Poiseuille’s law

6 min read Last updated April 9, 2026

Why it matters in anaesthesia

  • Explains how small changes in cannula/airway radius cause large changes in flow and resistance
    • IV access: choosing a wider, shorter cannula markedly increases achievable flow for rapid transfusion/fluids
    • Airway: narrowing (secretions, bronchospasm, kinked tube) dramatically increases resistance and work of breathing
  • Links pressure generation to flow: for a given driving pressure, flow falls as viscosity rises or length increases
    • Blood viscosity rises with haematocrit and hypothermia → reduced flow for same pressure
  • Helps interpret when Poiseuille does NOT apply (turbulence, collapsible tubes, non-Newtonian fluids)
    • High flows through narrow devices (ETT, nebulisers) often turbulent → resistance increases more than Poiseuille predicts

Practical implications (high-yield)

  • To increase flow: increase radius, shorten length, increase pressure gradient, reduce viscosity
    • Doubling radius increases flow 16-fold (if laminar and other factors constant)
    • Halving length doubles flow
  • For rapid infusion, a pressure bag increases ΔP, warming fluids reduces viscosity (small effect for crystalloids, more relevant for blood)
    • Two cannulae in parallel reduce resistance (add flows), can outperform one larger line depending on sizes

Statement of Poiseuille’s law

  • For steady, incompressible, Newtonian fluid in laminar flow through a long, straight, rigid cylindrical tube: volumetric flow rate is proportional to pressure gradient and radius^4, and inversely proportional to viscosity and length
  • Flow equation: Q = (ΔP · π · r^4) / (8 · η · L)
  • Resistance form: R = ΔP / Q = (8 · η · L) / (π · r^4)

Definitions and units (exam-friendly)

  • Q: volumetric flow rate (m^3·s⁻1) (clinically often mL·min⁻1)
  • ΔP: pressure difference between ends (Pa) (1 mmHg ≈ 133 Pa)
  • r: internal radius (m), L: length (m)
  • η: dynamic viscosity (Pa·s), water at 20°C ≈ 1 mPa·s, blood ≈ 3–4 mPa·s (varies with Hct, temperature, shear rate)
  • R: resistance (Pa·s·m⁻3) (clinical analog: mmHg per (L·min⁻1))

Key proportionalities to memorise

  • Q ∝ ΔP
  • Q ∝ r^4 (dominant effect)
  • Q ∝ 1/η
  • Q ∝ 1/L

Assumptions/conditions for validity

  • Laminar flow (Reynolds number below critical, typically &lt, 2000 in straight tubes, but depends on geometry)
  • Newtonian fluid (constant viscosity independent of shear rate)
    • Blood is non-Newtonian at low shear rates (apparent viscosity changes), but approximates Newtonian in many large-vessel/high-shear contexts
  • Rigid, straight, long tube with constant circular cross-section, fully developed flow (entrance effects negligible)
  • Incompressible fluid (reasonable for liquids, gases may deviate especially with large pressure changes)

Velocity profile and shear (often examined)

  • Laminar flow in a tube has a parabolic velocity profile: maximum velocity at centre, zero at wall (no-slip condition)
  • Mean velocity v̄ = Q / (πr^2), vmax = 2·v̄ for Poiseuille flow
  • Wall shear stress τw relates to viscosity and velocity gradient, clinically relevant to endothelial shear and haemolysis in high-shear devices

Clinical applications and examples

  • IV cannulae: flow is extremely sensitive to internal radius, short, wide cannulae (e.g., 14G/16G) deliver much higher flow than long, narrow lines
    • Central venous catheters are longer and have smaller lumens than large-bore peripheral cannulae → often lower maximum flow despite central location
  • Airway resistance: small reductions in radius (bronchospasm, oedema, secretions, ETT narrowing) markedly increase resistance, work of breathing rises
    • In practice, airway flow may become turbulent at higher flow rates, making resistance increase more steeply than Poiseuille predicts
  • Regional anaesthesia: flow of local anaesthetic through fine spinal needles is low due to small radius and length, injection pressure rises
  • Transfusion: blood viscosity and temperature affect flow, warming blood reduces viscosity and improves flow for a given pressure gradient

Relationship to Reynolds number and turbulence

  • Re = (ρ · v · D) / η (ρ density, v mean velocity, D diameter). Higher Re increases likelihood of turbulence
  • Factors promoting turbulence: increased velocity/flow, increased diameter, increased density, decreased viscosity, abrupt changes in geometry (connectors, bends)
  • In turbulent flow, pressure drop is more proportional to Q^2 than Q (non-linear), Poiseuille no longer holds

Test yourself…

State Poiseuille’s law and give the equation for flow through a tube.

