Fluid velocity profile shape calculation

[chibifibi]

Hello,

I have two (2) questions:

I understand that for a fully-developed velocity profile for a fluid, the shape it takes resembles a parabola. However, it does not start out this way before entering the pipe/launder/duct/etc. A visual of this transition is provided below:

Reference: https://en.wikipedia.org/wiki/Entrance_length_(fluid_dynamics)

Q1) Is there an equation, or set of equations, that help determine the shape of the velocity profile as it changes shape as it travels down the pipe/launder/duct/etc.?

Q2) Is there a different set of equations to then take the final flow profile from this pipe/launder/duct/etc. and apply to it the wall-shear effects of a much larger pipe/launder/duct/etc.? Meaning, the flow suddenly enters a much larger space after being developed in a smaller space. Would I be able to determine the shape of the velocity profile as it travels and transforms in this new space? (I am assuming that both the larger and the smaller pipes/launders/ducts/etc. are axisymmetric.)

Please let me know if anything needs clarification.

Thank you.

 

[SuGroB]

https://calcdevice.com/fluid dynamic/242.html
Here you can calculate the steady-state flow velocity profile in the pipe. Equations are also available.
Computation is carried out for any point given by the distance from the inner surface of the pipe.
And a few more fluid dynamics problems are presented on this site.