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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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Q = π Δp D⁴ / (128 μ L), plus pressure, diameter, and viscosity solves, + O3 mpmath tabulated flow.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Q = π Δp D⁴ / (128 μ L), plus pressure, diameter, and viscosity solves, + O3 mpmath tabulated flow.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (flow, solve, alias, awkward, invalid-domain) · Matrix
Deferred
Not Darcy f, not Colebrook, and not Q=Av.
Numerical scope
O2: Hagen–Poiseuille vs a separate-module identity (≤2 ULP). Not Darcy f, not Colebrook, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated Hagen–Poiseuille vectors. Not Darcy f, not Colebrook, and not Q=Av.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
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 · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Q = π Δp D⁴ / (128 μ L), plus pressure, diameter, and viscosity solves, + O3 mpmath tabulated flow.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3
Verification revision
2026-09-26.o2-o3 · 1/1 property · digest 1b4d56fc52d7
Legacy regression
17/17 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
4/4 golden · 9/9 CVP boundary · 10/10 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 5/5 CVP contract · Manifest
Sources
  • ISO 80000-4 — Mechanics — Dynamic viscosity
  • White, F.M. Fluid Mechanics — Hagen–Poiseuille flow
Sources
Evidence
2 legacy golden · 10 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 10/10 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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

1

Enter Δp, D, μ, L

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

2

Read Q

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

3

Check the identity

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

Example calculations

Common configurations with formula and result.

ϟ

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
ϟ

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

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 5/5 · 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: Alternate Depth Calculator, Bernoulli Equation Calculator, Broad-Crested Weir Calculator, Cavitation Number Calculator, Chezy Formula Calculator, Colebrook Friction Factor Calculator, Concentric Reducer Calculator, Critical Depth Calculator . Explore all Fluid Mechanics.

→ Alternate Depth Calculator Compute both rectangular depths at the same specific energy from E, Q, and width b. Runs locally. Not specific energy from y+v; not specific force M; not hydraulic jump y2; not critical depth yc as the product. › → Bernoulli Equation Calculator Incompressible Bernoulli along a streamline: solve P2, v2, or h2. Runs locally in your browser. › → Broad-Crested Weir Calculator Compute rectangular broad-crested weir discharge Q = C_d (2/3)^{3/2} b √g H^{3/2}. Default C_d=0.85. Runs locally. Not a thin-plate weir, not Manning, not Q=Av, and not an orifice. › → Cavitation Number Calculator Compute the cavitation number σ = (p − p_v) / (½ ρ v²). Runs locally. Not NPSH, not Thoma NPSH/H, not Weber, and not Euler Eu=Δp/(ρ v²). › → Chezy Formula Calculator Compute open-channel mean velocity V = C √(R S). Runs locally. Not Manning n, not Q=Av, not Hazen–Williams, and not Darcy. › → Colebrook Friction Factor Calculator Solve the Colebrook–White equation for Darcy friction factor f from Re and ε/D. Turbulent pipes Re≥4000. Runs locally. Not laminar 64/Re and not a Moody-chart lookup. › → Concentric Reducer Calculator Compute concentric-reducer loss h = K v₂² / (2g) with K = 0.5 (1 − A₂/A₁) referenced to the downstream speed. Runs locally. Not sudden expansion, not gradual contraction α, not vena-contracta C_c, not catalog K, and not Q=Av. › → Critical Depth Calculator Compute rectangular critical depth yc = (q²/g)^{1/3}. Runs locally. Not a Froude wrap, not specific energy, not a hydraulic jump, and not Q=Av. ›

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.