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Let me create a very artificial experimental set up. Take a bathtub the size of Delaware and suspend it a mile above the ground. Fill it with water (though I'm not sure to what depth - and it might matter). Now pull the plug out.

The effect is that you have a waterfall with an endless supply of water, high above the ground, and, in my idealization, the stream of water is an arbitrarily thick, cylinder of water.

The question is, how fast does the water travel at the center of the waterfall? Obviously, I'm not looking for a numerical answer here, but I just want to understand the limits. Water at the edges of the waterfall are subject to wind resistance and will therefore quickly reach terminal velocity. But water at the center of the stream is removed from the effects of air resistence, so if the stream was thick enough and the waterfall high enough, would the center of the waterfall be unboundedly fast?

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If you just want to get rid of air resistace, make the water fall in vacuum. The water on the outside surface is affected by air resistance and wind velocity. If everything is idealized the water should behave as if it's in a vacuum. I don't know how cohesion would affect the fall in this situation –  raindrop Feb 16 '13 at 17:11
    
@Raindrop By your name you should be an authority on this question! Thanks for your input. –  John Berryman Feb 16 '13 at 17:44
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@Raindrop, liquid water can't exist in vacuum. Some of the water will evaporate, some will freeze, no water will remain liquid. Details here: askville.amazon.com/water-evaporate-vacuum/… –  Soonts Feb 17 '13 at 5:15
    
Soonts is right –  raindrop Feb 17 '13 at 5:26
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It wont be unboundedly fast e.g. due to the viscosity of the water.

In addition, I'd like to point out that you won't have the cylinder of water. Water will accelerate with gravity as it falls. The density of water is constant. The total flow of the water at different height is constant as well. So the area of the water's cross-section will decline.

For thin stream of water from a kitchen tap, surface tension force is enough to keep the water flowing in one stream, but the thickness of the scream decreases, take a look here.

However, as you increase the initial radius of your stream, the water mass (~R^2) grows while the pressure gradient created by surface tension forces (~1/R) declines. That's why the cylinder will pretty quickly separates into multiple streams, and eventually into droplets..

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*the total amount of volume passing through a certain height is constant. The water spreads out wallpaperpimper.com/wallpaper/Landscape/Lake_and_Waterfall/…. btw the last part of your answer reminds me of rain... –  raindrop Feb 17 '13 at 5:28
    
Yep. That makes sense. Thanks! –  John Berryman Feb 17 '13 at 14:38
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