Bernoulli principle

Where you meet it in anaesthesia

Variable orifice devices: flowmeters (rotameters), Venturi devices, air-entrainment masks, nebulisers
- Pressure drop across a constriction can be used to entrain air/oxygen and generate a predictable FiO2 (Venturi mask) or to drive entrainment (jet nebuliser).
Suction and aspiration phenomena: airway suction catheters, surgical suction, scavenging interfaces
- High-velocity gas stream can create sub-atmospheric side-port pressure (Venturi effect) and entrain surrounding gas/fluids.
Airway/upper airway physiology: dynamic airway collapse, wheeze, stridor
- In narrowed segments, velocity rises and lateral (static) pressure falls, potentially promoting airway narrowing (conceptual link; real airways are compliant and flow may be turbulent).
Cardiovascular: murmurs and stenotic jets (aortic stenosis), pressure recovery downstream of stenosis
- High velocity through a stenosis corresponds to lower static pressure at the throat; some pressure is recovered downstream as velocity falls (not all—losses due to turbulence/viscosity).
Measurement: Pitot-static tube (concept), Doppler echocardiography (velocity → pressure gradient)
- Modified Bernoulli in echo: ΔP ≈ 4v² (mmHg) across a stenosis (assumes negligible viscous losses and proximal velocity).

How it changes clinical decisions

Venturi masks: choose device by required FiO2 and total flow; ensure patient inspiratory flow does not exceed device total flow (otherwise entrains extra room air and FiO2 falls).
- Higher total flow devices are more stable for tachypnoeic patients.
Jet ventilation/entrainment: understand that high driving pressure can entrain additional gas and increase delivered volume/pressure unpredictably if outflow is impeded.
- Obstruction to egress (e.g., laryngeal surgery) risks barotrauma because entrained flow + driving flow cannot escape.

Statement of Bernoulli principle

For steady, incompressible, non-viscous flow along a streamline, the total mechanical energy per unit volume is constant.
Bernoulli equation: P + ½ρv² + ρgh = constant (along a streamline).
- P = static pressure; ½ρv² = dynamic pressure; ρgh = hydrostatic term.

Derivation outline (what you should be able to explain in a viva)

Start from conservation of energy for a fluid element moving from point 1 to point 2: work done by pressure forces equals change in kinetic + potential energy.
Assumptions: no energy added/removed (no pump/turbine), no viscous dissipation, constant density, steady flow, along a streamline.
If height change negligible (same level), then P + ½ρv² = constant → higher velocity implies lower static pressure.

Link to continuity equation

Continuity (incompressible): Q = Av = constant along a tube (steady flow).
If area decreases, velocity increases; via Bernoulli, static pressure decreases at the constriction (ideal flow).

Venturi effect vs Bernoulli

Venturi effect: practical consequence of Bernoulli + continuity in a constricted section causing a pressure drop that can be used for entrainment or measurement.
Venturi meter: uses pressure difference between wide and narrow sections to infer flow rate (with discharge coefficient to account for losses).

Real fluids: why Bernoulli often overestimates pressure recovery

Viscosity causes energy loss (dissipation as heat); turbulence increases losses; flow separation downstream of a stenosis reduces pressure recovery.
Extended Bernoulli: P1 + ½ρv1² + ρgh1 = P2 + ½ρv2² + ρgh2 + losses ± pump work.
Compressibility: for gases at high velocities/large pressure drops, density changes and simple Bernoulli becomes inaccurate.

Anaesthetic device examples

Venturi mask: oxygen passes through a jet/orifice → high velocity → low static pressure at side ports → entrains air; mixing gives fixed FiO2 if total flow exceeds patient demand.
- Air:O2 entrainment ratio determines FiO2 (e.g., 1:1 ≈ 60%, 3:1 ≈ 40%, 9:1 ≈ 28% approximate).
Jet nebuliser: driving gas through a jet creates low pressure that lifts liquid and shears it into aerosol; baffles remove large droplets.
Rotameter (variable area flowmeter): float rises until drag + buoyancy balance weight; flow relates to float height; not a direct Bernoulli device but relies on pressure drop/velocity changes around the float.
- Calibrated for specific gas (density/viscosity effects).

Key equations and quick manipulations

If h1 = h2: P1 − P2 = ½ρ(v2² − v1²).
Using continuity v2 = (A1/A2) v1 (incompressible). Substitute into Bernoulli to relate ΔP to area ratio.
Dynamic pressure q = ½ρv² (units Pa).

State Bernoulli’s equation and define each term.

Give the equation, then interpret it as conservation of energy per unit volume along a streamline.

P + ½ρv² + ρgh = constant (along a streamline for steady, incompressible, non-viscous flow).
P = static pressure; ½ρv² = kinetic energy per unit volume (dynamic pressure); ρgh = potential energy per unit volume.

