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.
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.
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.
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.
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.
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.
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.
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.
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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 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). Use this as a practical reminder before finalizing the result.
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.
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.
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.
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.