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 4th 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.