What this calculator does
Bernoulli's equation says that in a steady, frictionless flow the sum of pressure, kinetic energy per unit volume and potential energy per unit volume stays constant along a streamline. What that means practically is that where a fluid speeds up, its pressure falls.
It explains a lot of everyday behaviour: why a shower curtain pulls inward, why a narrowing pipe shows a pressure drop, and a good part of why a wing generates lift. It also has firm limits, since it assumes no friction and no turbulence.
The formula
The sum of static pressure, half the density times speed squared, and density times gravity times height is equal at both points. Rearranging for the pressure at point two gives the result, with the contributions from the speed change and the height change reported separately.
| Term | Meaning |
|---|---|
| Static pressure | The ordinary pressure the fluid exerts, which is what a gauge in the pipe wall reads. |
| Dynamic pressure | The half rho v squared term: the pressure equivalent of the flow's kinetic energy. |
| Streamline | The path a fluid element follows. The equation applies along one, not between separate flows. |
The inputs explained
| Field | What to enter |
|---|---|
| Pressure at point 1 (Pa) | Pressure at the first point, in pascals. |
| Fluid density (water = 1000) (kg/m³) | Fluid density in kilograms per cubic metre. Water is 1,000. |
| Speed at point 1 (m/s) | Flow speed at the first point, in metres per second. |
| Elevation at point 1 (m) | Elevation of the first point, in metres. |
| Speed at point 2 (m/s) | A number, measured in m/s. Starts at 5. |
| Elevation at point 2 (m) | A number, measured in m. Starts at 2. |
When to use it
Understanding a pressure drop in a pipe
Where a pipe narrows the flow must speed up, and Bernoulli says the pressure must fall to pay for it.
Explaining lift on a wing
Faster flow over the upper surface corresponds to lower pressure there, which is part of the explanation for lift.
Sizing a venturi or flow meter
Venturi meters work by measuring the pressure drop across a deliberate constriction and inferring the flow rate.
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 does speeding the flow up change the pressure?
The same starting conditions with a range of speeds at the second point.
| Speed at point 2 | Pressure at point 2 (kPa) | Change from speed term |
|---|---|---|
| 2 m/s | 180.387 kPa | 0 Pa |
| 4 m/s | 174.387 kPa | -6,000 Pa |
| 5 m/s | 169.887 kPa | -10,500 Pa |
| 6 m/s | 164.387 kPa | -16,000 Pa |
Questions
Does faster flow really mean lower pressure?
Along a single streamline in a steady, frictionless flow, yes. The energy has to come from somewhere, and it comes out of the pressure. The statement is often misapplied across different flows, where it does not hold.
When does the equation break down?
Whenever friction or turbulence is significant, which in real piping is often. It also assumes incompressible flow, so it fails for gases moving at a substantial fraction of the speed of sound.
Is this the full explanation of lift?
No, and the popular version of it is frequently wrong. Bernoulli correctly relates speed and pressure, but the usual claim that air must traverse the wing in equal time is false. Lift is better explained by the wing turning the airflow downward.
What is the height term for?
It accounts for elevation change, which matters for anything flowing up or down a significant height. For horizontal pipe runs the two heights are equal and the term cancels out.
For whether the flow is smooth enough for these assumptions, see the Reynolds number calculator. For flow in an open channel, see the open channel flow calculator.