# Poiseuille's Law Calculator

Created by Łucja Zaborowska, MD, PhD candidate
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Dec 09, 2022

The Hagen-Poiseuille's law calculator is a multi-task tool for computing not only the flow rate of laminar flow in a long, cylindrical pipe of constant cross sectional, but also the fluid/airway resistance and the pressure change.

Feeling confused? 🤯

Move onto our article below for a simple explanation of Hagen-Poiseuille's equation in biophysics and some useful examples, such as the water pipe flow velocity profile or the blood flow in blood vessels.

## What is Poiseuille's equation? (also Hagen-Poiseuille equation)

Hagen-Poiseuille law is a simple formula that we use in fluid dynamics calculations.

We already mentioned that this equation describes the laminar flow in a cylindrical container - in simple words, try to imagine a straight pipe with water flowing through it. 🚰

The Poiseuille's law equation (Hagen-Poiseuille law) describes how much water flows though the pipe in one second, depending on:

• The fluid's viscosity (thickness of the liquid, its ability to move against the friction of the pipe's walls);

• The pipe's length;

• The pipe's radius (or diameter); and

• The difference in pressure between the beginning and the end of the pipe. Our differential pressure calculator can help you find this value for some selected systems.

## Poiseuille flow equation and its applicability

Since we already know what Poiseuille's equation is, it's time to find out when we have to use it.

You guessed right - the Poiseuille's law can be used in plumbing, since pipes are involved (flow resistance equation). Humans, however, are indeed creative creatures:

• You can use this equation when estimating blood flow in blood vessels, and to describe the different processes in the human body that involve fluids (in that case the Poiseuille's law equation acts as a blood flow equation);

• We can use it to estimate the airflow resistance in human airways. This is incredibly important in the research on asthma and ;

• We also use Poiseuille's flow equation to describe the work of the kidney and the pressure in tubules and glomeruli;

• Poiseulle's law equation was essential in the creation of artificial kidney and hemodialysis machines; and

• We can also use the Hagen-Poiseuille equation in engine design 🚀

Would you like to know more? Check one of our flowing calculators:

## How to use Poiseuille's law calculator?

Our Poiseuille's equation calculator requires only 4 simple steps (or 3 if you want to compute flow resistance):

1. Enter the dynamic viscosity ($\mu$):

It's basic unit is $\mathrm{Pa \cdot s}$. For different pressures units, check the pressure converter.

1. Enter the radius of the pipe - it's equal to half of its diameter.

2. Enter the length of the pipe.

3. Find the pressure change (only required for the flow rate).

If you don't know the pressure change ($Δp$), click the advanced mode button and enter your initial and final pressures, or subtract the final pressure ($p_2$) from the initial pressure ($p_1$) yourself, following the formula below:

Δp = p_1-p_2

💡 Our calculator allows you to enter whole equations into the blank fields. Try to enter e.g. $10 - 5$. We will calculate everything for you automatically.

The results of the Poiseuille's law calculator will include both the flow rate and the resistance.

## Flow rate - equation & example

Here's the Poiseuille flow equation:

$Q = \frac{\pi \cdot Δp \cdot r^4}{8\cdot \mu\cdot l}$

where:

• $Q$ - Flow rate ($\mathrm{m^3/s}$);
• $π$ - Our famous and beloved constant pi, equals to $3.14159..$;
• $μ$ - Dynamic viscosity ($\mathrm{Pa \cdot s}$);
• $r$ - Radius of the pipe;
• $l$ - Length of the pipe; and
• $Δp$ - Pressure change ($\mathrm{Pa}$).

We'd like to discover the change in blood flow when the blood vessel contract using the blood flow equation (a specific case of the Poiseuille's law equation).

This situation is very straightforward to imagine - you're relaxing in a hot, Finnish sauna, and you know you'll have to leave and walk out onto the freezing snow. Your skin's blood vessels will contract - but how will it affect your skin's blood flow?

🙋 In some situations, for example with turbulent fluids, Poiseuille's law can be replaced with the more empirical Darcy-Weisbach equation: we detailed it at the Darcy-Weisbach calculator.

