The Physics of Fluids

summary 11/27

Today we discussed our results from last week's experiments probing how water flow rate through a piece of tube is affected by the height difference between the tube outlet and the surface of the water in the reservoir tub. We found a nice linear relationship, but also an interesting inconsistency. The linear trend would predict a significant flow rate even when the tube outlet is exactly at or slightly above the water level, but we know from experiments that that is not possible (we checked today). There must have been some systematic error in the experiments or an incomplete understanding of the physics that remain a mystery.

This result put us in a position to finally discuss the driving force for flow. Up to now, we have been talking about why the water flows in vague terms of gravity, incompressibility, air pressure, and tube angle. 

Aside: the angle the tube made with the floor was a contentious issue last week: the analog of a block sliding down a ramp seemed certainly relevant, suggesting that angle ought to make a difference. But we had also seen at least one experiment before suggesting the shape of the tube between the inlet and outlet didn't matter so much. Fortunately, last week's experiments fixed the tube shape to a straight configuration so that the height difference (our main independent variable) also corresponded to a specific angle: the linear dependence on height corresponds to a sinusoidal dependence on angle. But is one more "fundamental" than the other?

Back to the main story: why does the flow rate increase with increasing height difference? We talked about the concept of "hydrostatic pressure", which is the reason why our ears pop when we dive deep under water or fly high up in the air. It also means submarines can be crushed like a soda can if they wander too deep in the ocean. The same phenomenon is at work in our water reservoir and through our tube. Our experimental results together with this understanding help us describe the origin or driving force of the water flow: a difference of water pressure between the tube inlet and the tube outlet. The water pressure at the tube inlet (the reservoir outlet) is the height to which the water is filled times water's density times the earth's gravitational acceleration. As long as the distance water has to travel UP the tube does not exceed that distance, the water will flow in proportion to the difference in height.

We now have 2 more variables to explore: tube length and tube diameter. Today's experiments dealt with tube length. We hypothesized, based on our new understanding, that tube length shouldn't matter, except to the extent that we can only achieve large height differences using longer tubes. We argued that friction should not be significant.

But we quickly realized our hypothesis was missing something. The longer the tube we used, the slower the water seemed to flow...significantly! Next week we'll go over the results quantitatively and see if we can find a rule for how tube length affects flow rate from the data. Remember, our ultimate goal is not only to describe our data using mathematical equations, but to see if we can predict the mathematical equations themselves without any knowledge of experimental data!