Poiseuille's Law Calculator
Find the volumetric flow rate through a rigid circular tube from the pressure drop, radius, viscosity and length — or solve for any of the five variables. The r⁴ dependence makes this the most sensitive law in haemodynamics.
| Radius | Flow rate Q | % of current Q | Resistance Rh | Note |
|---|
| Quantity | Small unit | Mid unit | Large unit |
|---|
r ↑ → Q explodes
Flow scales with the fourth power of radius. A 19% increase in radius (1.19⁴ ≈ 2) doubles flow. A 10% narrowing (0.9⁴ ≈ 0.66) cuts flow by a third — the reason vasoconstriction is so powerful.
ΔP ↑ → Q linearly
Flow is directly proportional to the pressure drop. Doubling the driving pressure doubles the flow, with no other consequence — this is what makes Poiseuille's law so easy to linearise.
η ↑ → Q falls inversely
Viscosity enters only in the denominator. Doubling viscosity halves the flow. Blood viscosity depends on haematocrit and shear rate — the Fåhræus–Lindqvist effect makes it drop in vessels below ~300 µm.
The laminar-flow caveat
Poiseuille's law assumes laminar, steady, fully developed, incompressible Newtonian flow in a rigid circular tube with no slip at the wall. In arteries, turbulence (Re > 2000) and pulsatility break every one of these assumptions.
How to Use This Calculator
Poiseuille's law connects the flow rate through a tube to the pressure drop, the fourth power of the radius, the fluid's viscosity and the tube length.
1. Enter four of the five variables
Use presets or type values, choosing a convenient unit for each — flow (µL/s, mL/s, L/min), pressure (Pa, kPa, mmHg), radius (µm, mm, cm), viscosity (µPa·s, mPa·s, Pa·s) and length (mm, cm, m).
2. Choose what to solve for
Q, ΔP, r, η and L are all algebraically solvable from the other four — including the radius via a fourth root.
3. The solved field goes read-only
It's tagged "SOLVED", greyed out, and updated live as you change anything else.
4. Explore the charts
See how flow changes with radius (the r⁴ curve), with pressure (a straight line), or view the parabolic velocity profile inside the tube.
5. Check the reference tables
See how Q and Rh change across a range of radii, your values converted into every unit, and a hydraulic-resistance grid.
6. Switch to Advanced
See sensitivity to each input, mean and peak velocity, wall shear stress, Reynolds number, and entrance length.
What This Tool Offers
A complete workbench for laminar flow in a circular tube — the same calculation used in haemodynamics, respiratory physiology, microfluidics and pipeline engineering.
Five-way solve
Solve for flow, pressure drop, radius (fourth root), viscosity or length from the other four values.
Three interactive views
Flow-vs-radius curve, flow-vs-pressure line, and a parabolic velocity-profile diagram, all driven by your live values.
Per-field unit switching
Q, ΔP, r, η and L each switch independently between small, mid and large units, with values converted automatically.
Reference tables
A radius-sensitivity table showing the r⁴ law in numbers, plus a live any-unit conversion table for your exact values.
Sensitivity check
See how a +1% change in each input moves the result — and why radius dominates by a factor of four.
Companion quantities
Hydraulic resistance, mean and peak velocity, wall shear stress, Reynolds number (blood) and entrance length.
What Is Poiseuille's Law?
Poiseuille's law (also called the Hagen–Poiseuille equation) describes steady, laminar flow of a Newtonian fluid through a rigid circular tube. It says the volumetric flow rate Q is proportional to the pressure drop ΔP and to the fourth power of the radius r, and inversely proportional to the fluid's dynamic viscosity η and the tube's length L. The r⁴ term is the headline feature: small changes in radius produce enormous changes in flow. In physiology this is why a 20% narrowing of an artery can halve perfusion, and why vasodilation is such an effective way to increase blood supply.
Formula and Variables
| Symbol | Meaning | Notes |
|---|---|---|
| Q | Volumetric flow rate | m³/s (or mL/s, L/min) |
| ΔP | Pressure drop along the tube | Pa (or kPa, mmHg) |
| r | Internal radius | m (or mm, µm) — enters as r⁴ |
| η | Dynamic viscosity | Pa·s (or mPa·s = cP) |
| L | Tube length | m (or mm, cm) |
| Rh = 8ηL / (πr⁴) | Hydraulic resistance | Pa·s/m³ — the "Ohm's law" of fluid flow |
| v̄ = Q / (πr²) | Mean velocity | m/s |
| vmax = 2v̄ | Peak (centreline) velocity | m/s — parabolic profile |
| τw = 4ηQ / (πr³) | Wall shear stress | Pa — sensed by endothelial cells |
Typical Values
| Quantity | Typical value | Notes |
|---|---|---|
| Blood viscosity | 3–4 mPa·s | Whole blood at 37 °C, high shear |
| Plasma viscosity | ~1.2 mPa·s | Slightly above water |
| Water viscosity | 1.00 mPa·s | At 20 °C |
| Aorta radius | ~12 mm | Peak Re ≈ 4000 (transitional) |
| Capillary radius | ~4 µm | Single-file red cells; Re ≪ 1 |
| Mean arterial pressure drop | ~90 mmHg | Aorta → vena cava |
| Cardiac output | ~5 L/min | ≈ 83 mL/s |
Frequently Asked Questions
Two powers come from geometry, two from the velocity profile. A wider tube has more cross-sectional area (r²) and, for the same pressure gradient, a higher mean velocity. In laminar flow the velocity profile is parabolic, and the mean is half the peak; integrating that parabola over the cross-section adds another r². The product r² · r² = r⁴. It is why a 10% narrowing reduces flow by ~34%, not 10%.
Five assumptions: laminar flow (Re < ~2000), steady flow, fully developed profile (L ≫ entrance length), incompressible Newtonian fluid, and a rigid circular tube with no slip. Real arteries are pulsatile, elastic and taper; real blood is non-Newtonian at low shear. For large vessels and high flow rates, use the Navier–Stokes equations or a Womersley-type pulsatile model instead.
Rearranging Poiseuille's law as ΔP = Q · Rh gives Rh = 8ηL/(πr⁴), the fluid analogue of Ohm's law. In physiology the resistance of the whole circulation is quoted in mmHg·min/L or Wood units — normal systemic vascular resistance is about 10–20 Wood units.
A parabola: v(r') = vmax·(1 − (r'/r)²), where r' is the radial distance from the centreline. The velocity is zero at the wall (no-slip condition) and highest at the centre, where vmax = 2·v̄. The mean velocity is exactly half the peak — a useful shortcut when only one is measured.
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
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