Velocity profile in pipe flow:\nWhen a liquid flows through a circular pipe, how does the local velocity vary from the wall to the centreline (typical laminar or turbulent cases)?

Introduction / Context:
The no-slip condition at a solid boundary requires fluid velocity relative to the wall to vanish at the wall. Away from the wall, velocity increases toward the pipe centreline, reaching a maximum. This holds for both laminar (parabolic profile) and turbulent (flatter, but still with zero at the wall) flows, making the general qualitative statement robust.

Given Data / Assumptions:

• Circular pipe, steady internal flow.
• No-slip boundary condition applies.
• Incompressible fluid; fully developed region considered.

Concept / Approach:
For laminar flow, the exact solution of the Navier–Stokes equations yields a parabola: u(r) = umax * (1 − (r/R)^2), where u = 0 at r = R (wall) and u = umax at r = 0 (centre). For turbulent flow, the mean profile is blunter (log-law in the inner/outer layers), yet still satisfies u = 0 at the wall and u = umax at the centreline. Therefore, the most precise of the given choices is “maximum at the centre and zero near the walls.”

Step-by-Step Solution:

Verification / Alternative check:
Pitot traverses show stagnation reading rising from wall toward centre; in laminar theory, umax = 2 * v_mean; in turbulent, umax ≈ 1.2–1.3 * v_mean depending on Reynolds number, corroborating the qualitative pattern.

Why Other Options Are Wrong:

• Maximum near walls: Violates no-slip.
• Zero at centre: Physically impossible for pressure-driven pipe flow.
• “Minimum near walls” is true but imprecise; the exact boundary value is zero, making option D superior.

Common Pitfalls:
Assuming plug flow for turbulent cases; ignoring viscous sublayer presence where velocity ramps from 0 at the wall.

Final Answer:
maximum at the centre and zero near the walls