Bernoulli Equation Calculator
Solve for the pressure at one point on a streamline from the speed and height at another.
Pressure change (P₂ − P₁): -16,000 Pa — the faster/higher point has lower pressure.
Bernoulli’s equation says pressure, speed, and height trade off along a streamline: where a fluid speeds up, its pressure drops. For water (ρ = 1000 kg/m³) speeding from 2 m/s to 6 m/s at the same height, starting at 200,000 Pa, the pressure falls by 16,000 Pa to 184,000 Pa.
What Bernoulli’s equation describes
For a steady, incompressible, non-viscous flow, the sum of pressure energy, kinetic energy, and potential energy stays constant along a streamline. So if one term grows, another must shrink. This calculator holds that total fixed between two points and solves for the pressure P₂ at the second point given the pressure, speed, and height at the first. To go the other way — force per unit area on a wall — use the pressure calculator.
P = pressure (Pa), ρ = density (kg/m³), v = speed (m/s), g = 9.81 m/s², h = height (m)
Worked example
Water (ρ = 1000 kg/m³) flows through a horizontal pipe that narrows, speeding up from v₁ = 2 m/s to v₂ = 6 m/s. The upstream pressure is P₁ = 200,000 Pa and the height is unchanged (h₁ = h₂):
- 1 Rearrange for the unknown pressure. P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(h₁ − h₂).
- 2 Compute the kinetic term. ½ × 1000 × (2² − 6²) = 500 × (4 − 36) = 500 × (−32) = −16,000 Pa.
- 3 Compute the potential term. Height is unchanged, so ρg(h₁ − h₂) = 1000 × 9.81 × 0 = 0 Pa.
- 4 Add the terms to get P₂. P₂ = 200,000 + (−16,000) + 0 = 184,000 Pa — 16,000 Pa lower because the water sped up.
The three energy terms per unit volume
Each term has units of pressure (Pa = J/m³); Bernoulli keeps their sum constant along a streamline.
| Term | Expression | Represents |
|---|---|---|
| Pressure | P | Static pressure energy of the fluid |
| Kinetic | ½ρv² | Energy of motion (dynamic pressure) |
| Potential | ρgh | Gravitational energy from elevation |
Assumptions and intuition
Bernoulli’s equation only holds when the flow is incompressible (constant density), non-viscous (no friction losses), and steady (not changing in time), and only along a single streamline. Real pipes have friction, so downstream pressure is usually a little lower than the ideal result here.
The speed–pressure trade-off is the whole story behind a Venturi meter, where a constriction speeds the fluid and the pressure drop reveals the flow rate, and behind aerodynamic lift, where faster flow over a curved surface leaves lower pressure above it. Squeeze a flow faster and its pressure must fall to keep the total energy fixed.