*Michael Fowler, UVa*

Some of the most interesting results of hydrodynamics, such as the sixteen-fold 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 _{}, and acceleration _{}

We shall use *square brackets* [] to denote the
dimensions of a quantity, for example, for velocity, we write _{} Force must have the
same dimensions as mass times acceleration, so _{} 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 _{} _{}, (the left hand side is force per unit area, the right hand *v*_{0}/*d* is the velocity
gradient) that

_{}

How
can thinking dimensionally help us find the flow rate *I* through a
pipe? Well, the flow itself, say in
cubic meters per second, has dimensions _{} What can this flow
depend on?

*The
physics of the problem is that the pressure difference _{} between the ends of
the pipe of length L is doing work overcoming the viscous force*. The

Therefore,

_{}

where
*f* is some function we don’t know, but we *do* know that the two
sides of this equation must match dimensionally, so *f* must have the same
dimensions as *I*, that is, _{}

Now
_{}, a pressure, has dimensions _{} so _{} has dimensions _{}

The
other variables in *f* have dimensions _{} (from above) and [*a*]
= *L*.

The
game is to put these three variables (or powers of them) together to give a
function *f* having the dimensions of flow, that is, _{}, otherwise the above equation must be invalid.

The
first thing to notice is that there is no *M* term in flow, and none in *a* either, so _{} and _{} must appear in the
equation in such a way that their *M*
terms cancel, that is, one divides the other.

We
know of course that increased pressure increases the flow, so they must appear
in the combination _{}*.* This gets rid
of *M*.
The next task is to put this combination, which has itself dimensions _{} together with [*a*]
= *L*, to get a quantity with the
dimensions of flow, _{} The unique choice is
to multiply_{}by *a*^{4}.

We therefore conclude that the flow rate through a circular pipe must be given by:

_{}

This
is certainly much easier than solving the differential equation and integrating
to find the flow rate! The catch is the
unknown constant *C* in the equation—we
can’t find that without doing the hard work. However, we *have*
established from this dimensional argument that the flow rate increases by a
factor of 16 when the radius is doubled.

It
should be noted that this conclusion *does*
depend on the validity of the assumptions made—in particular, that the flow is
uniform and in straight streamlines. At
sufficiently high pressure, the flow becomes turbulent. When this happens, the pressure causes the
fluid to bounce around inside the pipe, and the flow pattern will then depend
also on the *density* of the fluid, which was irrelevant for the slow
laminar flow, and the reasoning above will be invalid.

*Exercise*: derive the depth dependence of the steady
flow of a wide river under gravity.
(Note: The appropriate flow rate is cubic meters per second *per meter
of width of the river*.)

So
dimensional analysis cannot give overall dimensionless constants, but can
predict how flow will change when a physical parameter, such as the pressure or
the size of the pipe, is altered. We’ve
shown above how it rather easily gives a nonobvious result, the *a*^{4} dependence of flow on
radius, which we found earlier with a good deal of work. But as we shall see, dimensional analysis can
also illuminate the essential physics of flow problems where exact mathematical
analysis is far more difficult, such as Stokes’ Law in the next lecture, and
help us understand how the nature of fluid flow changes at high speeds.

© 2006 Michael Fowler