ANSWERS: 3

Ah. Good question. Here are some thoughts on the matter, made by someone who is not an expert on fluid flow. There are several important forces one might consider here when we try to assemble a model for a physical process like this. The forces I would think of that might have some affect on the water as it flows are: 1. Gravity 2. Surface tension 3. Air resistance 4. Vanderwaals forces 5. Internal frictional forces in the flow 6. Turbulence There, I've made a list of the possible forces that might impact water flow. Not all of these forces will be important. The art of modeling is to find the important factors in your system and to understand how they will come into play. Then look to see if the model describes reality. Should you try to improve your model? Can you add in these other factors to get a better model, to improve your understanding of the system? My first candidate is gravity. Imagine a water faucet that points directly downward towards the ground. As the water flows out, we know that it will tend to flow in a stream that thins out, and I'll conjecture that the acceleration of gravity causes the water to speed up as it falls towards he ground. So try a different thought experiment. Imagine now a faucet that flows in the space station, without the influence of gravity. I would expect that the water now flows at a constant rate. Each molecule of water undergoes no acceleration due to gravity. The difference in these two thought experiments is the acceleration of gravity. Consider two molecules in the water stream (M1 and M2.) M1 is just a wee bit ahead of M2 in the stream, so it starts accelerating towards the ground just a second ahead of M2. If you want, think of two cannon balls dropped from a tower. One a second before the first. It will turn out that the distance between the two balls is not constant (at least while they are both falling.) A simple model for the distance traveled by a dropped object in a gravitational field is D = g t^2 where g is the acceleration due to gravity and t is time. So if the first ball dropped is dropped one second before the first ball, then the distance traveled by ball the second ball will be D2 = g (t1)^2 We can see that the distance between the two balls increases with time. D  D2 = g (2t  1) So ball number 1 gains ground on ball number 2! The same thing happens in the water stream. As two molecules accelerate under gravity, they must spread apart. This causes the water stream to thin out, since we just showed that the volume of water must decrease at any location in the stream. In the lack of gravity, this simple model predicts that the water stream will stay at a constant width. The simple model, as I have described it, whereby gravity causes the stream to accelerate as it falls. There are other forces to contend with. Surface tension causes the water stream to stay roughly circular as it falls. As the stream gets very thin however, surface tension will eventually cause the stream to break into separate drops. Lets keep on adding "terms" to this model. How about friction? There is friction of the water along the sides of the pipe. So the water does not exit the faucet at a perfectly uniform velocity in laminar flow. Water also will interact with the air as it falls. There will be turbulence in the outer layer of the water stream, so those molecules will probably fall just a bit slower. As the water exits the faucet, Vanderwaals forces will attract some of those molecules to the faucet on the edges of the stream. Again, this will impart some turbulence to the stream. Is there an aerator at the faucet tip? Overall though, my gut says that the two important factors to consider here are gravity and surface tension. The point is to start with the most obvious forces. Come up with a cogent argument for how they enter into a model. Then look to see if this model explains the behavior that you see. Look to see if other terms will be significant. Whenever possible, test your ideas.

Sawdust, thank you for your answer. I believe that you are right on! I was trying to find the answer so that I could explain it to my students. Thanks for the details and the math.

The water speeds up because it's falling by gravity. Since flow is constant, the faster the stream the thinner it gets to maintain constant flow (volume per second).
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