Find the volumetric flow rate through a pipe from its radius, pressure difference, viscosity, and length, solve for a missing variable, or see how sensitive flow is to radius — Q = πr⁴ΔP/(8µL).
Pipe & fluid properties
1 cm = 0.01 m. Use the internal radius, not diameter.
Water ≈ 0.001 Pa·s at room temperature.
Volumetric flow rate
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Enter radius, pressure difference, viscosity, and length to compute.
Flow rate (L/min)—
Radius sensitivityr⁴ law
Solve for a missing variable
Use scientific notation, e.g. 4e-7 for 4×10⁻⁷.
Solved value
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Choose a variable, set the target flow rate, and enter the other three values.
See how flow rate scales with radius
Q ∝ r⁴, so flow rate is extremely sensitive to radius — a doubled radius means 16× the flow, and a halved radius means 1/16th the flow, even though ΔP, µ, and L never changed.
Flow rate at current radius
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Enter radius, pressure difference, viscosity, and length to see the sensitivity table.
Flow rate vs. radius multiplier
Radius
Flow rate
vs. current
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Each row scales the current radius by the listed multiplier; flow rate scales by the multiplier to the 4th power.
Jean Léonard Marie Poiseuille derived this equation in the 1840s while studying blood flow, and it remains the standard tool for predicting how fast a fluid moves through a pipe under laminar conditions.
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Walk-through
How to Use This Calculator
3 steps▸
1
Enter the pipe and fluid properties
On the Flow Rate tab, enter the pipe's internal radius (m), the pressure difference driving the flow (Pa), the fluid's dynamic viscosity (Pa·s), and the pipe's length (m). All four inputs must be greater than zero for a laminar-flow result.
2
Read the flow rate and the r⁴ note
The result card shows the volumetric flow rate Q in both m³/s and L/min, plus an interpretation explaining how sensitive Q is to radius specifically — because Q scales with r⁴, small changes in radius dominate the result far more than equal-percentage changes in pressure, viscosity, or length.
3
Solve for a missing variable or explore radius sensitivity
On the Solve tab, pick the variable you don't know (radius, pressure difference, viscosity, or length), set a target flow rate, and fill in the other three — the calculator rearranges Q = πr⁴ΔP/(8µL) to find it. On the Radius Sensitivity tab, the table shows exactly how flow rate changes at 0.5×, 0.75×, 1.25×, 1.5×, and 2× the current radius.
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Reference
Formula & Methodology
3 formulas▸
Hagen-Poiseuille equation
Q = πr⁴ΔP / (8µL)
Volumetric flow rate Q (m³/s) equals π times pipe radius r (m) to the 4th power, times the pressure difference ΔP (Pa) across the pipe's length, divided by 8 times the dynamic viscosity µ (Pa·s) times the pipe length L (m).
The r⁴ sensitivity relation
Q₂/Q₁ = (r₂/r₁)⁴
Because flow rate depends on the 4th power of radius, the ratio of two flow rates equals the ratio of their radii raised to the 4th power — a 2× radius gives 16× the flow, a 0.5× radius gives 1/16th the flow, entirely independent of ΔP, µ, or L.
Validity condition
Laminar flow only (low Reynolds number)
Poiseuille's law assumes steady, laminar flow of an incompressible Newtonian fluid through a rigid, straight cylindrical pipe. It breaks down once the flow becomes turbulent — check the Reynolds number if you're unsure which regime applies.
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Glossary
Key Terms Explained
7 terms▸
Poiseuille's law ↗An equation (also called the Hagen-Poiseuille equation) describing the volumetric flow rate of a Newtonian fluid in steady, laminar flow through a rigid cylindrical pipe: Q = πr⁴ΔP/(8µL).
Laminar flow ↗Flow that moves in smooth, parallel layers with minimal mixing. Poiseuille's law only applies in this regime — it does not describe turbulent, chaotic flow.
Volumetric flow rate ↗The volume of fluid passing a cross-section per unit time, denoted Q, usually measured in m³/s or converted to more intuitive units like liters per minute (L/min).
Viscosity ↗A fluid's resistance to flow or shear, denoted µ (dynamic viscosity) and measured in pascal-seconds (Pa·s). Higher viscosity means more resistance and, all else equal, lower flow rate.
Pressure gradient ↗The pressure difference ΔP driving fluid through the pipe, divided by the pipe length — the force per unit area that pushes the fluid forward against viscous drag.
Radius ↗The internal radius r of the pipe (half the internal diameter). Flow rate is proportional to r⁴, making radius by far the most sensitive variable in Poiseuille's law.
Hagen-Poiseuille ↗The full name of the law, credited to Gotthilf Hagen and Jean Léonard Marie Poiseuille, who independently derived the relationship for laminar pipe flow in the 1840s.
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Scenarios
Real-World Examples
3 worked examples▸
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Water through a small pipe
A 1 cm radius pipe carrying water under a modest pressure difference
Radius 0.01 mPressure difference 1,000 PaViscosity 0.001 Pa·sLength 1 m
Q = π × 0.01⁴ × 1,000 / (8 × 0.001 × 1) ≈ 3.93 × 10⁻³ m³/s, or about 236 L/min — water's very low viscosity means even a modest 1,000 Pa pressure difference drives a large flow through a 2 cm diameter pipe. (In practice, real flow at this rate would be turbulent, not laminar — Poiseuille's law is applied here as an idealized formula; see the Limits section.)
