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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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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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Thickness is not enough of an explanation speaking as an old drilling fluid engineer. Is the fluid in turbulant or laminar flow? With out using a rheometer on the sample and knowing the characteristics of your thick fluid the question can not be answered. See link I haved used before that will explain this to you and enlighten you but not give you your answer. Thick fluid will not flow the same through out as you do realize. I do believe one of your previous answers to be incorrect. Not all of this thick water will fall at 32 ft per second sqaure due to the fluid characteristics.http://www.drillinghandbook.com/category/engineering-calculations

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