Waterfall Calculations

A challenge, for our budding young engineers

Given are two problems on free falling liquid, the first being straightforward. The second I  found not so easy. Let us for terminology take water as our fluid.

We consider a waterfall, a free flowing falling stream as a (balanced) steady-state system i.e. unchanging with time. Reasonable assumptions may be made, such as uniform density and that cross-sections are circular. Our model is simplified and we will need only elementary (high-school level) mathematics and physics.
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The first question is, is there a temperature difference between the top and the bottom of a waterfall? Why? Can you calculate it?
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For the second, would the diameter d increase, or decrease with increasing h (distance as measured from above) like a “funnel”? Why?

Our problem is to find the diameter of the stream at a given h. It is given by 

   d = 2 SqRt ( pi.A ) where A = V / [ SqRt (2gh) ].

A is the area of the cross section at h, V the volume of water passing through this section in one second. SqRt here stands for the square root.

As a hint the following: Use conservation of energy, conservation of mass, and an equation that relates to the geometry.
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In fact if you open the tap just a little you can already see the effect. It would actually take very simple experiments to test our findings.

All remarks are welcome as contributions, as well as those that confirm or disprove my own solutions. If you want we could always have a discussion on the blogs.