24
5
SIMILITUDE
Q
n
=
f
Q
(
Q
1
,
Q
2
,
· · ·
,
Q
n
−
1
)
.
(83)
There
are
n
variables
here,
but
suppose
also
that
there
are
only
k
independent
ones;
k
is
equivalent
to
the
number
of
physical
unit
types
encountered.
The
theorem
asserts
that
there
are
n
−
k
dimensionless
groups
β
i
that
can
be
formed,
and
the
functional
equivalence
is
reduced
to
β
n
−
k
=
f
ψ
(
β
1
,
β
2
,
· · ·
,
β
n
−
k
−
1
)
.
(84)
Example.
Suppose
we
have
a
block
of
mass
m
resting
on
a
frictionless
horizontal
surface.
At
time
zero,
a
steady
force
of
magnitude
F
is
applied.
We
want
to
know
X
(
T
),
the
distance
that
the
block
has
moved
as
of
time
T
.
The
dimensional
function
is
X
(
T
)
=
f
Q
(
m,
F,
T
),
so
n
=
4.
The
(MKS)
units
are
[
X
(
·
)]
=
m
[
m
]
=
kg
[
F
]
=
kgm/s
2
[
T
]
=
s,
and
therefore
k
=
3.
There
is
just
one
nondimensional
group
in
this
relationship;
β
1
assumes
only
a
constant
(but
unknown)
value.
Simple
term-cancellation
gives
β
1
=
X
(
T
)
m/F
T
2
,
not
far
at
all
from
the
known
result
that
X
(
T
)
=
F
T
2
/
2
m
!
Example.
Consider
the
ﬂow
rate
Q
of
water
from
an
open
bucket
of
height
h
,
through
a
drain
nozzle
of
diameter
d
.
We
have
Q
=
f
Q
(
h,
d,
π,
µ,
g
)
,
where
the
water
density
is
π
,
and
its
absolute
viscosity
µ
;
g
is
the
acceleration
due
to
gravity.
No
other
parameters
affect
the
ﬂow
rate.
We
have
n
=
6,
and
the
(MKS)
units
of
these
quantities
are:
[
Q
]
=
m
3
/s
[
h
]
=
m
[
d
]
=
m
[
π
]
=
kg/m
3
[
µ
]
=
kg/ms
[
g
]
=
m/s
2
There
are
only
three
units
that
appear:
[length,
time,
mass],
and
thus
k
=
3.
Hence,
only
three
non-dimensional
groups
exist,
and
one
is
a
unique
function
of
the
other
two.
To