Poiseuille's Law Calculator — Pipe Flow Rate
Use the Hagen-Poiseuille equation to find the volumetric flow rate through a cylindrical tube given the tube radius and length, pressure difference and fluid viscosity. Valid for steady, laminar, Newtonian flow.
mm
mm
Pa
cP
Q = π r⁴ ΔP / (8 η L) — Hagen-Poiseuille equation
1,192.823 mL/min
flow rate- 1
r⁴ — radius to the 4th power
(1.5 mm × 10⁻³)⁴ = 5.063e-12 m⁴The r⁴ dependence makes radius the dominant factor by far - 2
Numerator: π × r⁴ × ΔP
π × 5.063e-12 × 1,000 Pa = 1.590e-8 - 3
Denominator: 8 × η × L
8 × 1.000e-3 Pa·s × 0.1 m = 8.000e-4 - 4
Flow rate Q (mL/min)
(1.590e-8 ÷ 8.000e-4) × 10⁶ × 60 = 1,192.823
How does this calculator work?
Q = π r⁴ ΔP / (8 η L). Flow rate scales as the fourth power of radius — doubling r increases Q by 16×. Enter radius (mm), length (mm), pressure difference (Pa) and viscosity (cP) for the flow rate in mL/min. Valid only for laminar Newtonian flow (Re < 2 300).
Formula
How this is calculated
Poiseuille's law (more precisely the Hagen-Poiseuille equation) describes steady, laminar flow of a Newtonian fluid through a straight, rigid, circular tube. The volumetric flow rate Q depends on four parameters: tube radius r (raised to the fourth power), pressure difference ΔP between inlet and outlet, dynamic viscosity η, and tube length L. Doubling the radius increases flow 16-fold — the r⁴ dependence makes radius by far the most influential parameter.
The velocity profile across the tube cross-section is parabolic: maximum (v_max = r²ΔP / 4ηL) at the centre-line and zero at the wall (no-slip condition). The mean velocity is exactly half the maximum, so Q = v_mean × πr².
The law applies strictly to laminar flow. Turbulence begins above a Reynolds number of about 2 300; above that threshold actual flow rates are lower than the Hagen-Poiseuille prediction and the Darcy-Weisbach equation applies. This calculator does not check Re — always verify that Re < 2 300 for your conditions. Common applications include microfluidics, blood flow in capillaries, IV infusion lines and hydraulic tubing.
Frequently asked questions
The r⁴ term comes from two effects: a larger radius adds more cross-sectional area (r² factor) and simultaneously reduces the velocity gradient near the wall — the faster central region is relatively larger — contributing another r² factor. In biological and engineering systems this makes even a small increase in tube diameter enormously effective.
The equation assumes: laminar flow (Re < ~2 300), Newtonian fluid (constant viscosity), rigid straight tube, and fully developed flow far from the inlet. Blood is non-Newtonian (shear-thinning at low shear rates), real tubes are flexible and curved, and turbulence can occur in large arteries — so Poiseuille's law is an approximation in those contexts.
Hydraulic resistance R = 8ηL / (πr⁴) is the fluid analogue of electrical resistance. By Ohm's law analogy, ΔP = R × Q — a higher resistance means more pressure is needed for the same flow rate. Resistances in series add directly; resistances in parallel add as reciprocals, just like electrical resistors.
Also known as
TG we-Calculate Editorial Team. (2026). Poiseuille's Law Calculator — Pipe Flow Rate [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/poiseuilles-law-calculator
TG we-Calculate Editorial Team. "Poiseuille's Law Calculator — Pipe Flow Rate." TG we-Calculate. 2026. https://we-calculate.com/calculator/poiseuilles-law-calculator.
TG we-Calculate Editorial Team, "Poiseuille's Law Calculator — Pipe Flow Rate," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/poiseuilles-law-calculator
@misc{wecalculate_poiseuilles_law_calculator, title = {Poiseuille's Law Calculator — Pipe Flow Rate}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/poiseuilles-law-calculator}}, year = {2026}, note = {TG we-Calculate} }
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