Velocity profile in a pipe: value at the centreline In steady, fully developed internal flow through a pipe, the velocity of the fluid particle at the centre of the circular section is:

Introduction / Context:
Understanding velocity distributions in pipe flow is foundational for computing discharge, shear stress, and head loss. Whether the regime is laminar or turbulent, the no-slip condition dictates zero velocity at the wall and a peak at the centreline for steady, fully developed flows in a round pipe.

Given Data / Assumptions:

• Flow is steady and fully developed.
• Pipe is circular and straight.
• No swirl; single-phase Newtonian fluid.

Concept / Approach:

No-slip condition enforces u = 0 at the wall. Momentum balance and symmetry about the pipe axis create a monotonic profile rising from the wall to the axis. In laminar flow the profile is parabolic with u_max = 2 * u_avg at the centreline. In turbulent flow the profile is fuller (flatter) but still peaks at the centreline due to symmetry and wall shear effects.

Step-by-Step Solution:

Verification / Alternative check:

For laminar Poiseuille flow: u(r) = u_max * (1 − (r/R)^2), clearly maximum at r = 0. Turbulent empirical profiles (e.g., 1/nth-power laws) are also maximal at the centreline.

Why Other Options Are Wrong:

Minimum occurs at the wall, not at the centre; uniform velocity (option c) would violate no-slip; zero at the centre contradicts both theory and experiments.

Common Pitfalls:

Confusing average velocity with centreline velocity; assuming plug flow in all cases.

Final Answer:

Maximum