Poiseuille’s law

6 min read Last updated April 8, 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 < 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
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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