Internal laminar flow in a circular pipe: What is the ratio of maximum velocity to average velocity for a fully developed viscous (laminar) flow?
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A0.5
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B0.75
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C1.25
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D2.00
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E1.50
Answer
Correct Answer: 2.00
Explanation
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:
Velocity distribution: v(r) = vmax * (1 − (r/R)^2).Compute Q = ∫_A v dA = 2π ∫_0^R v(r) r dr = 2π vmax ∫_0^R (1 − (r/R)^2) r dr.Evaluate integral → Q = (π R^2) * (vmax / 2).Thus v_avg = Q / (π R^2) = vmax / 2 ⇒ vmax / v_avg = 2.00.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