Poiseuille Flow Calculator
Calculate laminar flow rate through a pipe with the Hagen-Poiseuille equation, from pressure drop, radius, dynamic viscosity, and pipe length.
Poiseuille Flow Calculator
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Formula: Q = πΔPr⁴ / (8μL)
Worked example — ~3.93×10⁻⁴ m³/s
Formula
Q = πΔPr⁴ / (8μL)
Laminar flow rate through a pipe. Flow increases with the 4th power of radius.
Worked Examples
Example 1: Small pipe
Problem:ΔP=1000Pa, r=0.01m, μ=0.001 Pa·s, L=1m
Solution:Q=π×1000×10⁻⁸/(8×0.001×1)
Result:~3.93×10⁻⁴ m³/s
Frequently Asked Questions
What is Poiseuille's law and when does it apply?
Poiseuille's law describes steady laminar flow of a viscous fluid through a cylindrical pipe: Q = πΔPr⁴/(8μL). It applies when the Reynolds number Re = ρvD/μ is below ~2100 (laminar regime). Above Re ≈ 4000 the flow becomes turbulent and the formula no longer holds.
How do you calculate volumetric flow rate using Poiseuille's law?
Plug in: Q = π × ΔP × r⁴ / (8 × μ × L). For ΔP = 1000 Pa, r = 0.01 m, μ = 0.001 Pa·s, L = 1 m: Q = π × 1000 × 10⁻⁸ / (8 × 0.001 × 1) ≈ 3.93×10⁻⁴ m³/s or 0.393 L/s.
What units are used in Poiseuille flow calculations?
Pressure drop ΔP in Pascals (Pa), pipe radius r in metres (m), dynamic viscosity μ in Pascal-seconds (Pa·s; water ≈ 0.001 Pa·s at 20°C), pipe length L in metres (m). The result Q is in m³/s; multiply by 1000 to get litres per second (L/s).
Why does pipe radius have such a large effect on flow rate?
Flow rate scales with the fourth power of radius (r⁴). Doubling the pipe radius increases flow by 2⁴ = 16 times. This means a 10% reduction in radius (e.g. from arterial plaque) reduces flow to 0.9⁴ ≈ 66% of normal — explaining why partial blockages significantly impair blood flow.
How do you confirm the flow is actually laminar?
Compute the Reynolds number Re = rho x v x D / mu, using mean velocity v = Q/(pi x r-squared) and diameter D = 2r. Check the worked example: Q = 3.93 x 10^-4 m-cubed/s in a 0.01 m radius pipe gives v = 1.25 m/s, so for water Re = 1000 x 1.25 x 0.02 / 0.001 = about 25,000. That is firmly turbulent, so the prediction would not hold in practice - always test Re afterwards.
What is hydraulic resistance, and how do pipe sections combine?
Rewriting the law as Q = delta-P / R_hyd defines hydraulic resistance R_hyd = 8 x mu x L / (pi x r^4), which behaves exactly like electrical resistance with pressure playing the role of voltage. Segments joined in series add their resistances; branches in parallel add reciprocals. That analogy makes it practical to model branching networks such as capillary beds or manifolds without solving the flow field directly.
What does the velocity profile look like across the pipe?
Laminar viscous flow forms a parabolic profile: velocity is highest on the axis and falls to zero at the wall because of the no-slip condition. The centreline maximum is v_max = delta-P x r-squared / (4 x mu x L), and the average velocity is exactly half of that. Multiplying the mean velocity by the cross-sectional area pi x r-squared reproduces the flow rate the calculator returns.
What is the difference between dynamic and kinematic viscosity?
Poiseuille's law needs dynamic viscosity mu, in Pa s. Kinematic viscosity is nu = mu / rho, in m-squared/s, and appears mainly in Reynolds number work. Water at 20 C has mu of about 0.001 Pa s and density near 998 kg/m-cubed, giving nu of about 1.0 x 10^-6 m-squared/s. Older sources use poise, where 1 poise = 0.1 Pa s and 1 centipoise = 0.001 Pa s.
How does temperature change the predicted flow rate?
Flow rate is inversely proportional to viscosity, and liquid viscosity drops steeply as temperature rises - water's roughly halves between 20 C and 50 C - so the same pressure drop pushes about twice as much hot water through the same pipe. Gases behave the opposite way, becoming more viscous when heated, because their viscosity comes from molecular momentum transfer rather than intermolecular attraction.
When does entrance length invalidate the result?
The parabolic profile takes distance to develop after fluid enters a pipe, roughly L_e = 0.05 x Re x D in laminar flow. If the tube is short compared with that, much of it is still in the developing region where wall shear is higher, so the real pressure drop exceeds the Poiseuille prediction. Bends, valves and sudden contractions add further losses the formula does not include.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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