Engineering · Fluid mechanics

Hagen–Poiseuille law for laminar pipe flow

For steady laminar Newtonian flow in a long straight circular tube, Poiseuille relates volume flow to pressure drop and viscosity.

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Formula

Q = \frac{π r ^{4} Δ p}{8 η L}

Q in m³/s; r and L in m; Δp in Pa; η in Pa·s

Variables & units

Variables & units
Symbol	Meaning	Unit
$Q$	Volume flow through the tube	m³/s
$r$	Inside tube radius	m
$Δp$	Pressure drop along the tube	Pa
$η$	Dynamic viscosity	Pa·s
$L$	Tube length	m

LaTeX

Q=\frac{\pi r^4\Delta p}{8\eta L}

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Derivation & background

The solution assumes steady, fully developed laminar flow of an incompressible Newtonian liquid through a straight circular tube with no-slip at the wall. Radius enters to the fourth power: halving it reduces flow to one-sixteenth at otherwise fixed conditions. The simple relation does not apply to turbulent flow, noncircular ducts or entrance regions.

Exam blueprint

Validity range

Derivation steps

For steady laminar Newtonian flow in a long straight circular tube, Poiseuille relates volume flow to pressure drop and viscosity.

1State the model assumptions and list quantities with units.
2Substitute into Q = πr⁴Δp/(8ηL); convert quantities to SI first.
3For water with η=1.0 mPa·s, r=1 mm, L=0.20 m and Δp=100 Pa, Q≈1.96×10⁻⁷ m³/s.

Rearrangements

Base equation for the target quantity

Q=\frac{\pi r^4\Delta p}{8\eta L}

Preserve signs, boundary conditions and units when rearranging.

Task variant

Work through the numerical example for “Hagen–Poiseuille law for laminar pipe flow” and check its unit and magnitude.

For water with η=1.0 mPa·s, r=1 mm, L=0.20 m and Δp=100 Pa, Q≈1.96×10⁻⁷ m³/s.

Common mistakes

Substituting a related quantity that shares the same symbol.

Use variable definitions and check: Q in m³/s; r and L in m; Δp in Pa; η in Pa·s.

Exam context

Analyze laminar pressure-driven flow in capillaries and narrow tubes.

Formula cluster

Fluid mechanics

For steady laminar Newtonian flow in a long straight circular tube, Poiseuille relates volume flow to pressure drop and viscosity.

Re = ρvL/η = vL/ν

Reynolds number

Q = A v̄; A₁v̄₁ = A₂v̄₂

Volume flow rate and continuity for incompressible flow

hf = fD(L/D)(v̄²/2g)

Darcy–Weisbach equation for pipe-friction loss

Worked example

For water with η=1.0 mPa·s, r=1 mm, L=0.20 m and Δp=100 Pa, Q≈1.96×10⁻⁷ m³/s.

Applications

Analyze laminar pressure-driven flow in capillaries and narrow tubes.

How do you calculate with Hagen–Poiseuille law for laminar pipe flow?

Here is how to work through a typical Hagen–Poiseuille law for laminar pipe flow (Q = πr⁴Δp/(8ηL)) task step by step:

1
Task
Work through the numerical example for “Hagen–Poiseuille law for laminar pipe flow” and check its unit and magnitude.
Solution path
For water with η=1.0 mPa·s, r=1 mm, L=0.20 m and Δp=100 Pa, Q≈1.96×10⁻⁷ m³/s.

Frequently asked questions about Hagen–Poiseuille law for laminar pipe flow

Why does Q depend on r⁴?+

The parabolic laminar pipe velocity profile is set by shear and the no-slip boundary condition. Integrating that profile over the circular area produces the fourth power of radius. Halving radius cuts flow to one-sixteenth if pressure gradient and other quantities stay fixed. Poiseuille flow assumes a Newtonian incompressible fluid, steady fully developed laminar motion and a long straight circular tube with no slip. Radius enters to the fourth power, so small radius errors matter greatly. The relation does not apply to turbulent or entrance-region flow.