Key statement and formulae expected in a viva.

  • For laminar flow of an incompressible Newtonian fluid through a long rigid cylindrical tube: Q = (ΔPπr^4)/(8ηL)
  • Resistance form: R = ΔP/Q = (8ηL)/(πr^4)
What assumptions must be satisfied for Poiseuille’s law to apply?

Examiners usually want 4–6 clear assumptions.

  • Laminar, steady flow with fully developed velocity profile
  • Fluid is incompressible and Newtonian (constant viscosity)
  • Tube is long, straight, rigid, with constant circular radius, no significant entrance/exit effects
  • No-slip condition at the wall
A cannula’s internal radius is doubled with all else constant and flow remains laminar. By what factor does flow increase for the same ΔP?

This is a common calculation/MCQ.

  • Q ∝ r^4, so doubling r increases Q by 2^4 = 16 times
If the length of a tube is doubled (same radius, viscosity, and ΔP), what happens to flow?
  • Q ∝ 1/L, so doubling length halves flow
Derive or explain the resistance relationship from Poiseuille’s law.
  • Start with Q = (ΔPπr^4)/(8ηL). Rearrange: ΔP/Q = (8ηL)/(πr^4). Define R = ΔP/Q.
Describe the velocity profile in laminar flow through a tube and relate maximum to mean velocity.
  • Parabolic profile: velocity is zero at the wall (no-slip) and maximal at the centre
  • For Poiseuille flow: vmax = 2 × v̄ (mean velocity)
How does Poiseuille’s law help you choose between a large-bore peripheral cannula and a central venous catheter for rapid fluid resuscitation?

A frequent FRCA viva theme: apply physics to clinical choice.

  • Flow increases strongly with radius and decreases with length, many CVC lumens are narrower and much longer than short peripheral cannulae
  • Therefore, a short wide peripheral cannula (e.g., 14–16G) often provides higher maximum flow than a standard CVC lumen, for the same driving pressure
  • If higher ΔP is used (pressure bag/rapid infuser), flow increases proportionally provided flow remains laminar, in practice turbulence may limit gains in narrow lumens
Why can Poiseuille’s law overestimate flow through an endotracheal tube during high minute ventilation?
  • At high flows, Reynolds number increases and flow may become turbulent, especially with connectors, bends, and narrowing
  • In turbulence, pressure drop becomes non-linear (approximately proportional to Q^2), so much higher pressures are required than Poiseuille predicts
State the Reynolds number equation and list factors that increase Reynolds number.
  • Re = (ρ · v · D) / η
  • Re increases with: higher density (ρ), higher velocity/flow (v), larger diameter (D), lower viscosity (η)
  • Abrupt changes in geometry (bends, junctions, roughness) promote transition to turbulence even at lower Re
Previous FRCA-style written question: ‘Explain why a small reduction in airway radius causes a large increase in the work of breathing.’

Structure: define resistance, apply r^4, then clinical consequences and limitations.

  • For laminar flow, airway resistance R ∝ 1/r^4, so small decreases in radius markedly increase resistance
  • To maintain the same flow (minute ventilation), a larger pressure gradient is needed (ΔP = Q·R), increasing inspiratory effort/work of breathing
  • In acute obstruction/high flows, turbulence may occur, further increasing pressure requirements beyond Poiseuille predictions
Previous FRCA-style viva: ‘You are asked to set up rapid transfusion. Using Poiseuille’s law, what practical steps increase flow?’
  • Increase radius: use the widest available cannula/large-bore access, consider multiple lines in parallel
  • Decrease length: choose short peripheral cannulae, minimise extension tubing and unnecessary connectors
  • Increase ΔP: pressure bag/rapid infuser, raise fluid bag height (small effect compared with pressurisation)
  • Reduce viscosity: warm blood/fluids, recognise higher Hct and hypothermia increase viscosity
  • Avoid turbulence: smooth connectors, avoid sharp bends/kinks, recognise that at very high flows turbulence may be unavoidable

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