04 // Interactive Formula Compendium // ENGINEERINGP₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Bernoulli's Fluid Dynamics Equation
Bernoulli's equation states that along a steady, incompressible, frictionless streamline, the sum of static pressure (P), dynamic pressure (½ρv²), and hydrostatic potential (ρgh) remains constant.
Formula Reference
Main Formula
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Rearranged Forms
P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(h₁ − h₂)
H = P/(ρg) + v²/(2g) + z = constant
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| P2 | Downstream Static Pressure | Pa |
| P1 | Upstream Pressure | Pa |
| rho | Fluid Density | kg/m³ |
| v1 | Upstream Velocity | m/s |
| v2 | Downstream Velocity | m/s |
| h1 | Upstream Elevation | m |
| h2 | Downstream Elevation | m |
Inputs & Parameters
[Pa]
[kg/m³]
[m/s]
[m/s]
[m]
[m]
Calculated Result (Downstream Static Pressure)
P2 =263,420Pa
Formula Notes & Background
- Derived by integrating Euler's momentum equation −(1/ρ)∇P − g∇z = v(dv/ds) along a streamline.
- Dividing every term by ρg converts pressures into hydraulic head meters (pressure head + velocity head + elevation head).
Step-by-Step Examples
#1: Water (ρ = 1000 kg/m³) flows from a pipe at P₁ = 250 kPa, v₁ = 2 m/s, h₁ = 5 m into a constriction at v₂ = 6 m/s, h₂ = 2 m.
P₁ = 250,000 Pav₁ = 2 m/s, v₂ = 6 m/sh₁ = 5 m, h₂ = 2 m
P₂ = 250,000 + 500(4 − 36) + 1000(9.80665)(3) = 250,000 − 16,000 + 29,420
→ P₂ ≈ 263,420 Pa (263.42 kPa)
#2: Horizontal Venturi meter (h₁ = h₂) with air (ρ = 1.225 kg/m³) accelerating from 15 m/s to 45 m/s at P₁ = 101,325 Pa.
ρ = 1.225 kg/m³v₁ = 15 m/s, v₂ = 45 m/sΔh = 0
P₂ = 101,325 + 0.6125(225 − 2025) = 101,325 − 1,102.5
→ P₂ = 100,222.5 Pa (ΔP = −1.10 kPa suction)
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Frequently Asked Questions
When does incompressible Bernoulli's equation break down?+
Above Mach 0.3 in gases (compressibility effects become significant) or in long pipes with high viscous friction (use the extended energy equation with Darcy-Weisbach head loss).
What is stagnation pressure?+
The total pressure P₀ = P + ½ρv² measured by a Pitot tube when fluid is decelerated isentropically to zero velocity.
Why does pressure drop in a pipe constriction?+
By mass continuity (A₁v₁ = A₂v₂), velocity increases in the narrower cross-section, converting static pressure energy into kinetic dynamic pressure.