# Poiseuille's Law Calculator

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:

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 m
^{3}/s or ft^{3}/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:

where:

- $Q$ — Flow rate in m
^{3}/s or ft^{3}/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:

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:

where:

- $R$ — Flow resistance in Pa * s/m
^{3}; - $\Delta p$ — Pressure change usually in Pa or psi;
- $Q$ — Flow rate in m
^{3}/s or ft^{3}/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:

- Begin by entering the fluid's
**dynamic viscosity in Pa * s.** - Next, in the
`Radius of the pipe (r)`

field, input the**radius of the pipe.**This is equivalent to half of the diameter. - The next dimension you need to enter is the
**length of the pipe.** - With this information, the calculator can determine the fluid's
**volumetric flow rate**. - If you also want to calculate the volumetric flow rate, simply enter the
**pressure difference**in the`Pressure change (Δp)`

row. - 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.