Poiseuille's Law Calculator
Enter any four of the five Hagen-Poiseuille variables and solve for the fifth: volumetric flow rate, pressure drop, tube radius, tube length, or dynamic viscosity. Switch units across all inputs, see the Reynolds number and flow regime, and step through the full derivation with your exact numbers.
Formula
Worked example
Water (mu = 0.001002 Pa·s) flows through a 5 mm radius tube, 0.1 m long, under 100 Pa pressure: Q = (pi * 100 * 0.005^4) / (8 * 0.001002 * 0.1) = 1.23e-5 m3/s = 12.3 mL/s. Average velocity = Q / (pi * r^2) = 0.157 m/s. Re = (2 * 998 * 1.23e-5) / (0.001002 * pi * 0.005) = 1558 - confirmed laminar.
What is Poiseuille's law?
Poiseuille's law (formally the Hagen-Poiseuille equation, published by Jean Poiseuille in 1840 and independently by Gotthilf Hagen) describes the volumetric flow rate of a viscous, incompressible, Newtonian fluid through a long, straight, rigid cylindrical tube under laminar conditions. The formula is Q = (pi * dP * r^4) / (8 * mu * L), where Q is the volumetric flow rate (m3/s), dP is the pressure difference from inlet to outlet (Pa), r is the inner radius (m), mu is the dynamic viscosity (Pa·s), and L is the tube length (m). The fourth-power dependence on radius is the most important consequence: doubling the radius increases flow 16-fold for the same pressure, so even small changes in pipe diameter or vessel caliber have enormous effects on throughput.
Hydraulic resistance and the Ohm analogy
The Hagen-Poiseuille equation can be rewritten as dP = R * Q, where R = (8 * mu * L) / (pi * r^4) is the hydraulic resistance. This mirrors Ohm's law (V = I * R) in electrical circuits: pressure drop is analogous to voltage, volumetric flow rate to current, and hydraulic resistance to electrical resistance. Like resistors, hydraulic resistors in series add linearly (R_total = R1 + R2 + ...) while parallel paths combine as 1/R_total = 1/R1 + 1/R2 + .... This analogy is exploited in microfluidics, anesthesiology, respiratory physiology (airway resistance), and renal physiology (glomerular filtration rate).
Velocity profile and average vs peak velocity
Under fully developed laminar Poiseuille flow, the velocity across the tube cross-section follows a paraboloid: v(r_pos) = v_max * (1 - (r_pos/r)^2), where r_pos is the radial position from the centerline. The centerline (maximum) velocity is exactly twice the cross-sectional average: v_max = 2 * v_avg = 2 * Q / (pi * r^2). This parabolic profile is only valid well downstream of the tube entrance; the entry length for fully developed flow is approximately L_entry = 0.06 * Re * diameter. Measurements near the inlet will deviate from Poiseuille predictions.
Assumptions, limits, and when the formula breaks down
Poiseuille's law holds only when: (1) the fluid is Newtonian (constant viscosity, unlike blood at low shear rates or polymer melts); (2) flow is laminar - Reynolds number Re = (2 * rho * Q) / (mu * pi * r) below about 2300 for circular pipes; (3) the tube is rigid, straight, and circular; (4) the flow is steady and fully developed (past the hydrodynamic entry length); (5) no-slip at the wall. In clinical applications, blood can be treated as Newtonian in large vessels but not in capillaries (Fahraeus-Lindqvist effect). Turbulent flow (Re > 4000) requires Darcy-Weisbach with a friction factor, not this equation.
Common fluid viscosities and typical Reynolds numbers
| Fluid | Viscosity (Pa·s) | Flow regime at Re < 2300 |
|---|---|---|
| Air | 1.81 x 10^-5 | Laminar up to ~0.34 m/s avg velocity |
| Water | 1.00 x 10^-3 | Laminar up to ~0.46 m/s avg velocity |
| Blood (whole) | 3.0 x 10^-3 | Laminar up to ~1.4 m/s avg velocity |
| Engine oil (SAE 10) | 3.0 x 10^-2 | Laminar up to ~14 m/s avg velocity |
| Glycerin | 1.412 | Typically laminar in most practical flows |
| Honey | 2 to 10 | Almost always laminar |
Approximate dynamic viscosity at 20 C and the Reynolds number threshold for laminar flow in a 5 mm radius tube with 100 Pa pressure drop over 0.1 m.
Frequently asked questions
Why does radius have such a large effect on flow rate?
Flow rate scales with the fourth power of the radius. This means a 10% increase in radius (a factor of 1.1) increases flow by 1.1^4 = 1.46, or 46%. Conversely, a 50% reduction in radius (stenosis or vasoconstriction) reduces flow to 0.5^4 = 1/16th of the original. In medicine, this explains why even mild arterial narrowing dramatically reduces organ perfusion, and why bronchospasm severely restricts airflow.
What is hydraulic resistance and how does it relate to Ohm's law?
Hydraulic resistance R = (8 * mu * L) / (pi * r^4) measures how much a tube resists flow for a given pressure. It is directly analogous to electrical resistance: pressure drop (dP) is like voltage, flow rate (Q) is like current, so dP = R * Q mirrors V = I * R. This means you can analyse branching pipe networks using the same series/parallel rules as electrical circuits.
When does Poiseuille's law NOT apply?
The law breaks down for turbulent flow (Re > ~2300-4000), non-Newtonian fluids such as blood in very small vessels, flows near the tube entrance before the velocity profile becomes parabolic, pulsatile or unsteady flows, curved or irregular tubes, and compressible gases at high velocities. If you get Re > 2300, switch to the Darcy-Weisbach equation with an appropriate friction factor.
How do I use this calculator to solve for tube radius?
Set the Solve for dropdown to Tube radius (r), then enter the target flow rate, desired pressure drop, tube length, and fluid viscosity. The calculator rearranges the Hagen-Poiseuille formula to r = ((8 * mu * L * Q) / (pi * dP))^(1/4) and returns the required radius. This is useful for pipe sizing in microfluidics or biomedical device design.
What is the Reynolds number and why does it matter here?
The Reynolds number Re = (2 * rho * Q) / (mu * pi * r) is a dimensionless ratio of inertial forces to viscous forces. Below Re ~2300 the flow is laminar and Poiseuille's law applies accurately. Between 2300 and 4000 the flow is transitional and unpredictable. Above 4000 the flow is turbulent and the Hagen-Poiseuille equation is invalid - use Darcy-Weisbach instead.
How is blood flow described by Poiseuille's law?
In large and medium arteries, blood behaves approximately as a Newtonian fluid and Poiseuille's law predicts flow well. Cardiac output, peripheral vascular resistance, and the effect of arterial stenosis are all analysed using these principles. In capillaries (diameter ~5-10 micrometers, comparable to red blood cell size), the Fahraeus-Lindqvist effect makes apparent viscosity lower than in bulk, so the simple Poiseuille formula overestimates resistance there.
What is the difference between average velocity and peak velocity?
For laminar Poiseuille flow, the velocity profile across the tube is a paraboloid. The peak velocity occurs at the centreline and is exactly twice the cross-sectional average: v_max = 2 * v_avg = 2 * Q / (pi * r^2). Measuring centreline velocity with a Doppler probe therefore slightly overestimates bulk flow unless you apply the factor of 0.5.