Fluid velocity profile shape calculation

[dmayorga]

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:

Development_of_fluid_flow_in_the_entrance_region_of_a_pipe.jpg

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 fully-developed 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 much larger space after being fully 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?

Please let me know if anything needs clarification.

Thank you.