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Poiseuille's Law
Volumetric flow rate of a viscous fluid (e.g., blood) through a cylindrical vessel.
Interpretation
Poiseuille’s law Q = πΔPr⁴/(8ηL) describes laminar flow in a tube. Flow is proportional to r⁴, so small radius changes dramatically affect flow. It explains vascular resistance, catheter design, and airflow in airways. A cornerstone of haemodynamics and respiratory physiology.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| Q | Flow rate | m³/s |
| ΔP | Pressure difference | Pa |
| r | Vessel radius | m |
| η | Viscosity | Pa·s |
| L | Vessel length | m |
What it means
Poiseuille’s law is a cornerstone of haemodynamics, describing the volumetric flow rate of a Newtonian fluid through a rigid cylindrical tube under steady, laminar flow conditions. The equation Q = π ΔP r⁴ / (8 η L) reveals that flow is directly proportional to the fourth power of the radius, making it exquisitely sensitive to changes in vessel calibre. This principle explains why minor arterial stenosis dramatically reduces blood flow and why vasodilators are so effective in increasing perfusion. Clinically, Poiseuille’s law underpins the understanding of vascular resistance, blood pressure regulation, and the pathophysiology of diseases like atherosclerosis, where plaque accumulation narrows lumens. It also guides the design of intravenous catheters and haemodialysis access, where flow optimisation is critical. In respiratory physiology, the same law applies to airflow in bronchi, influencing the work of breathing in asthma and COPD. However, real blood is non-Newtonian (shear-thinning), and vessels are elastic and branching, so the law is an idealisation. Nevertheless, it provides a robust conceptual framework for predicting the effects of radius changes, pressure gradients, and viscosity on flow. Clinicians use this understanding to interpret haemodynamic measurements, titrate vasoactive drugs, and plan revascularisation procedures. The law’s simplicity and clinical utility make it an essential teaching tool in medical and physiological education.
Worked example
Poiseuille's Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| ΔP | 1000 Pa |
| r | 0.01 m |
| η | 0.001 Pa·s |
| L | 1.0 m |
| Parameter | Value |
|---|---|
| r | 0.015 m |
Common mistakes
- Radius r: Use the fourth power of the radius – small changes in radius have a huge effect on flow. Doubling radius increases flow 16‑fold.
- Pressure drop ΔP: The pressure difference driving the flow – not the absolute pressure.
- Viscosity η: For blood, viscosity depends on shear rate (non‑Newtonian) and haematocrit. Use the appropriate value for the conditions.
- Length L: The length of the tube/vessel – in metres.
- Units: Q in m³/s, ΔP in Pa, r in m, η in Pa·s, L in m – all SI. Convert cm, mmHg, etc., as needed.
- Assumptions: Laminar, steady, incompressible, Newtonian flow in a rigid, straight, circular pipe. Many biological flows are pulsatile and non‑Newtonian.
Applications
Poiseuille's law describes the volumetric flow rate (Q) of an incompressible, Newtonian fluid through a cylindrical pipe, showing that Q is proportional to the pressure drop (ΔP) and the fourth power of the radius (r), inversely proportional to viscosity (η) and length (L). This law is fundamental to understanding blood flow in the cardiovascular system, as it explains how small changes in vessel radius dramatically affect flow. Clinicians use it to understand the haemodynamic consequences of vasoconstriction, stenosis, or aneurysm formation. In medical device design, Poiseuille's law guides the sizing of catheters, cannulas, and intravenous tubing to achieve desired flow rates. It also underpins the principles of fluid administration and the design of infusion pumps and dialysis systems.
- Haemodynamic analysis of vascular stenosis and aneurysms
- Design and sizing of catheters, cannulas, and IV tubing
- Calculation of resistance to flow in dialysis and extracorporeal circuits
- Understanding the effects of vasodilation and vasoconstriction on blood pressure
- Optimisation of fluid delivery in infusion therapy and parenteral nutrition
Frequently Asked Questions
Poiseuille's law describes the volumetric flow rate (Q) of a viscous, incompressible fluid through a cylindrical pipe under laminar flow conditions: Q = (π·ΔP·r⁴) / (8·η·L), where ΔP is the pressure difference, r is the tube radius, η is dynamic viscosity, and L is the tube length. It is fundamental to understanding blood flow in the circulatory system.
- Steady, laminar flow (low Reynolds number).
- Newtonian fluid (constant viscosity).
- Incompressible fluid.
- Rigid, straight, circular tube of constant radius.
- No slip at the wall.
- Fully developed flow (entry effects neglected).
The flow rate is proportional to r⁴, meaning that small changes in radius have a dramatic effect on flow. For example, doubling the radius increases flow 16‑fold. This makes vasodilation and vasoconstriction powerful mechanisms for regulating blood flow in the body.
Flow rate is inversely proportional to viscosity (Q ∝ 1/η). Increased blood viscosity (e.g., in polycythaemia or hyperviscosity syndromes) reduces flow for the same pressure gradient, increasing cardiac workload. Conversely, conditions like anaemia (low haematocrit) reduce viscosity and increase flow.
The flow resistance is defined as R = ΔP / Q. From Poiseuille's law, R = 8ηL / (πr⁴). This is analogous to electrical resistance (Ohm's law: V = IR). The resistance depends strongly on radius, making it the primary determinant of vascular resistance.
It describes flow in large arteries (where blood is approximately Newtonian and flow is mostly laminar). However, in small arterioles and capillaries, blood behaves as a non‑Newtonian fluid (due to red blood cell aggregation), and vessel walls are elastic, so the law must be modified. Still, it provides a useful framework for understanding pressure‑flow relationships.
A stenosis reduces the radius r, increasing resistance by 1/r⁴. Even a 50% reduction in diameter increases resistance 16‑fold, dramatically reducing flow for the same pressure gradient. This is why arterial blockages cause significant ischaemia. The law also explains why pressure drops across a stenosis.
- Blood is non‑Newtonian at low shear rates (e.g., in microcirculation).
- Blood vessels are elastic and change diameter with pressure.
- The flow is often pulsatile, not steady.
- Branches and bifurcations are not straight tubes.
- In small vessels, the Fåhræus‑Lindqvist effect reduces apparent viscosity.
- Estimating pressure gradients across vascular stenoses (e.g., using Doppler ultrasound).
- Calculating cardiac output from pressure and resistance measurements.
- Designing vascular grafts and stents (to minimise resistance).
- Understanding the haemodynamic consequences of anaemia, polycythaemia, or vasodilators.
They are essentially the same. The Hagen‑Poiseuille equation is often written to solve for pressure drop (ΔP) given a flow rate: ΔP = (8ηLQ) / (πr⁴). Poiseuille's law is the same rearranged to solve for Q. Both refer to the same physical relationship.