Laminar pipe flow (Hagen–Poiseuille): Select the correct head loss due to viscosity for laminar flow in a circular pipe (d = diameter, l = length, v = mean velocity, μ = dynamic viscosity, w = specific weight).

Introduction / Context:
In the laminar regime (Re < 2000 for pipes), head loss varies linearly with flow rate and viscosity. The Hagen–Poiseuille relation can be expressed in head form using specific weight, yielding a simple proportionality to length and velocity, and inversely to the square of diameter.

Given Data / Assumptions:

• Steady, incompressible, fully developed laminar flow in a circular pipe.
• No slip at the wall; constant μ.
• v is mean (average) velocity.

Concept / Approach:
Darcy–Weisbach with laminar friction factor f = 64/Re leads to h_f = f * (l/d) * (v^2 / (2g)). Using Re = ρ * v * d / μ and w = ρ * g, rearrangement yields h_f = 32 * μ * v * l / (w * d^2). This is equivalent to the classical Hagen–Poiseuille pressure-drop formula divided by w to convert to head.

Step-by-Step Solution:

Verification / Alternative check:
For water at 20°C (μ ≈ 1.0e−3 Pa·s), small d and long l give appreciable head loss, matching practical experience in microchannels and viscous liquids.

Why Other Options Are Wrong:

• 16*…/ (wd): Wrong constants and diameter power.
• 4…/(wd): Not derived from laminar friction factor; incorrect scaling.
• 64…/(w*d^2): Double the correct constant.

Common Pitfalls:
Using turbulent formulas in laminar range; forgetting v is mean velocity; mixing pressure drop and head loss without dividing by w.

Final Answer:
h_f = (32 * μ * v * l) / (w * d^2)