Poiseuille Flow and Hydraulic Resistance
Hagen-Poiseuille flow is steady, fully developed, incompressible Newtonian laminar flow through a long straight circular tube driven by a pressure difference.
Why this shows up in the exam
Flow through capillaries and needles · Comparing pipes in series or parallel · Viscosity measurement from pressure-driven flow
Learn the idea
Laminar flow through a long circular tube is extremely sensitive to radius because resistance varies as the inverse fourth power. No-slip at a tube wall produces a parabolic speed profile. Widening a tube helps every layer and creates much more conducting area, so even a small radius change strongly changes flow.
🧠 Memory hook: For laminar tubes, radius rules to the fourth power.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- Q = pi r⁴ Delta P/(8 eta L) — Poiseuille volume rate under fully developed laminar tube-flow assumptions
- R_h = 8 eta L/(pi r⁴) — hydraulic resistance defined by Delta P = R_h Q
- R_series = sum(R_i) — series resistance for the same steady volume rate through consecutive tube segments
How to approach it
- 1Verify laminar circular-tube conditions
- 2Replace each segment by a hydraulic resistance
- 3Use common flow in series and common pressure drop in parallel
Common slip-ups that cost marks
- •Using diameter to the fourth power without converting constants
- •Applying Poiseuille's law to turbulent or entrance-region flow
- •Adding flow rates in series instead of pressure drops
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A wire 1 m long and cross-sectional area 2 mm^2 extends by 1 mm under a 200 N load. Find Young modulus.
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