Hagen-Poiseuille Equation | Equation, Derivation and Rearrangements

Core idea

Overview

This equation describes laminar flow conditions where the fluid moves in parallel layers with no disruption between them. It relates the pressure drop across the length of a pipe to the radius of the pipe and the viscosity of the fluid. The result provides the rate at which the fluid volume passes through the cross-section per unit time.

When to use: Use this equation when analyzing laminar flow of a viscous, incompressible Newtonian fluid through a pipe with a constant circular cross-section.

Why it matters: It is essential for understanding blood flow in the circulatory system, designing lubrication systems, and analyzing flow in microfluidic devices.

Symbols

Variables

Q = Volumetric Flow Rate, R = Pipe Radius, $μ$ = Dynamic Viscosity, $P$ _1 = Inlet Pressure, $P$ _2 = Outlet Pressure

Q

Volumetric Flow Rate

m^{3} / s

R

Pipe Radius

m

μ

Dynamic Viscosity

P a \cdot s

P_{1}

Inlet Pressure

Pa

P_{2}

Outlet Pressure

Pa

Δ P

Pressure Difference

Pa

L

Pipe Length

m

Walkthrough

Derivation

Derivation of Hagen-Poiseuille Equation

This derivation determines the volumetric flow rate of a Newtonian fluid through a cylindrical pipe by integrating the velocity profile derived from the Navier-Stokes equations.

The fluid is incompressible and Newtonian.
The flow is laminar, steady, and fully developed.
The pipe is a straight, rigid cylinder with a constant circular cross-section.
There is no-slip at the pipe walls.

1

Force Balance on a Fluid Element

We consider a cylindrical fluid element of radius r and length L. For steady flow, the pressure force pushing the fluid must be balanced by the shear stress force acting on the surface of the element.

τ (2 π r L) = (P_{1} - P_{2}) π r^{2}

Note: This assumes the pressure gradient is constant along the length of the pipe.

2

Expressing Shear Stress

Using Newton's law of viscosity, we relate the shear stress to the velocity gradient. Rearranging the force balance equation allows us to solve for the velocity gradient in terms of the pressure drop.

τ = - μ \frac{d v}{d r} = \frac{( P _{1} - P _{2} ) r}{2 L}

Note: The negative sign indicates that velocity decreases as the radius increases.

3

Integrating for Velocity Profile

Integrating the velocity gradient with respect to r and applying the no-slip boundary condition (v=0 at r=R) yields the parabolic velocity profile.

v (r) = \frac{( P _{1} - P _{2} )}{4 μL} (R^{2} - r^{2})

Note: This shows that the velocity is maximum at the center of the pipe (r=0).

4

Calculating Volumetric Flow Rate

The total volumetric flow rate Q is found by integrating the velocity profile over the entire cross-sectional area of the pipe using cylindrical coordinates.

Q = \int_{0}^{R} v (r) 2 π r d r

Note: The term 2πr dr represents the area of a thin ring at radius r.

5

Final Integration

Performing the integration results in the final Hagen-Poiseuille equation, relating flow rate to pipe geometry, fluid viscosity, and pressure drop.

Q = \int_{0}^{R} \frac{( P _{1} - P _{2} )}{4 μL} (R^{2} - r^{2}) 2 π r d r = \frac{R ^{4} π}{8 μ} (\frac{P _{1} - P _{2}}{L})

Note: Note the strong dependence on the pipe radius ( $R^{4}$ ).

Result

Q = \int_{0}^{R} \frac{( P _{1} - P _{2} )}{4 μL} (R^{2} - r^{2}) 2 π r d r = \frac{R ^{4} π}{8 μ} (\frac{P _{1} - P _{2}}{L})

Source: Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport Phenomena.

Free formulas

Rearrangements

Solve for $μ = \frac{R ^{4} π ( P _{1} - P _{2} )}{8 Q L}$

Dynamic Viscosity

μ = \frac{R ^{4} π ( P _{1} - P _{2} )}{8 Q L}

Rearrange the Hagen-Poiseuille equation to solve for the fluid's dynamic viscosity.

Difficulty: 3/5

Solve for $Δ P = \frac{8 Q μL}{R ^{4} π}$

Pressure Difference

Δ P = \frac{8 Q μL}{R ^{4} π}

Rearrange the Hagen-Poiseuille equation to find the pressure difference (ΔP = P₁ - P₂) required for a specific flow.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph shows a linear relationship between volumetric flow rate (Q) and pressure difference ($\Delta\mathcal{P}$). As the pressure difference increases, the volumetric flow rate increases directly and proportionally. For a student, this means that doubling the pressure difference will double the flow rate, assuming other factors remain constant. The most important feature is this direct proportionality, clearly illustrating how pressure drives fluid flow in a pipe.

Graph type: linear

Why it behaves this way

Intuition

Imagine a fluid moving through a long, straight straw. The fluid near the center moves the fastest, while the fluid touching the walls is stationary due to friction (the no-slip condition). This creates a parabolic velocity profile where the 'core' of the liquid slides through a sleeve of slower-moving layers. The total volume squeezed through per second depends heavily on how wide the straw is and how 'thick' or sticky the fluid feels.

