Poiseuilles Law Calculator - Hagen-Poiseuille Pipe Flow Solver

A poiseuilles law calculator is a fluid-mechanics tool that applies the Hagen-Poiseuille equation to laminar flow of a Newtonian fluid in a straight cylindrical pipe, returning volumetric flow rate, pressure drop, viscosity, radius, or length depending on the variable you mark as the unknown.

• • Pipe-flow sizing: Estimate how much fluid a given pipe radius and pressure drop can deliver.
• • Pressure-drop budgeting: Work backward from a target Q to the pressure drop the upstream system must supply.
• • Required-radius problems: Solve for the radius that meets a flow-rate target without exceeding an allowable dP.
• • Biomedical flow check: Estimate volumetric flow through a vessel of known radius and length for a textbook haemodynamics problem.

The Hagen-Poiseuille relation is Q = pi r^4 dP / (8 mu L). It says that the volumetric flow rate Q in a cylindrical pipe scales with the fourth power of the inner radius r and the pressure difference dP, and inversely with the dynamic viscosity mu and the pipe length L.

The calculator lets you supply four of the five Poiseuille variables plus fluid density, then pick which unknown to solve for. The result returns the chosen variable in its natural unit (m^3/s, Pa, Pa.s, or m) plus the average velocity, the implied Reynolds number, and a laminar / transitional / turbulent regime flag so you can confirm Poiseuille's law applies to your inputs.

Before trusting the result, confirm the flow stays in the laminar regime with the Reynolds number calculator, which uses the same density, viscosity, and pipe-radius inputs you already have.

The calculator rearranges the Hagen-Poiseuille equation for the variable you selected and reports the average velocity and Reynolds number implied by your inputs so you can confirm the flow is in the laminar regime where the law applies.

• Q: Volumetric flow rate, in cubic metres per second (m^3/s).
• r: Inner pipe radius, in metres (m).
• dP: Pressure drop between inlet and outlet, in pascals (Pa).
• mu: Dynamic viscosity of the fluid, in pascal-seconds (Pa.s).
• L: Pipe length between pressure measurements, in metres (m).
• rho: Fluid mass density, in kg/m^3. Used to compute the average velocity and Reynolds number.

The fourth-power radius term is why pipe-radius engineering is so sensitive: doubling r multiplies Q by sixteen at the same pressure drop. The average velocity v_avg = Q / (pi r^2) is the plug-flow speed, and Re = rho v_avg D / mu tells you whether the laminar assumption still holds.

According to Wikipedia, the Hagen-Poiseuille equation is the exact analytical solution of the Navier-Stokes equations for steady, laminar, incompressible, Newtonian flow in a straight cylindrical pipe with no-slip walls.

According to Wikipedia, The Hagen-Poiseuille equation Q = pi r^4 dP / (8 mu L) gives the volumetric flow rate of a Newtonian fluid in steady, laminar flow through a cylindrical pipe of radius r and length L.

As published by Wikipedia, Dynamic viscosity mu is the proportionality factor between shear stress and shear rate in a Newtonian fluid; its SI units are pascal-seconds (Pa.s).

For ideal inviscid flow the energy balance is given by the Bernoulli equation calculator, which adds the dynamic and elevation heads on top of the viscous pressure loss solved here.

Four ideas behind the poiseuilles law calculator that are worth understanding before you trust the numbers.

These four ideas reappear throughout fluid-mechanics homework and biomedical-engineering coursework. Knowing that radius is r^4 and viscosity is a material property is the key to avoiding common Poiseuille mistakes.

When the regime flag turns turbulent, the friction factor calculator gives the Darcy friction factor that augments Poiseuille's law with realistic pipe-wall shear losses.

Use the poiseuilles law calculator in five steps.

1. 1 Pick the unknown: Open the Solve For menu and choose the variable to return: Q, dP, mu, r, or L. The default is Q for a volumetric-flow-rate solve.
2. 2 Enter dP and radius: Type the pressure drop dP in pascals and the inner pipe radius r in metres. Most lab pipes fall between 1 and 1000 Pa of pressure drop and between 1 mm and 100 mm of radius.
3. 3 Enter fluid properties: Type mu in Pa.s (water at 20 C is 0.001; blood plasma at 37 C is about 0.0015) and rho in kg/m^3 (water 1000; blood about 1060).
4. 4 Enter pipe length: Type the straight length L of the pipe section between the two pressure measurements. Curved sections, fittings, and entrance effects should be added as equivalent length or handled with a separate loss term.
5. 5 Read the result and regime: The primary output shows the solved variable in its natural unit; the secondary outputs report the average velocity, pipe area, diameter, and Reynolds number so you can confirm the laminar assumption.

For a water pipe with dP = 100 Pa, r = 0.005 m, mu = 0.001 Pa.s, L = 1 m, and rho = 1000 kg/m^3, leave Solve For on the default Q and read the result. The calculator returns about 2.45e-5 m^3/s with v_avg ~ 0.31 m/s and Re ~ 3125 (transitional).

To cross-check the resulting average velocity against a free-stream or particle motion problem, plug the same speed into the kinematics motion calculator.

Practical reasons to use this poiseuilles law calculator instead of solving the Hagen-Poiseuille equation by hand.

• • One tool for all five unknowns: Switch the Solve For menu to rearrange for Q, dP, mu, r, or L without re-deriving the algebra each time.
• • Built-in regime check: The Reynolds number and laminar / transitional / turbulent flag are computed automatically so you can confirm Poiseuille's law applies.
• • Works for any Newtonian fluid: Set mu and rho to the actual values for water, oil, blood plasma, or air instead of hard-coding a single fluid.
• • Hand-check friendly precision: Six-decimal volumetric flow rate and four-decimal velocity match the precision expected in a fluid-mechanics class.
• • Quick textbook sanity check: Recreate worked examples in seconds and compare your handwritten answer to the calculator before turning in homework.
• • Connects to adjacent calculators: The same viscosity and pipe-radius inputs feed reynolds-number and bernoulli-equation solvers, so a laminar pipe problem becomes a one-stop study.

The calculator is intentionally narrow: it does one laminar Newtonian pipe problem well. For turbulent flow, non-circular ducts, non-Newtonian fluids, or fittings, use a friction-loss treatment on top of the Hagen-Poiseuille result.

The same pressure drop times volumetric flow rate that drives Poiseuille flow also reports hydraulic power through the work energy power calculator, giving you a quick pump-size check.

What changes the answer the poiseuilles law calculator returns, and what it cannot capture.

• • Poiseuille's law assumes steady, laminar, incompressible, Newtonian flow in a straight cylindrical pipe with no-slip walls. It does not capture turbulent dissipation, entrance effects, fittings, or pulsatile flow.
• • Solving for the pipe radius requires the numerator 8 mu L Q / (pi dP) to be positive. If the user inputs force that numerator to zero or negative, the calculator returns null.
• • The calculator uses a single mu and rho for the whole pipe, so temperature- or pressure-driven viscosity changes along the flow are not modelled. The result is an idealised laminar pipe answer.

According to Wikipedia, for flow in a pipe of diameter D, laminar flow occurs when the Reynolds number Re is below about 2300 and turbulent flow occurs when Re is above about 2900. That is why the calculator's regime flag is the easiest way to tell at a glance whether Poiseuille's law is the right model for your inputs.

According to Wikipedia, For flow in a pipe of diameter D, laminar flow occurs when Re < 2300 and turbulent flow occurs when Re > 2900.

To convert the dynamic viscosity mu factor between poise, centipoise, and the pascal-seconds used by this calculator, use the poise stokes converter first.