Internal laminar flow in a circular pipe: What is the ratio of maximum velocity to average velocity for a fully developed viscous (laminar) flow?

Introduction / Context:

In laminar flow through a circular pipe (Hagen–Poiseuille regime), the velocity profile is parabolic: v(r) = vmax * (1 − (r/R)^2). Recognizing the relationship between maximum and mean velocities is important for flow rate calculations and Reynolds-number-based regime identification.

Given Data / Assumptions:

• Incompressible Newtonian fluid.
• Steady, fully developed, laminar flow in a circular pipe of radius R.
• No slip at the wall.

Concept / Approach:

The volumetric flow rate Q is the area integral of the velocity profile across the section. For the parabolic profile, the section-average velocity v_avg = Q / A equals half the centerline (maximum) velocity.

Step-by-Step Solution:

Verification / Alternative check:

Dimensionless profile checks and standard textbooks confirm vmax = 2 v_avg for laminar pipe flow; in turbulent flow the ratio is smaller (profile is flatter).

Why Other Options Are Wrong:

• 0.5 and 0.75 invert or underestimate the laminar peaking.
• 1.25 is typical of mildly turbulent flattened profiles, not laminar.

Common Pitfalls:

• Confusing laminar and turbulent profiles; for turbulent flow, vmax/v_avg is typically ~1.1–1.3 depending on Reynolds number and roughness.

Final Answer:

2.00