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The r⁴ effect, isolated
Same pipe as above, but the radius is halved to 0.005 m
Radius 0.005 mPressure difference 1,000 PaViscosity 0.001 Pa·sLength 1 m
Q = π × 0.005⁴ × 1,000 / (8 × 0.001 × 1) ≈ 2.45 × 10⁻⁴ m³/s (about 14.7 L/min) — exactly 1/16th of the original flow rate, even though nothing else changed. Halving the radius, not the pressure or length, is what crushed the flow.
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Blood flow approximation
A small arteriole modeled as a rigid pipe carrying blood
Radius 0.0001 m (0.1 mm)Pressure difference 1,330 Pa (~10 mmHg)Viscosity 0.0035 Pa·s (blood)Length 0.01 m
Q ≈ π × 0.0001⁴ × 1,330 / (8 × 0.0035 × 0.01) ≈ 1.49 × 10⁻⁹ m³/s (about 0.09 mL/min). This is a textbook approximation only — blood is not a perfectly Newtonian fluid and vessels aren't perfectly rigid, so real physiological flow diverges from the ideal formula, especially in small vessels.
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Reference
Cite This Calculator
APA & MLA▸
Use either format to cite this calculator in a paper, report, or resource list.
Jean Léonard Marie Poiseuille derived this equation in the 1840s while studying blood flow, and it remains the standard tool for predicting how fast a fluid moves through a pipe under laminar conditions. Q = πr⁴ΔP/(8µL) looks simple, but its most important lesson is easy to miss at a glance: flow rate depends on the pipe radius to the 4th power, making radius the single most powerful lever in the whole equation.
How the Poiseuille's Law Calculator works
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The calculator applies Q = πr⁴ΔP/(8µL) directly: pipe radius r (m) raised to the 4th power, times the pressure difference ΔP (Pa) driving the flow, divided by 8 times dynamic viscosity µ (Pa·s) times pipe length L (m). The Solve tab rearranges this same equation algebraically to isolate any one of the four inputs given a target flow rate and the other three — solving for radius requires taking a 4th root, which is the trickiest rearrangement but handled automatically.
Inputs and what they mean
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Radius (m) is the pipe's internal radius, not diameter — a common source of factor-of-16 errors when radius is accidentally entered as diameter. Pressure difference (Pa) is the pressure drop from one end of the pipe to the other, the force actually pushing fluid through. Dynamic viscosity (Pa·s) is a property of the fluid itself — water is about 0.001 Pa·s at room temperature, while honey or blood plasma run far higher. Pipe length (m) is the straight-line distance the fluid travels. Of these four, radius dominates: doubling it multiplies flow by 16×, while doubling pressure only doubles flow.
Limits and edge cases
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Poiseuille's law assumes steady, laminar flow of an incompressible Newtonian fluid through a rigid, straight, cylindrical pipe with no-slip walls — it does not apply once flow becomes turbulent (check with a Reynolds Number Calculator if unsure), and it does not account for entrance effects near a pipe inlet, pipe curvature, or elastic vessel walls. The blood-flow example above is a textbook approximation only: blood is a non-Newtonian, shear-thinning fluid, and real blood vessels are elastic and branching, so clinical blood-flow calculations use more specialized models. Treat this calculator as an idealized engineering estimate, not a substitute for domain-specific modeling in biomedical or high-precision industrial contexts.
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Questions
Frequently Asked Questions
6 questions▸
What is the formula for Poiseuille's law?+
Q = πr⁴ΔP/(8µL), where Q is volumetric flow rate (m³/s), r is the pipe's internal radius (m), ΔP is the pressure difference across the pipe (Pa), µ is dynamic viscosity (Pa·s), and L is pipe length (m).
Why does radius matter so much more than the other variables?+
Flow rate is proportional to radius raised to the 4th power (r⁴), while it's only proportional to the first power of pressure difference, viscosity, and length. That means halving the radius cuts flow rate by 16×, but halving the pressure only cuts flow rate by 2×.
Does Poiseuille's law only apply to laminar flow?+
Yes. The equation assumes steady, laminar flow of an incompressible Newtonian fluid through a rigid, straight cylindrical pipe. Once the Reynolds number rises into the turbulent regime, Poiseuille's law no longer accurately predicts flow rate.
Can this calculator be used for blood flow?+
Only as a rough approximation. Real blood is a non-Newtonian, shear-thinning fluid and blood vessels are elastic rather than rigid, so Poiseuille's law captures the general trend (radius dominates) but not the precise physiological flow rate — clinical work relies on more specialized hemodynamic models.
What units does the calculator use?+
Radius and length are in meters (m), pressure difference is in pascals (Pa), viscosity is in pascal-seconds (Pa·s), and the resulting flow rate is shown in both cubic meters per second (m³/s) and liters per minute (L/min).
Can I solve for radius instead of flow rate?+
Yes — the Solve tab lets you pick radius, pressure difference, viscosity, or length as the unknown, enter a target flow rate and the other three values, and the calculator rearranges the equation (solving for radius involves a 4th root) to find the missing value.
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