Solutions for homework set #1:
Problem #1:
Consider a spherical particle falling freely in a large fluid body. It is found experimentally that
the terminal settling velocity
u
of the particle is dependent on: (1) particle diameter D; (2) the
buoyant weight of the particle
W
(the difference between the gravity and buoyant forces acting
on the particle); (3) fluid density
ρ
, and (4) fluid viscosity
μ
.
It is assumed that the velocity
u
can be mathematically expressed as products of powers of the
above variables (i.e. similar to the example of dimensional analysis given in the previous
lectures).
ab
c
d
uC
D
W
ρμ
=
Obtain the relationship for
u
using dimensional analysis in which the set of linear equations is
solved in terms of the exponent
c
.
Show that the respective dimensionless groups obtained by solving the set of linear equations in
terms of the exponent
b
and
c
are correlated.
Answer:
If
c
d
D
W
=
Equating dimensions:
Solving in terms of c:
a = 1
b = 1+c
d = (1+2c)
11
2
1
2
2
cc
c
c
c
D
W
WW
C
D
DW
W
ρ
μμ
μρ
μ
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 dicarlo

Click to edit the document details