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.