For fully developed laminar flow of a Newtonian fluid in a smooth circular pipe, what is the ratio of the cross-sectional average velocity to the centerline (maximum) velocity?

Introduction:
Velocity profiles in internal laminar flow are parabolic. The relationship between average velocity (useful for flow rate calculations) and maximum velocity (at the centerline) is a fixed ratio that underpins Reynolds number estimation, residence time calculations, and pressure-drop correlations for Newtonian fluids.

Given Data / Assumptions:

• Steady, incompressible, fully developed laminar flow.
• Newtonian fluid in a smooth, circular pipe.
• No slip at the wall boundary.

Concept / Approach:
The laminar solution of the Navier–Stokes equations in a round tube gives a parabolic profile: u(r) = umax * (1 - (r/R)^2). Integrating over the cross-section yields an average velocity u_avg = umax / 2. Hence, the ratio u_avg / umax = 0.5 is universal for this ideal case.

Step-by-Step Solution:

Verification / Alternative check:
Hagen–Poiseuille law and classic textbook derivations confirm the parabolic profile and the 0.5 ratio, independent of fluid properties so long as the flow remains laminar and Newtonian.

Why Other Options Are Wrong:

• 0.75, 0.87, 0.37: Do not result from the parabolic laminar solution; values greater or less than 0.5 would imply non-Newtonian behavior or non-laminar conditions.

Common Pitfalls:
Applying the 0.5 ratio to turbulent or non-Newtonian flows; those require different profiles and factors.

Final Answer:
0.5