
can be used to confirm that equations are dimensionally correct.
e.g.
=
(dv) / (dy)
(ML
1
T
2
)
=
(ML
1
T
1
)(LT
1
)(L
1
)
Example:
H
=
f (v,
, D,
LHS
=
L
V
=
m/s
=
LT
1
=
kg/m
3
=
ML
3
D
=
m
=
L
=
Pa.s
=
M.T
1
L
1
H
=
K (v
a
x
b
x
D
c
x
We now solve for “a, b, c and d” using the fact that the dimensions on the LHS must equal the
dimensions on the RHS
)
d
)
Buckingham
Theorem
This theorem is one method of determining appropriate dimensionless groups, from a number of
given variables.
It states that:

in a physical problem involving “n” quantities / variables

in which there are “m” fundamental dimensions (M,L,T)

the quantities may be arranged in (nm) independent dimensionless groups or parameters
These are called
groups.
Each
group is formed from the product of 3 repeating variables raised to a power and one of the
remaining nonrepeating variables raised to the power of unity.
The choice of repeating variables is guided by the following considerations:
1.
Each repeating variable must contain between them all the fundamental dimensions.
2.
The repeating variables should describe a size characteristic, a fluid property characteristic
and kinematic characteristic, d,
and v respectively.
3.
The repeating set must contain three variables that cannot themselves be formed into a
dimensionless group.
e.g.
a)
both l and d cannot be chosen as they can be formed into the dimensionless
group
l/d.
b)
p,
and v cannot be used since
p/
v
2
is dimensionless.
Experiment:
Consider a sphere of diameter
d
which is towed towards the left at a velocity
V
through a fluid of
density
and dynamic viscosity
.
The fluid could be air, water, oil, etc.
The force required to
tow the sphere is simply that required to overcome the force on the sphere due to friction and
pressure forces.
If the sphere is brought to a rest and the fluid given a velocity V to the right then
the sphere experiences the same force as the towing force, but in the opposite direction.
This force
is called the
Drag Force D
.