💡 So how does the body decrease the blood vessel radius? Blood vessels are connected only to the sympathetic nervous system - its fibers cause the muscles that encircle the vessels to contract. This way your body can send the blood flow to more important organs, such as the brain or heart.

What do we know?

• Our original vessels' radius was $1\ \mathrm{mm} = 0.001\ \mathrm{m}$;
• The pressure at the beginning of the vessels system was equal to $120\ \mathrm{mmHg} = 15998.7\ \mathrm{Pa}$;
• The pressure at the end was equal to $80\ \mathrm{mmHg} = 10665.8\ \mathrm{Pa}$;
• Our dynamic viscosity is $0.005\ \mathrm{Pa \cdot s}$;
• Total length of our vessels is $2\ \mathrm{m}$; and
• Our vessels contracted by $3/4$ - they're $1/4$ of their original size.

Let's create the original blood flow equation:

$\begin{split} Q &= \frac{3.14\cdot (15998.7-10665.8\ \mathrm{Pa})\cdot(0.001\ \mathrm{m})^4}{8\cdot 0.005\ \mathrm{Pa\cdot s}\cdot 2\ \mathrm{m}}\\ &= \frac{3.14 \cdot 5332.9 \cdot (0.001 m)^4}{0.08\ \mathrm{m}}\\ &=\frac{3.14 \cdot 5332.9 \cdot 0.000000000001 m^3}{0.08}\\ &= 0.00000020942\ \mathrm{m^3/s} = 209.42\ \mathrm{mm^3/s} \end{split}$

So how does the blood flow change with the radius being 1/4 of its original size?.

$r_2 = \frac{1}{4}\cdot 0.0001\ \mathrm{m} = 0.000025\ \mathrm{m}$

Then:

$\begin{split} Q_2 & = \frac{3.14 \cdot 5332.9 \cdot (0.000025 m)^4}{80\ \mathrm{m}}\\ &= \frac{3.14 \cdot 5332.9 \cdot 3.90625\times10^{-19}\ \mathrm{m^3}}{80})\\ & = 0.000000000000818\ \mathrm{m^3/s} \\ &= 0.000818\ \mathrm{mm^3/s} \end{split}$

Let's calculate the ratio of $Q$ and $Q_2$ :

$0.20942\ \mathrm{mm^3/s} : 0.000818\ \mathrm{mm^3/s}$

That is equal to:

$256:1$

The blood flow will change by a factor $256$ if the radius reduces by a factor of $4$.

Why? Radius in our equation is taken to the power of $4$, so we could also use the following formula: $4^4= 256$.

## Flow resistance - equation & examples

Here's the Hagen-Poiseuille equation for flow resistance that we use in this Poiseuille's law calculator:

$R = \frac{8\cdot \mu\cdot l}{\pi\cdot r^4}$

Where:

• $R$ - Resistance ($\mathrm{Pa \cdot s/ m^3}$);
• $π$ - Pi again;
• $μ$ - Dynamic viscosity ($\mathrm{Pa\cdot s}$);
• $r$ - Radius of the pipe; and
• $l$ - Length of the pipe.

Now when we know what Poiseuille's equation is, let's calculate the radius of an airway with a flow resistance of $0.0315\ \mathrm{Pa \cdot s/ m^3}$, length of $0.5\ \mathrm{m}$, and viscosity of $2\ \mathrm{Pa \cdot s}$.

$\begin{split} 0.315\ \mathrm{Pa\cdot s/m^3} &= \frac{8\cdot 2\ \mathrm{Pa\cdot s}\cdot 0.5\ \mathrm{m}}{3.14\cdot r^4}\\ 0.0315\ \mathrm{1/m^4} &= \frac{8}{3.14\cdot r^4}\\ 0.0315\ \mathrm{1/m^4} &= \frac{2.548}{r^4}\\ 0.0315\cdot r^4 = 2.548\ \mathrm{m^4}\\ r^4&=81\ \mathrm{m^4}\\ r&=3\ \mathrm{m} \end{split}$
Łucja Zaborowska, MD, PhD candidate
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 /
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