Poiseuille's Law Calculator
The HagenPoiseuille's law calculator is a multitask 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 HagenPoiseuille'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 HagenPoiseuille equation)
HagenPoiseuille 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 (HagenPoiseuille law) describes how much water flows though the pipe in one second, depending on:
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 HagenPoiseuille 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):
 Enter the dynamic viscosity (μ, /mi/):
It's basic unit is Pa * s. For different pressures, check the pressure converter.

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

Enter the length of the pipe.

Find the pressure change (only required for the flow rate).
If you don't know the pressure change, click the advanced mode
button and enter you initial and final pressures, or subtract the final pressure from the start pressure yourself, following the formula below:
pressure change (Δp) = pressure 1  pressure 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 = (π * Δp * r^{4}) / ( 8 * μ * l)
where:
 Q  Flow rate (m³/s);
 π  Our famous and beloved constant pi, equals to 3.14159...;
 μ  Dynamic viscosity (Pa * s);
 r  Radius of the pipe;
 l  Length of the pipe; and
 Δp  Pressure change (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?
💡 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 mm = 0.001 m;
 The pressure at the beginning of the vessels system was equal to 120 mmg Hg = 15998.7 Pa;
 The pressure at the end was equal to 80 mmHg = 10665.8 Pa;
 Our dynamic viscosity is 0.005 Pa * s;
 Total length of our vessels is 2 m; and
 Our vessels contracted by 3/4  they're 1/4 of their original size.
Let's create the original blood flow equation:
Q = (3.14 * (15998.710665.8 Pa) * (0.001 m)^{4}) / ( 8 * 0.005 Pa * s * 2 m)
Q = (3.14 * 5332.9 * (0.001 m)^{4}) / 0.08 m)
Q = (3.14 * 5332.9 * 0.000000000001 m^{3}) / 0.08 )
Q = 0.00000020942 m³/s = 209.42 mm³/s
So how does the blood flow change with the radius being 1/4 of its original size?.
r_{2} = 1/4 * 0.0001 m = 0.000025 m
Q_{2} = (3.14 * 5332.9 * (0.000025 m)^{4}) / 80 m)
Q_{2} = (3.14 * 5332.9 * 3.90625e19 m^{3}) / 80 )
Q_{2} = 0.000000000000818 m³/s = 0.000818 mm³/s
Let's calculate the ratio of Q and Q_{2} :
0.20942 mm³/s : 0.000818 mm³/s
256 : 1
The blood flow will change by a factor of 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 HagenPoiseuille equation for flow resistance that we use in this Poiseuille's law calculator:
R = (8 * μ * l)/ (π * r^{4})
Where:
 R  Resistance (Pa * s/ m³);
 π  Our famous and beloved constant pi, equals to 3.14159...;
 μ  Dynamic viscosity (Pa * 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 Pa * s/ m³, length of 0.5 m, and viscosity of 2 Pa * s.
0.0315 Pa * s/m³ = (8 * 2 Pa * s * 0.5 m) / (3.14 * r^{4})
0.0315 / m^{4} = 8 / (3.14 * r^{4})
0.0315 / m^{4} = 2.548 / r^{4}
0.0315 * r^{4} = 2.548 m^{4}
r^{4} = 81 m^{4}
r = 3 m