What assumptions are required for Bernoulli’s equation to apply, and which are most often violated in clinical devices?

Examiners often want a list plus practical consequences.

Assumptions: steady flow, incompressible fluid, inviscid (no frictional losses), along a streamline, no energy added/removed (no pumps/turbines), uniform gravitational field.
Common violations: viscosity and turbulence (energy loss), compressibility for gases at high velocities/large pressure drops, non-steady flow (respiration), flow not confined to a single streamline (mixing).

Explain how a Venturi mask delivers a fixed FiO2 using Bernoulli’s principle.

A frequent FRCA viva topic: relate jet velocity, pressure drop, entrainment, and total flow.

Oxygen passes through a small jet/orifice → velocity increases → static pressure at side ports falls (Bernoulli).
The low pressure entrains room air through side ports; the entrainment ratio (air:O2) sets the approximate FiO2 after mixing.
FiO2 remains predictable only if the device total flow exceeds patient peak inspiratory flow; otherwise additional room air is entrained at the mask and FiO2 falls.

A common written question: ‘As fluid velocity increases, pressure decreases.’ Is this always true? Discuss.

The mark-scoring points are ‘static vs total pressure’ and ‘losses’.

In ideal Bernoulli flow at constant height, an increase in velocity is associated with a decrease in static pressure along a streamline (P + ½ρv² = constant).
Total pressure (stagnation pressure) is conserved only if there are no losses; in real flows, viscosity/turbulence reduce total pressure downstream.
If an external pump adds energy, both velocity and static pressure can increase; if height changes, hydrostatic term matters.

Differentiate static pressure, dynamic pressure, and stagnation pressure. How would you measure them?

Often examined with Pitot-static concept.

Static pressure: thermodynamic pressure of the fluid; measured via a side port flush with the wall aligned parallel to flow.
Dynamic pressure: q = ½ρv²; represents kinetic energy per unit volume.
Stagnation pressure: pressure when flow is brought to rest isentropically at a point; P0 = P + ½ρv² (if same height, no losses). Measured with a Pitot tube facing the flow.

A Venturi meter has cross-sectional areas A1 and A2 (A2 < A1). Derive an expression linking flow to the measured pressure difference (outline only).

You can score well with continuity + Bernoulli + mention of discharge coefficient.

Continuity: Q = A1v1 = A2v2 → v2 = (A1/A2)v1.
Bernoulli (same height): P1 + ½ρv1² = P2 + ½ρv2² → ΔP = ½ρ(v2² − v1²).
Substitute v1 = Q/A1 and v2 = Q/A2 to obtain Q = A2 * sqrt( (2ΔP/ρ) / (1 − (A2/A1)²) ). In practice multiply by discharge coefficient Cd (<1).

Previous FRCA-style theme: Why might a Venturi device fail to deliver the expected FiO2 in a tachypnoeic patient?

This is about total flow and entrainment limits.

If patient inspiratory flow exceeds device total flow, additional room air is entrained around the mask, reducing FiO2 below the nominal value.
Incorrect assembly/blocked entrainment ports reduces air entrainment and can increase FiO2 (and reduce total flow).
Back-pressure (e.g., long tubing, kinks) can alter jet performance and entrainment.

Explain ‘pressure recovery’ after a stenosis and why catheter-measured gradients can differ depending on where you measure.

A classic application in cardiology/echo and fluid dynamics.

At the stenosis throat: velocity is high and static pressure is low (conversion of pressure energy to kinetic energy).
Downstream: as area increases, velocity falls and some kinetic energy converts back to static pressure (pressure recovery).
Not all pressure is recovered due to viscous and turbulent losses; measured pressure gradient depends on sampling location relative to the vena contracta and recovery region.

Previous FRCA-style theme: Discuss limitations of applying Bernoulli to gas flow in anaesthetic circuits.

Mention compressibility and turbulence; relate to clinical relevance.

Gases are compressible; if pressure changes are large, density changes and incompressible Bernoulli is inaccurate.
Circuit flows can be turbulent (high Reynolds number, sharp bends, connectors), causing energy losses not captured by ideal Bernoulli.
Unsteady flow (respiratory cycle, ventilator waveforms) violates steady-flow assumption; use time-averaged or more complex models.

Echo uses ΔP ≈ 4v² (mmHg). Where does this come from, and what are the key assumptions?

This is the modified Bernoulli equation used in Doppler echocardiography.

From Bernoulli (same height): ΔP = ½ρ(v2² − v1²). If proximal velocity v1 is small compared with jet velocity v2, then ΔP ≈ ½ρv².
Using blood density ρ ≈ 1060 kg·m⁻³ and converting Pa to mmHg gives ΔP(mmHg) ≈ 4v² where v is in m·s⁻¹.
Assumes negligible viscous losses and that measured Doppler velocity represents peak jet velocity at the vena contracta.