Geometry of Waterfalls

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.