Physics 103

**Flowing Water: Poiseuille or Bernoulli?**

The clocks of the Hellenistic world were mostly water clocks. Ancient Chinese clocks were also water clocks. These are not simple devices! As soon as you think about a jar with a hole in the bottom, through which water can drain, it occurs to you, won’t the water drain faster at first, when there is a lot of water in the jar, and more slowly later, when the jar is almost empty? That is true! The height of water in the jar will not be a simple linear function of time, as we now understand time. That means that a water clock is not a simple clock.

There is reason to think, though, that it might be a *logarithmic clock *(just like radioactive
decay, oddly enough). The reason is that
under some circumstances , namely very slow flow
through a very narrow pipe, the volume rate of outflow should just be
proportional to the pressure at the bottom of the jar. In
this case the height of water in the jar goes down at a rate proportional to
pressure, which in turn is proportional to height. It is analogous to the number of radioactive
nuclei in a sample going down, by radioactive decay, at a rate proportional to
the number of radioactive nuclei.
“Height of water” plays the role of “number of radioactive nuclei.” The
water drains fast at first because the pressure is high, just as a radioactive
sample loses radioactive nuclei fast at first because there are many of them,
all liable to decay.

We therefore expect the decay of height H to be of exponential form

_{}

Thus the *logarithm*
of H should be a linear function of time *t*. This is something one can check
experimentally, of course. Just measure
H as a function of t, and plot appropriately to see if the data fall on a
straight line. If they do, we can
determine the decay rate alpha from the slope of the line.

Slow viscous flow through a pipe is called Poiseuille flow. This
is the case that we were describing above.
In Poiseuille flow, the volume current I is
proportional to pressure difference _{}:

_{}

_{}_{}

Here L is the length of the tube, R is its radius,
and _{} is the viscosity of
the water, about 0.001 in SI units (Pa-s). The volume of water in the jar is V=AH,
where A is the cross-sectional area of the jar.
Putting this all together we have a prediction for the decay rate alpha
(rate at which height decays),

_{}

To check common sense, consider that the height
would decay faster if g were bigger (because it is gravity driving the fluid to
fall), and if the fluid were denser (it would weigh more). It would decay slower if L, the length of the
pipe, were greater, because there is friction all along the pipe, and also
slower if A, the area of the jar, were bigger, because then the same volume of
outflow would change the height less. It
drains slower if the fluid has higher viscosity – that makes sense, the
internal friction in the fluid would be greater. The most dramatic dependence is on the radius
R of the pipe, which appears to the 4^{th} power! If R is cut in half, the decay rate is only
1/16 what it was, apparently.

Another possible fluid behavior is called Bernoulli flow. In this case, viscosity is unimportant, and the fluid does not dissipate energy appreciably. The flow velocity through the oulet pipe is only proportional to the square root of pressure, not to the pressure itself, and the fluid drains according to the law

_{}

We can check for this behavior by graphing the
square root of H vs t. The parameter _{} for the rate of
outflow, in terms of the cross-sectional area A of the jar and the
cross-sectional area _{} of the outflow pipe,
is

_{}

To check common sense, note that if the jar is wider (larger A), then the rate is slower, and if the pipe is wider (larger R), then the rate is faster. Also if gravity were stronger, the outflow would be faster.

Real fluid flow could be one or the other of these
possibilities, or it could be something else – perhaps something intermediate
between the two, or if the flow is fast enough, it could be turbulent. Measure how the height H in the water clock
depends on time t for a suitable outflow pipe, and compare your data with
the two different predictions, Poiseuille and
Bernoulli. If the actual behavior
seems to be like either of these two predicted behaviors, check to see if the
rate parameter (_{} or _{}) is roughly the predicted value.

As always, write a paragraph summarizing what you did, and what it means.

_{}