Why is flow proportional to R⁴?

In a parabolic velocity profile, both the cross-sectional area (∝ R²) and the average velocity (∝ R²) increase with radius, giving Q ∝ R⁴. This is why atherosclerosis (artery narrowing) dramatically reduces blood flow.

When does Poiseuille flow break down?

When Re > 2300 (turbulent transition), for non-Newtonian fluids, for compressible gases at high pressure gradients, for very short tubes (entrance effects), and for pulsatile flow (blood, engine oil pumps).

Is blood really Newtonian?

No. Blood is a non-Newtonian shear-thinning fluid due to red blood cells. Poiseuille's law gives approximate results for large vessels (aorta, major arteries) where shear rates are high and viscosity is approximately constant.

What is flow resistance?

Flow resistance R_flow = ΔP/Q = 8µL/(πR⁴). It is analogous to electrical resistance: higher resistance means less flow for the same pressure, just as higher electrical resistance means less current for the same voltage.

How does this relate to Darcy-Weisbach?

The Darcy-Weisbach equation generalizes to turbulent flow. For laminar flow, the Darcy friction factor f = 64/Re, and Darcy-Weisbach reduces to Poiseuille's law.

Can I use this for gas flow?

Only for low-speed, incompressible gas flow where the pressure drop is small compared to absolute pressure (ΔP/P < 10%). For larger pressure drops, compressibility effects require different equations.

Poiseuille's Law Calculator

Calculate laminar flow rate, pressure drop, velocity profile, and flow resistance in cylindrical tubes using Poiseuille's equation. Essential for blood flow and pipe sizing.

Solve For

Dynamic Viscosity µ (mPa·s / cP)

Tube Length L (m)

Tube Radius R (m)

Pressure Drop ΔP (Pa)

Flow Rate

48.4582 mL/s

2.9075 L/min

Pressure Drop

1,333.0 Pa

10.00 mmHg

Avg Velocity

1.7139 m/s

Q / (πR²)

Max Velocity

3.4277 m/s

2 × avg (parabolic)

Reynolds Number

2,938

⚠ Turbulent

Flow Resistance

2.751e+7 Pa·s/m³

8µL/(πR⁴)

Wall Shear Stress

7.9980 Pa

ΔP·R / (2L)

Power Dissipated

64.5947 mW

ΔP × Q

Flow Regime

Re = 2,938 — Turbulent (Poiseuille invalid!)

Applications of Poiseuille's Law

Application	Radius	Flow	Notes
Blood flow (aorta)	12.5 mm	~80 ml/s	Pulsatile; blood is non-Newtonian
Blood (capillary)	4 µm	~0.6 nl/s	Single red blood cell passage
Water supply pipe	50-150 mm	1-50 L/s	Turbulent at high flow (Poiseuille invalid)
IV infusion	0.5-1 mm	~0.5-3 ml/min	Gravity-driven; laminar
Microfluidics	10-100 µm	nL-µL/s	Always laminar (Re << 1)
Oil pipeline	0.15-0.5 m	100s L/s	Must check Re; may be turbulent

Planning notes, formulas, and examples

About the Poiseuille's Law Calculator

Poiseuille's law (Hagen-Poiseuille equation) describes the steady, laminar, viscous flow of an incompressible fluid through a long cylindrical tube. The volumetric flow rate is proportional to the pressure gradient and the fourth power of the radius — meaning that doubling the tube diameter increases flow by 16 times.

This calculator solves for flow rate given pressure drop, or pressure drop given flow rate. It also computes the average and maximum (centerline) velocities, flow resistance, wall shear stress, and power dissipated. The Reynolds number is checked to verify that the laminar flow assumption is valid.

Poiseuille's law is the foundation of hemodynamics (blood flow analysis), IV infusion rate calculations, microfluidics channel design, and lubrication theory. The law fails for turbulent flow (Re > 2300), compressible fluids, and non-Newtonian fluids like blood in large vessels.

Preset buttons load parameters for blood flow in arteries, water pipes, IV drip tubing, oil pipelines, and microfluidic channels. The reference table shows real-world applications with typical dimensions and flow rates.

When This Page Helps

Poiseuille's law is one of the most-used equations in fluid mechanics, from medical blood flow analysis to industrial pipe sizing and microfluidic chip design.

This calculator handles the conversions between common units and checks whether the laminar flow assumption is valid, which is the main step people miss when they do the algebra by hand. It is most useful when you want to compare how radius, viscosity, and tube length change the same flow problem.

How to Use the Inputs

Choose whether to solve for flow rate or pressure drop.
Enter the fluid dynamic viscosity in mPa·s (cP).
Enter the tube length and radius in meters.
Enter the pressure drop (Pa) or flow rate (mL/s) as appropriate.
Check that the Reynolds number confirms laminar flow.
Read the flow rate, velocity, resistance, and wall shear stress.

Formula used

Q = πR⁴ΔP / (8µL).
v_avg = Q / (πR²). v_max = 2·v_avg.
R_flow = 8µL / (πR⁴) (flow resistance).
τ_wall = ΔP·R / (2L).
Re = ρ·v·2R / µ. Valid for Re < 2300.

Example Calculation

Result: Q = 0.60 mL/s, v_avg = 0.021 m/s

Q = π × (0.003)⁴ × 1333 / (8 × 0.0035 × 0.25) = 6.0×10⁻⁷ m³/s = 0.60 mL/s. This models blood flow in a medium-sized artery.

Tips & Best Practices

A 50% artery stenosis (R → 0.5R) reduces flow to 1/16 (6.25%) of normal — this is why blockages are dangerous.
For IV infusions, raising the bag height increases ΔP and thus flow rate linearly.
In microfluidics, Reynolds numbers are often << 1, making Poiseuille flow an excellent assumption.
Convert mmHg to Pa: multiply by 133.322 (1 mmHg = 133.322 Pa).
The parabolic velocity profile means fluid at the center moves at twice the average speed.

When The Law Applies

Poiseuille's law assumes a long, straight, cylindrical tube with laminar flow and a Newtonian fluid. When those assumptions break, the results can still be a useful estimate, but they are no longer exact.

What The Outputs Mean

Flow rate increases with the fourth power of radius, so a small change in tube size has a large effect. Pressure drop, wall shear stress, and resistance all move together, which makes the calculator useful for comparing tube sizes rather than looking at a single value in isolation.

Practical Checks

Use the Reynolds number and the fluid type before trusting the result. If the flow is turbulent, pulsatile, or strongly non-Newtonian, this formula is the wrong tool and the output should be treated as a rough guide only.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

In a parabolic velocity profile, both the cross-sectional area (∝ R²) and the average velocity (∝ R²) increase with radius, giving Q ∝ R⁴. This is why atherosclerosis (artery narrowing) dramatically reduces blood flow.