Q

Volumetric Flow Rate

The total volume of fluid passing through a cross-section of the pipe per unit of time; essentially how fast the 'bucket' fills up.

R

Pipe Radius

The distance from the center to the wall. Because it is raised to the 4th power, doubling the radius increases the flow by 16 times, making it the most sensitive factor in the equation.

µ

Dynamic Viscosity

The 'internal friction' or thickness of the fluid. High viscosity (like honey) resists flow more than low viscosity (like water).

P1 - P2

Pressure Drop

The 'push' or driving force. Fluid only flows if there is a difference in pressure to overcome the resistive viscous forces.

L

Pipe Length

The distance the fluid must travel. Longer pipes create more total friction against the walls, which slows down the flow for a given pressure.

Signs and relationships

R^4: Positive and exponential; it signifies that widening the pipe dramatically reduces the resistance to flow by moving more fluid away from the high-friction walls.
(P1 - P2): Positive; flow always moves from high pressure (P1) to low pressure (P2). A larger difference results in a higher velocity.
8µL in the denominator: Inverse relationship; increasing the 'stickiness' (viscosity) or the distance (length) adds to the total resistance, thereby decreasing the flow rate.

Free study cues

Insight

Canonical usage

The Hagen-Poiseuille equation is used to calculate the volumetric flow rate (Q) of a Newtonian fluid in a pipe, given the pipe's dimensions, fluid viscosity, and pressure drop.

Common confusion

Students often confuse dynamic viscosity (Pa·s) with kinematic viscosity (m²/s), which is related by density (ν = μ/ρ).

Dimension note

This equation does not inherently produce a dimensionless quantity; all variables have physical units.

Unit systems

$Q$ m^3/s - Represents the volume of fluid passing a point per unit time.

$R$ m - The internal radius of the pipe.

muPa·s - Dynamic viscosity of the fluid. Also known as absolute viscosity.

$P 1$ Pa - Pressure at the inlet of the pipe.

$P 2$ Pa - Pressure at the outlet of the pipe.

deltaPPa - The difference between inlet and outlet pressure (P1 - P2).

$L$ m - The length of the pipe.

One free problem

Practice Problem

Calculate the flow rate Q ( $m^{3}$ /s) for a fluid with dynamic viscosity 0.001 Pa·s, a pipe radius of 0.01 m, a length of 2 m, and a pressure difference of 100 Pa.

Pipe Radius0.01 m

Dynamic Viscosity0.001 Pa·s

Inlet Pressure100 Pa

Outlet Pressure0 Pa

Pipe Length2 m

Solve for: $Q$

Hint: Ensure the pressure difference is calculated as (P1 - P2) and units are in SI.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In the volume of blood flowing through a specific vessel segment to assess cardiovascular function, Hagen-Poiseuille Equation is used to calculate Volumetric Flow Rate from Pipe Radius, Dynamic Viscosity, and Inlet Pressure. The result matters because it helps turn a changing quantity into a total amount such as area, distance, volume, work, or cost.

Study smarter

Tips

Ensure the flow is laminar by checking the Reynolds number.
Ensure the pipe is sufficiently long relative to its diameter to ignore entrance effects.
Check that units for pressure, length, and radius are consistent.

Avoid these traps

Common Mistakes

Applying the equation to turbulent flow conditions, where it is no longer valid.
Confusing the radius of the pipe with the diameter.
Failing to convert units for viscosity, resulting in incorrect pressure or flow values.

Common questions

Frequently Asked Questions

This derivation determines the volumetric flow rate of a Newtonian fluid through a cylindrical pipe by integrating the velocity profile derived from the Navier-Stokes equations.

Use this equation when analyzing laminar flow of a viscous, incompressible Newtonian fluid through a pipe with a constant circular cross-section.

It is essential for understanding blood flow in the circulatory system, designing lubrication systems, and analyzing flow in microfluidic devices.

Applying the equation to turbulent flow conditions, where it is no longer valid. Confusing the radius of the pipe with the diameter. Failing to convert units for viscosity, resulting in incorrect pressure or flow values.

In the volume of blood flowing through a specific vessel segment to assess cardiovascular function, Hagen-Poiseuille Equation is used to calculate Volumetric Flow Rate from Pipe Radius, Dynamic Viscosity, and Inlet Pressure. The result matters because it helps turn a changing quantity into a total amount such as area, distance, volume, work, or cost.

Ensure the flow is laminar by checking the Reynolds number. Ensure the pipe is sufficiently long relative to its diameter to ignore entrance effects. Check that units for pressure, length, and radius are consistent.

References

Sources

White, F. M. (2016). Fluid Mechanics. McGraw-Hill Education.
Munson, B. R., Young, D. F., & Okiishi, T. H. (2013). Fundamentals of Fluid Mechanics. Wiley.
NIST CODATA
IUPAC Gold Book
Wikipedia: Hagen–Poiseuille equation
White, Frank M. Fluid Mechanics. 8th ed., McGraw-Hill Education, 2016.
Britannica - Hagen-Poiseuille equation
Wikipedia - Hagen–Poiseuille equation