Laminar loss in a circular pipe: For a viscous liquid flowing at mean velocity v through a circular pipe of length L and diameter d, the head loss due to viscous effects is approximately what?

Introduction / Context:

In fully developed laminar flow inside a circular pipe, the Hagen–Poiseuille law governs the pressure drop and head loss. Recognizing the correct dependence on viscosity μ, length L, diameter d, and mean velocity v is important for microfluidics, lubrication problems, and low-Reynolds-number flows.

Given Data / Assumptions:

• Steady, incompressible, Newtonian fluid.
• Fully developed laminar flow (Re < 2100 typically).
• Gravity head g included to convert pressure loss to head loss.

Concept / Approach:

Start from Darcy–Weisbach form h_f = f (L/d) * v^2/(2g) with laminar friction factor f = 64/Re, where Re = ρ v d / μ. Substituting f and simplifying yields the laminar head-loss expression directly proportional to μ, v, and L, and inversely proportional to d^2.

Step-by-Step Solution:

Verification / Alternative check:

Using the Hagen–Poiseuille equation Δp = 32 μ v L / d^2 also leads to the same head loss since h_f = Δp/(ρ g).

Why Other Options Are Wrong:

• Option A is the general Darcy–Weisbach form but not a closed form for laminar flow unless f is replaced appropriately.
• Option C has incorrect diameter dependence.
• Option D is dimensionally inconsistent.

Common Pitfalls:

• Confusing mean velocity v with centerline velocity (which is 2v in laminar flow).

Final Answer:

h_f = 32 μ v L / (ρ g d^2)