What this calculator does
Poiseuille's law governs smooth flow through a pipe, and it contains one of the most dramatic relationships in physics: flow rate depends on the fourth power of the radius. Doubling the pipe radius multiplies the flow by sixteen.
That is why a small narrowing has such an outsized effect. A blood vessel or a pipe reduced to nine tenths of its radius carries only about two thirds of the flow at the same pressure, which is far more than the size change suggests.
The formula
Multiply π by the pressure difference and by the radius to the fourth power, then divide by eight times the viscosity times the pipe length. The result is volumetric flow rate in cubic metres per second.
| Term | Meaning |
|---|---|
| Volumetric flow rate (Q) | The volume passing a point per unit time. |
| Dynamic viscosity (η) | The fluid's resistance to shearing. Water is about 0.001 Pa·s; treacle is vastly higher. |
| Laminar flow | Smooth layered flow, which this law requires. It does not apply once flow becomes turbulent. |
The inputs explained
| Field | What to enter |
|---|---|
| Pressure difference (Pa) | The pressure difference driving the flow, in pascals. |
| Pipe radius (m) | The internal pipe radius in metres. This is the input the result is most sensitive to by far. |
| Dynamic viscosity (water ≈ 0.001) (Pa·s) | Dynamic viscosity in pascal seconds. |
| Pipe length (m) | A number, measured in m. Starts at 1. |
When to use it
Understanding why narrowing matters so much
A modest reduction in radius causes a large drop in flow, which is central to both plumbing and physiology.
Sizing a pipe run
Achieving a target flow at an available pressure sets a minimum radius, and the fourth power makes that choice unusually sharp.
Comparing fluids
Viscosity divides the result directly, so a fluid ten times thicker flows at a tenth the rate for the same pressure.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How dramatically does pipe radius change the flow?
The same pressure difference through pipes of increasing radius.
| Pipe radius | Flow rate (L/s) | Average flow velocity |
|---|---|---|
| 5 mm | 0.245 L/s | 3.125 m/s |
| 10 mm | 3.927 L/s | 12.500 m/s |
| 20 mm | 62.832 L/s | 50.000 m/s |
Questions
Why the fourth power rather than the second?
Two factors compound. A wider pipe has more cross-sectional area, which goes as radius squared, and the fluid in it also moves faster on average because more of it is far from the drag at the walls. Together they give the fourth power.
Does this apply to blood flow?
Approximately, and it is widely used in physiology to explain why small changes in vessel diameter have such large effects on flow and blood pressure. Blood is not a simple Newtonian fluid, so the law is an approximation there.
When does it stop working?
When flow becomes turbulent, which the Reynolds number predicts. In turbulent flow the pressure drop rises roughly with the square of velocity instead, and a different treatment is needed.
Why does length matter?
Because the pressure is being used up against viscous drag along the whole run. Doubling the length halves the flow for the same pressure difference, which is a simple inverse relationship.
For whether the flow is laminar enough for this law, see the Reynolds number calculator. For flow in an open channel instead, see the open channel flow calculator.