Solve Q = π·ΔP·r⁴ / (8·η·L) for steady laminar flow of a Newtonian fluid through a long, straight, circular pipe. Compute the flow rate, solve for any one of the five variables, check that the flow is actually laminar with the Reynolds number, or convert a target flow rate into pressure drop and head loss.
| Fluid | η (mPa·s) |
|---|---|
| Water, 20°C | 1.002 |
| Water, 40°C | 0.653 |
| Water, 100°C | 0.282 |
| Seawater, 20°C | 1.07 |
| Ethanol, 20°C | 1.074 |
| Mercury, 20°C | 1.526 |
| Diesel (approx) | 3.2 |
| Blood, 37°C (approx) | 3.5 |
| Air, 20°C | 0.0181 |
| Olive oil (approx) | 84 |
| Engine oil SAE 10W-30 (approx) | 200 |
| Glycerin, 20°C | 1410 |
Viscosity is strongly temperature-dependent; preset values are at about 20°C except where noted. Values marked approx vary with grade and source. 1 Pa·s = 1000 mPa·s = 1000 cP = 10 poise.
Poiseuille's law, also called the Hagen–Poiseuille equation, gives the volumetric flow rate of a viscous fluid through a long straight circular pipe in laminar flow: Q = π·ΔP·r⁴/(8·η·L), where ΔP is the pressure drop, r the pipe radius, η the dynamic viscosity and L the pipe length. Example: ΔP = 1000 Pa, r = 1 cm, η = 0.001 Pa·s (water at 20°C), L = 1 m gives Q = 0.00392699 m³/s = 3.927 L/s = 235.619 L/min = 62.2441 GPM, with an average velocity of 12.5 m/s.
Enter the pressure drop, the pipe radius or diameter, the fluid viscosity and the pipe length in the Flow rate Q tab. Note that flow scales with the fourth power of the radius: doubling the radius multiplies the flow rate by 16. That is why even a small increase in pipe diameter cuts the required pumping pressure dramatically.
Yes — the 5-way solver tab solves for any one of Q, ΔP, r, η or L from the other four. For example, asking for the radius that passes Q = 3.927 L/s with ΔP = 1000 Pa, η = 0.001 Pa·s and L = 1 m returns r = 10 mm, and solving for viscosity from a glycerin flow of 2.7851 mL/s returns η = 1.41 Pa·s.
Only for steady laminar flow of an incompressible Newtonian fluid in a long straight circular pipe, with no entrance or exit effects. In practice this means the Reynolds number must stay below about 2300. The worked example above (ΔP = 1000 Pa, r = 1 cm, water) gives Re = 250000, which is strongly turbulent — so Poiseuille's law does NOT apply there, and the calculator flags exactly this situation.
Take water at 20°C in a small tube: r = 2 mm, ΔP = 1000 Pa, L = 1 m, η = 0.001 Pa·s. Then v = 0.5 m/s, Q = 6.2832 mL/s and Re = 2000, which is below 2300, so the flow is laminar and Poiseuille's law is valid.
Use the Laminar check tab: it computes Re = ρ·v·D/η from your flow rate, diameter, density and viscosity. Below 2300 the flow is laminar and Poiseuille's law applies; 2300 to 4000 is transitional; above 4000 the flow is turbulent and Poiseuille's law over-predicts the flow rate. If Re is 2300 or higher, every result on this page is shown with a warning that it lies outside the valid range.
Head loss is h_L = ΔP/(ρ·g), the pressure drop expressed as a column of the flowing fluid. The ΔP from Q tab reports it directly: for water (ρ = 1000 kg/m³), a pressure drop of 1000 Pa equals 0.10197 m of head. Pick the fluid density to get the head in the correct fluid column.
Pressure: Pa, kPa, MPa, bar, psi, atm and mmHg. Length: mm, cm, m, in and ft. Viscosity: Pa·s, mPa·s, cP, µPa·s and poise. Flow rate: m³/s, L/s, mL/s, L/min, L/h, m³/h, GPM and ft³/s. Velocity: m/s, cm/s, mm/s and ft/s. Density: kg/m³, g/cm³ and lb/ft³. Reference: 1 GPM = 0.0630902 L/s = 3.78541 L/min, 1 bar = 100000 Pa, 1 psi = 6894.75729 Pa.
This tool is the laminar viscous-flow regime: pressure-driven flow computed from viscosity with Q = π·ΔP·r⁴/(8·η·L), valid only below Re 2300. The flow rate calculator at https://nrtechstudio.com/tools/flow-rate-calculator/ solves the general continuity equation Q = A·v and needs no pressure or viscosity data. The Darcy–Weisbach calculator at https://nrtechstudio.com/tools/darcy-weisbach/ handles head loss for turbulent flow. The pipe velocity calculator at https://nrtechstudio.com/tools/pipe-velocity/ converts between Q and average velocity.
Yes — high viscosity is exactly where Poiseuille's law shines, because the flow stays laminar. Example: glycerin (η = 1.41 Pa·s, ρ = 1260 kg/m³) with r = 1 cm, ΔP = 1000 Pa, L = 1 m gives Q = 2.7851 mL/s, v = 0.0088652 m/s and Re = 0.158443, deeply laminar. Blood (ρ = 1060 kg/m³, η = 3.5 mPa·s) with r = 2 mm, L = 0.5 m, ΔP = 2000 Pa gives Q = 7.1808 mL/s, v = 57.143 cm/s and Re = 692.245, also laminar. Blood is actually non-Newtonian, so treat that result as an approximation.
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