Aliquid flowing from a vertical pipe has a very definite shape as it flows from the pipe. to get the equation for this shape, assume that the liquid is in free fall once it leaves the pipe. just as it leaves the pipe, the liquid has speed v0 and the radius of the stream of liquid is r0.
find an equation for the speed of the liquid as a function of the distance y it has fallen.
combining this with the equation of continuity, find an expression for the radius of the stream as a function of y.

Calculating speed as function of distance y is fairly easy. Once water leaves pipe it is under free fall which means that it is accelerating with gravitational acceleration "a".
V=Vo + Vff   where Vff is speed gained due to free fall.



Calculating radius of stream is a little bit more complicated. Because water is accelerating as it falls it has to lower its radius. The reason for this is that water flow must be the same as one at the exit of pipe. Water flow is expressed in  liters/meter^3
Because of this we write condition:
V1/V2 = A2/A1 where  A2 and A1 are sections of water stream.
let V1 and A1 be the speed and section of water stream at the end of pipe.
A2 = V1*A1/V2

Wish i knew this to !