How can I tell whether the flow is laminar?+

First compute Reynolds number using pipe diameter as characteristic length and appropriate fluid properties. Poiseuille requires laminar, fully developed pipe flow; one transition threshold is not universal because of inlet, roughness and disturbances. Check geometry and boundary conditions too. Poiseuille flow assumes a Newtonian incompressible fluid, steady fully developed laminar motion and a long straight circular tube with no slip. Radius enters to the fourth power, so small radius errors matter greatly. The relation does not apply to turbulent or entrance-region flow.

Why does flow follow the pressure drop?+

In the equation, Q is proportional to Δp when Δp means inlet pressure minus outlet pressure. Positive flow then goes from higher to lower pressure. Reverse the pressure-difference definition and Q changes sign. Define orientation before calculating. Poiseuille flow assumes a Newtonian incompressible fluid, steady fully developed laminar motion and a long straight circular tube with no slip. Radius enters to the fourth power, so small radius errors matter greatly. The relation does not apply to turbulent or entrance-region flow.

How do I check the result and units for Hagen–Poiseuille law for laminar pipe flow?+

Write the quantity and unit from the variable list next to every value: Q in m³/s; r and L in m; Δp in Pa; η in Pa·s. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For water with η=1.0 mPa·s, r=1 mm, L=0.20 m and Δp=100 Pa, Q≈1.96×10⁻⁷ m³/s. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Poiseuille flow assumes a Newtonian incompressible fluid, steady fully developed laminar motion and a long straight circular tube with no slip. Radius enters to the fourth power, so small radius errors matter greatly. The relation does not apply to turbulent or entrance-region flow.

Which model limit matters for Hagen–Poiseuille law for laminar pipe flow?+

The background states this starting condition: The solution assumes steady, fully developed laminar flow of an incompressible Newtonian liquid through a straight circular tube. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Poiseuille flow assumes a Newtonian incompressible fluid, steady fully developed laminar motion and a long straight circular tube with no slip. Radius enters to the fourth power, so small radius errors matter greatly. The relation does not apply to turbulent or entrance-region flow.

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]OpenStax, University Physics Volume 1, §14.7 “Viscosity and Turbulence”, laminar tube flow.

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Hagen–Poiseuille law for laminar pipe flow

For steady laminar Newtonian flow in a long straight circular tube, Poiseuille relates volume flow to pressure drop and viscosity.

AdvancedUniversity/collegeBachelor foundationsSubject specialization

Free to browse · flashcards or a multiple-choice quiz · import from Essential

Symbol

Meaning

Unit

Q

Volume flow through the tube

m³/s

r

Inside tube radius

Δp

Pressure drop along the tube

η

Dynamic viscosity

Pa·s

L

Tube length

Derivation & background

Exam blueprint

Validity range

Derivation steps

For steady laminar Newtonian flow in a long straight circular tube, Poiseuille relates volume flow to pressure drop and viscosity.

1State the model assumptions and list quantities with units.
2Substitute into Q = πr⁴Δp/(8ηL); convert quantities to SI first.
3For water with η=1.0 mPa·s, r=1 mm, L=0.20 m and Δp=100 Pa, Q≈1.96×10⁻⁷ m³/s.

How do you calculate with Hagen–Poiseuille law for laminar pipe flow?

Here is how to work through a typical Hagen–Poiseuille law for laminar pipe flow (Q = πr⁴Δp/(8ηL)) task step by step:

Task

Work through the numerical example for “Hagen–Poiseuille law for laminar pipe flow” and check its unit and magnitude.

Solution path

For water with η=1.0 mPa·s, r=1 mm, L=0.20 m and Δp=100 Pa, Q≈1.96×10⁻⁷ m³/s.

Frequently asked questions about Hagen–Poiseuille law for laminar pipe flow

Why does Q depend on r⁴?+

How can I tell whether the flow is laminar?+

Why does flow follow the pressure drop?+

How do I check the result and units for Hagen–Poiseuille law for laminar pipe flow?+

Which model limit matters for Hagen–Poiseuille law for laminar pipe flow?+

Retain Hagen–Poiseuille law for laminar pipe flow for exams

Create a curated FSRS exam set for Q = πr⁴Δp/(8ηL): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential