Bernoulli Total Pressure⚠ unverified
Physics / Fluids · Total (stagnation) pressure along a streamline from Bernoulli's equation
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| pressure | p | Pa | 101325.0 | Static pressure |
| density | ρ | kg/m^3 | 1.225 | Fluid density |
| velocity | v | m/s | 10.0 | Flow velocity |
| height | h | m | 0.0 | Elevation (geometric height above datum) |
| g | g | m/s^2 | 9.81 | Gravitational acceleration |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| p0 | p0 | Pa | Total pressure |
The science & history
Understanding the Parameters
- $p$ — static pressure of the fluid particle.
- $\tfrac12\rho v^{2}$ — Dynamic Pressure; kinetic contribution.
- $\rho g h$ — hydrostatic head as pressure; important in free-surface and large elevation changes.
- $g$ — ≈ 9.81 m/s² on Earth (registry default here is sensible unlike some Mechanics calcs).
- $p_0$ — constant along a streamline under Bernoulli assumptions; pitot measures a related stagnation pressure.
Derivation (Approaching a Proof)
Euler’s equation along a streamline for steady inviscid flow integrates (with $\rho$ constant) to
$$\frac{p}{\rho} + \frac{v^{2}}{2} + g h = \mathrm{const}.$$
Multiplying by $\rho$ yields $p + \tfrac12\rho v^{2} + \rho g h = \mathrm{const} \equiv p_0$ (as stored energy per unit volume). Viscosity, unsteadiness, compressibility, and shaft work break the simple form.
History
Daniel Bernoulli’s Hydrodynamica (1738) and later Euler formalised energy conservation in ideal flow; pitot-static airspeed and pipe energy grades still rest on this relation.
Related Concepts: Dynamic Pressure, Pipe Head Loss, Continuity, Mass Flow Rate
Notes: Registry calculator bernoulli-total-pressure (unverified). Incompressible, inviscid,
steady streamline model.