Laminar Pipe Flow Is a Parabola, Fastest at the Center
The water in a pipe does not move at one speed everywhere. Drag a probe across the cross-section to find out what shape the speed makes when you measure it from wall to wall.
Water at the wall is held to zero by no-slip, and gets faster toward the center. Draw the speeds as arrows and they form a parabola, bulging most at the middle. Drag the driving push (the pressure difference) and the whole parabola grows. Viscous friction at the wall holds the edges back, so only the center is free to race ahead.
Laminar and turbulent flow differ in the very shape of the profile. Toggle between them. Laminar is a smooth parabola, but turbulent mixing flattens the middle into a blunt plug and drops off steeply only near the wall. The vigorous mixing of turbulence smears the speed out evenly.
The laminar parabola has one tidy fact: the maximum speed at the center is exactly twice the average speed over the section. Drag the driving push and this 2-to-1 ratio never changes. So when you compute flow rate from the mean speed, you can just halve the peak value measured at the center.
As an equation, u(r) = umax (1 − (rR)²). A parabola that is maximum at the center (r=0) and zero at the wall (r=R). Drag the radial position r and the speed follows the formula exactly, falling off quadratically toward the edge.
Blood in a vessel flows the same way. The layer against the vessel wall barely moves, while the red cells down the center go fastest. Drag the flow and the center cells' speed grows the most. When a vessel narrows the central velocity shoots up, while the slow layer near the wall is where nutrient exchange and clotting happen. In the next lesson we follow how much pressure this viscous friction eats along the pipe, as head loss.