Physics calculator

Hagen–Poiseuille Calculator

Compute laminar circular-pipe flow Q = π Δp D⁴ / (128 μ L). Runs locally. Not Darcy f, not Colebrook, and not Q=Av.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · 17/17 tests · Production surface contract 3/3 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
Model: Laminar circular-pipe discharge from Hagen–Poiseuille. Invert Δp, D, or μ.
Scope: Laminar Hagen–Poiseuille: Q = π Δp D⁴ / (128 μ L); circular, fully developed, Newtonian
Verification: Engine tested · 17/17 tests · Production surface contract 3/3 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: ISO 80000-4 — Mechanics — Dynamic viscosity
White, F.M. Fluid Mechanics — Hagen–Poiseuille flow
Sources
Evidence: 2 golden · 10 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production: Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status PASS (0 stale; 163 CURRENT) @ 2026-09-17T09:03:11.906Z
Semantic contract: PASS

Formulas

Core equations used by this calculator.

DischargeQ = π Δp D⁴ / (128 μ L)

Pressure dropΔp = 128 μ L Q / (π D⁴)

Mean speedv = 4Q / (π D²)

iSI circular pipe. This page is the laminar closed form. It does not take Darcy f, does not solve Colebrook, and is not Q=Av.

How to use

Enter Δp, D, μ, L

Pressure drop in pascals, inner diameter and length in metres, dynamic viscosity in Pa·s.

Read Q

Q = π Δp D⁴ / (128 μ L). Invert Δp, D, or μ on the other tabs.

Check the identity

Not Darcy hf=f L v²/(2gD). Not Colebrook. Not volumetric Q=Av.

Example calculations

Common configurations with formula and result.

2 mm capillary

Δp=1000 Pa, D=0.002 m, L=1 m, μ=0.001 Pa·s

Q = π × 1000 × (0.002)⁴ / (128 × 0.001 × 1)

Q = π × 1.25×10⁻⁷ ≈ 3.927×10⁻⁷ m³/s

Invert Δp

Same Q, D, L, μ

Δp = 128 μ L Q / (π D⁴)

Δp = 1000 Pa

Hagen–Poiseuille calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 17/17 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: Laminar Hagen–Poiseuille for a circular pipe: Q = π Δp D⁴ / (128 μ L). Fully developed, Newtonian, no-slip. Mean speed v = 4Q/(π D²) is reported.
What it calculates: Laminar circular-pipe discharge from Hagen–Poiseuille. Invert Δp, D, or μ.
Inputs: mode?
dP|Q
D
mu
L
Outputs: Q
dP
D
L
mu
v
mode
model
Formula: Q = π Δp D⁴ / (128 μ L)
Assumptions: Laminar Hagen–Poiseuille: Q = π Δp D⁴ / (128 μ L); circular, fully developed, Newtonian
Mean speed v = 4Q/(π D²) is reported, not an input
Not Darcy f, not Colebrook, not Q=Av, not Manning
Units: SI; Q in m³/s; Δp in Pa; D, L in m; μ in Pa·s
Boundary conditions: missing Δp, Q, D, μ, or L when required → MISSING_REQUIRED_INPUT
Δp, Q, D, μ, or L ≤ 0 → VALUE_MUST_BE_POSITIVE
f/Re/ε, n, A/v, K/Cd → MIXED_INPUT_ENCODING
darcy, colebrook, reynolds, or manning as mode → INVALID_MODE
Example: dP=1000 D=0.002 L=1 mu=0.001 → Q=π×1.25e-7
Validation cases: 3 published on this page · 17/17 tests · Production surface contract 3/3 · View evidence
dP=1000 D=0.002 L=1 mu=0.001 → Q=π×1.25e-7
mode=pressure Q=π×1.25e-7 D=0.002 L=1 mu=0.001 → dP=1000
dP=1000 D=0.002 L=1 mu=0.001 f=0.03 → MIXED_INPUT_ENCODING
Sources: ISO 80000-4 — Mechanics — Dynamic viscosity
Supports: μ has dimension pressure × time; Q volume/time; D and L length; Δp pressure
White, F.M. Fluid Mechanics — Hagen–Poiseuille flow
Supports: Fully developed laminar flow in a circular tube Q = π Δp D⁴ / (128 μ L)
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute laminar circular-pipe discharge from the Hagen–Poiseuille law.

Supported and not supported

Supported — Q from Δp, D, μ, L · invert Δp · invert D · invert μ · API physics.fluid.poiseuille

Not supported — Darcy f, Colebrook ε/Re, Q=Av, Manning n

Agent / API notes

Capability id: physics.fluid.poiseuille · tool id: hagen-poiseuille · pin 1.0.0.

{ "dP": 1000, "D": 0.002, "L": 1, "mu": 0.001 }

Modes: flow (default), pressure, diameter, viscosity. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/hagen-poiseuille. There is no /calc/fluid.

Other calculators in this family: Darcy–Weisbach, Colebrook, Reynolds number, Flow rate .

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Frequently asked questions

Key distinctions behind the calculation.

Is this Darcy–Weisbach?

No. hf = f L v² / (2 g D) is /calc/physics/darcy-weisbach and needs a given f. Sending f is MIXED_INPUT_ENCODING.

Is this Colebrook?

No. Turbulent-pipe f from Re and ε/D is /calc/physics/colebrook. Sending Re, ε, or rel_roughness is MIXED_INPUT_ENCODING.

Is this Q = A·v?

No. Volumetric flow from area and velocity is /calc/physics/flow-rate. Sending A or v is MIXED_INPUT_ENCODING. Mean speed is reported from Q and D.

Is this Manning?

No. Open-channel Q from n is /calc/physics/manning-formula. Sending n is MIXED_INPUT_ENCODING.

Where does this run?

Locally in the browser by default. REST and MCP call the same fluid-engine hagenPoiseuille wrapper.