Fluid Mechanics
Spring Semester 2008
1] (a) Consider a liquid in a cylindrical container in which both the liquid and the container are
rotating as a rigid body. The elevation difference h between the center of the liquid surface
and the rim of the liquid surface is a function of angular velocity , fluid density ,
gravitational acceleration g, and radius R. Use the method of repeating variables to find a
relationship between the parameters.
(b) Consider now a transient process where the container was originally at rest and rotation
started at time t = 0. After a certain period of time t = T the liquid will attain steady motion in
which both the liquid and the container are rotating as a rigid body. At any time t =  < T
from the start of rotation, the fluid velocity V (r ) at distance r from the axis of rotation was
found to be a function of time , angular velocity , liquid density , and liquid viscosity .
Use the method of repeating variables to find a relationship between these parameters. If, in
two separate experiments, honey and water are rotated in the same container at the same
angular velocity, determine from the obtained dimensional analysis results which of the two
liquids will attain steady motion faster.
(a) Solution
parameters.
Assumptions
We are to use dimensional analysis to find the functional relationship between the given
1 The given parameters are the only relevant ones in the problem.
Analysis
The step-by-step method of repeating variables is employed to obtain the nondimensional
parameters (the s).
Step 1 There are five parameters in this problem; n = 5,
h  f ,  , g , R 
List of relevant parameters:
n5
(1)
Step 2 The primary dimensions of each parameter are listed,
h


g
R
L 
t 
m L 
L t 
L 
1
1
1 2
1 3
1
Step 3 As a first guess, j is set equal to 3, the number of primary dimensions represented in the problem
(m, L, and t).
j 3
Reduction:
If this value of j is correct, the expected number of s is
Number of expected s:
k  n  j  53  2
Step 4 We need to choose three repeating parameters since j = 3. Following the guidelines outlined in this
chapter, we elect not to pick the viscosity. We choose
, , and R
Repeating parameters:
Step 5 The dependent  is generated:
1 
1  h a1  b1 R c1
 L  t
1
1
 m L  L 
a1
1 3
b1
1
c1

mass:
m   m 
0  b1
b1  0
time:
 t   t 
0  a1
a1  0
0  1  3b1  c1
c1  1
length:
b1
0
0
 a1
L   L L
0
1 3b1
Lc1

The dependent  is thus
1:
1 
h
R
The second Pi (the only independent  in this problem) is generated:
 2  
 2  g a2  b2 R c2
m   m 
mass:
 t   t
time:
length:
b2
0
0
2  a2
t
L   L L
0
1 3b2

Lc2

 L t
1 2
 t   m L   L 
1
a2
1 3
b2
1
c2

0  b2
b2  0
0  2  a2
a2  2
0  1  3b2  c2
c2  1
0  1  c2
which yields
2:
2 
g
2R
If we take  2 to the power –1/2 and recognize that R is the speed of the rim, we see that  2 can be
modified into a Froude number,
Modified 2:
 2  Fr 
R
gR
Step 6 We write the final functional relationship as
Relationship between s:
h
 f  Fr 
R
(2)
Discussion
In the generation of the first , h and R have the same dimensions. Thus, we could have
immediately written down the result, 1 = h/R. Notice that density  does not appear in the result. Thus,
density is not a relevant parameter after all.
(b) Solution
parameters.
We are to use dimensional analysis to find the functional relationship between the given