# Poiseuille's Law Calculator

Created by Gabriela Diaz
Last updated: Jun 30, 2022

With our Poiseuille's law calculator, you can easily determine the volumetric flow rate and resistance to flow of a fluid moving in a pipe.

This law, derived from the Darcy-Weisbach equation, describes the pressure drop experienced by a fluid that flows through a pipe. If you'd like to learn more about this subject, we highly recommend reading the accompanying text to find out:

• What is Poiseuille's law?;
• What is the relationship between the Darcy-Weisbach equation and Poiseuille's law equation?;
• How to calculate the volumetric flow rate; and
• The flow resistance equation from Poiseuille's law.

## What is Poiseuille's law? — About the Poiseuille's law or Hagen-Poiseuille equation

Poiseuille's law, also known as the Hagen-Poiseuille law or Hagen-Poiseuille equation, describes the pressure drop of an incompressible Newtonian fluid in laminar flow traveling across a cylindrical pipe of constant cross-sectional area. The pressure change, according to Poiseuille's law, is given by the expression:

$\small \Delta p = \cfrac{8 \, \mu \, L \, Q}{\pi \, r^4}$

where:

• $\Delta p$ ⁠— Pressure change in Pa or psi;
• $\mu$ — Dynamic viscosity of the fluid in Pa * s;
• $L$ ⁠— Lenght of the pipe in m or ft;
• $Q$ ⁠— Flow rate in m3/s or ft3/s; and
• $r$ ⁠— Radius of the pipe in m or ft.

This equation is derived from the Darcy-Weisbach formula for pressure loss — $ΔP=f \ L \ ρ \ V^2/2 \ D$. This formula yields the above formula for the specific conditions of an incompressible Newtonian fluid in laminar flow and a cylindrical pipe. If you'd like to go from Darcy-Weisbach to Poiseuille's law, remember that the friction factor in laminar flow is $f = Re/64$, and the Reynolds number equation should be the adjusted version for pipes $Re = \rho V D / \mu$.

Given the variables present in the Poiseuill'es law equation, this expression is also used for determining the[volumetric flow rate Q and flow resistance R. It is worth noting that $\Delta p$ and $Q$ are directly proportional. This makes sense since the greater the pressure difference, the greater the volumetric flow rate for the same pipe diameter. Remember that a flow moves from a high-pressure point to a lower-pressure point; otherwise, it will travel in the opposite direction.

Additionally, it's interesting to mention that this law has a wide range of applications that go from piping systems to blood vessels and the respiratory system. Being hemodynamics (the study of blood flow) the most popular application 💉Studying the blood flow with Poiseuille's law helps explain why constricted capillaries lead to higher blood pressure.

## Calculate flow rate with Poiseuille's law

To calculate the volumetric flow rate $Q$ using the Poiseuille's law, use the following expression:

$\small Q = \cfrac{\pi \, \Delta p \, r^4}{8 \, \mu \, L}$

where:

• $Q$ ⁠— Flow rate in m3/s or ft3/s;
• $\Delta p$ ⁠— Pressure change usually in Pa or psi;
• $r$ ⁠— Radius of the pipe in m or ft;
• $\mu$ — Dynamic viscosity of the fluid in Pa * s; and
• $L$ ⁠— Lenght of the pipe in m or ft.

## Calculate flow resistance with Poiseuille's law

There's an analogy between hydraulics and electricity. Poiseuilles' law is the equivalent to for electrical circuits$R = V/I$. In fluids, this resistance describes the difficulty that a fluid experiences when flowing through a pipe. The flow resistance equation is represented by the ratio of pressure change to flow rate:

$\small R = \cfrac{\Delta p}{Q}$

Where the pressure difference $\Delta p$ would be the equivalent to the electrical voltage $V$ and the flow rate $Q$ corresponds to the current $I$.

From Poiseuille's law, it's also possible to determine the resistance to flow $R$ using the following expression:

$\small R = \cfrac{8 \, \mu \, L}{\pi \, r^4}$

where:

• $R$ ⁠— Flow resistance in Pa * s/m3;
• $\Delta p$ ⁠— Pressure change usually in Pa or psi;
• $Q$ ⁠— Flow rate in m3/s or ft3/s;
• $\mu$ — Dynamic viscosity of the fluid in Pa * s;
• $L$ ⁠— Lenght of the pipe in m or ft;
• $r$ ⁠— Radius of the pipe in m or ft;
• $V$ ⁠— Electrical voltage for Ohm's law; and
• $I$ — Current for Ohm's law analogy.

This is the expression that we use in the Poiseuille's law calculator!

The Poiseuille equation for resistance shows that resistance is directly proportional to fluid viscosity and pipe length; the greater these, the higher the friction, and hence the greater the resistance to flow. Also, the diameter or radius of the pipe will affect the resistance. We can easily picture that the greater the diameter, the less resistance the flow will encounter, contrary to what happens in smaller diameter pipes.

## How to use the Poiseuille's law calculator

You'll see that determining the volumetric flow rate and flow resistance using Poiseulle's law calculator is quite straightforward:

1. Begin by entering the fluid's dynamic viscosity in Pa * s.
2. Next, in the Radius of the pipe (r) field, input the radius of the pipe. This is equivalent to half of the diameter.
3. The next dimension you need to enter is the length of the pipe.
4. With this information, the calculator can determine the fluid's volumetric flow rate.
5. If you also want to calculate the volumetric flow rate, simply enter the pressure difference in the Pressure change (Δp) row.
6. The calculator will display the pressure change.

🙋 Use the Advanced mode of the calculator to determine the fluid's pressure change from the initial and end pressures.

Gabriela Diaz
Dynamic viscosity (μ)
Pa * s
in
Length of the pipe (l)
in
Pressure change (Δp)
Pa
Volumetric flow rate (Q)
/s
Resistance (R)
Pa * s /
People also viewed…

### Friction factor

This Darcy friction factor calculator finds the value of the friction factor given the Reynolds number and the relative roughness.

### Poise-stokes converter

This Poise-Stokes converter converts between kinematic and dynamic viscosity for a fluid of known density.