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Using Dimensions
Michael Fowler, UVa
Some of the most interesting results of hydrodynamics, such as the sixteenfold increase in flow
down a pipe on doubling the radius, can actually be found without doing any calculations, just
from dimensional considerations.
We symbolize the “dimensions”
mass
,
length
and
time
by
M
,
L
,
T
.
We then write the dimensions
of other physical quantities in terms of these.
For example, velocity has dimensions
1
LT
−
, and
acceleration
2
.
LT
−
We shall use
square brackets
[] to denote the dimensions of a quantity, for example, for velocity,
we write
[ ]
1
.
vL
T
−
=
Force must have the same dimensions as mass times acceleration, so
[ ]
2
.
FM
L
T
−
=
This “dimensional” notation does
not
depend on the units we use to measure
mass, length and time.
All equations in physics must have the same dimensions on both sides
.
We can see from the equation defining the coefficient of viscosity
,
η
0
/
FA v d
/
=
, (the left
hand side is force per unit area, the right hand
v
0
/
d
is the velocity gradient) that
[ ] [ ] [ ]
()
[ ] [ ] ( )
22
1
11
//
/
/
FAdv
M
L
T
L
L
L
T
M
L
T
.
−
−−
=⋅
=
⋅
=
−
How can thinking dimensionally help us find the flow rate
I
through a pipe?
Well, the flow
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 Fall '07
 MichaelFowler
 Fluid Dynamics, Lη

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