Laminar vs Turbulent Flow: EDAIC Physics Examples

Poiseuille's relationship predicts flow through a specified laminar-flow model. Reynolds number helps describe the tendency towards turbulence. They solve different problems: one relates flow to pressure and tube properties; the other compares inertial and viscous effects.

The original calculations here are educational physics models, not advice for selecting patient equipment or altering a clinical infusion.

State the model before using the fourth power

For steady, fully developed laminar flow of an incompressible Newtonian fluid through a rigid, straight, circular tube:

Q = πΔPr⁴/(8ηL).

Q is volume flow, ΔP the pressure difference along the tube, r internal radius, η dynamic viscosity and L length. Rearranging gives resistance R = ΔP/Q = 8ηL/(πr⁴). OpenStax derives the relationship and its assumptions.

The radius exponent is powerful, but it is conditional. A branching airway, changing flow, a flexible tube or non-Newtonian behaviour can limit the simple prediction. “Radius matters strongly” is sound reasoning; “every clinical flow change follows the fourth power exactly” is not.

Worked example: compare two ideal tubes

Tube A has internal radius 1.0 mm and length 100 mm. Under a specified pressure difference, a model fluid flows at 20 mL/min. Tube B has radius 1.5 mm and the same length. Assume identical fluid viscosity, pressure difference and laminar conditions.

The flow ratio is:

Q_B/Q_A = (r_B/r_A)⁴ = 1.5⁴ = 5.0625.

Therefore Q_B = 20 × 5.0625 = 101.25 mL/min.

You did not need to know viscosity numerically because it cancelled in the ratio. That cancellation is legitimate only because the exercise explicitly kept it constant.

Now make B twice as long, at 200 mm. The new ratio is 1.5⁴ × 100/200 = 2.53125, giving 50.63 mL/min. A larger radius and a greater length push flow in opposite directions; the final result combines them.

As a separate perturbation, double viscosity while leaving tube A unchanged. Predicted flow halves to 10 mL/min at the same pressure difference. If instead flow were held at 20 mL/min, twice the pressure difference would be required by this model.

Keep radius and diameter consistent

Doubling diameter also doubles radius, so the ratio calculation can use either consistently. Trouble arises when one measurement is a radius and the other a diameter. A 2 mm diameter tube has radius 1 mm; comparing its diameter directly with another tube's 1 mm radius creates a fictional twofold size difference and sixteenfold flow difference.

Also distinguish a device's external label from its internal flow dimension. Real connectors, extensions and changes of bore may contribute additional resistance. OpenAnesthesia discusses the clinical context of the Poiseuille model. The ideal tube example does not establish a manufacturer's actual flow rating.

Reynolds number asks another question

For a circular tube, one common form is:

Re = ρvD/η.

Here ρ is density, v mean speed, D internal diameter and η dynamic viscosity. Re is dimensionless. Higher values indicate a greater tendency towards turbulence, but transition depends on geometry and disturbances; a textbook threshold is not a universal boundary for every clinical system. OpenStax explains the Reynolds comparison.

For a fictional flow with ρ = 1,000 kg/m³, v = 0.05 m/s, D = 0.004 m and η = 0.001 Pa·s:

Re = 1,000 × 0.05 × 0.004 ÷ 0.001 = 200.

Keeping everything else fixed and increasing speed tenfold gives 2,000. The arithmetic identifies a major change in the balance of effects; it does not prove that an arbitrary device suddenly changes behaviour at exactly that value.

The subtle “all else equal” trap

Re is proportional to diameter when mean speed is held constant. But at a fixed volume flow Q, mean speed equals Q divided by cross-sectional area, so v varies inversely with D². Substituting that into Re makes Re proportional to 1/D when Q, density and viscosity are fixed.

Thus the statement “a wider tube always increases Reynolds number” is incomplete. At fixed speed, it rises; at fixed volume flow, it falls. Write the controlled variable beside every proportionality. OpenAnesthesia's flowmeter discussion connects flow regime with gas properties.

Original self-test

1. Doubling radius predicts sixteenfold flow if the Poiseuille assumptions and other inputs remain unchanged. True.
2. Doubling length doubles flow at fixed pressure difference. False: it halves flow.
3. Re has units of litres per minute. False: its units cancel.
4. Higher viscosity raises Re when the other inputs remain fixed. False: viscosity is in the denominator.
5. A diameter change has the same effect on Re whether speed or volume flow is fixed. False: those constraints produce different speed changes.

Use the physics and clinical measurement guide to connect these models with the broader syllabus. In the EDAIC question bank, make a habit of underlining “constant pressure”, “constant flow” or “constant speed” before judging a proportionality statement.

Sources

• OpenStax: Viscosity and Laminar Flow; Poiseuille's Law.
• OpenStax: The Onset of Turbulence.
• OpenAnesthesia: Venous Access and Poiseuille Law.
• OpenAnesthesia: Flowmeter, Gas Properties.

Prepared with AI assistance for examination revision; this article is not a patient-care protocol or a substitute for current primary guidance. Sources checked 7 October 2026.