Bernoulli Equation Calculator
Calculate pressure, velocity, or height at any point in a fluid flow using Bernoulli's equation (P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂). Applies to steady, incompressible, inviscid flow along a streamline. See also our Reynolds Number Calculator and Flow Rate Calculator.
How to Use Bernoulli's Equation
Bernoulli's equation is one of the most important principles in fluid dynamics. It describes the conservation of energy in a flowing fluid, stating that the total mechanical energy per unit volume remains constant along a streamline. The equation connects three forms of energy: pressure energy (static pressure), kinetic energy (dynamic pressure from fluid velocity), and potential energy (due to elevation).
To apply Bernoulli's equation, identify two points along the same streamline in the flow. At each point, you need to know (or solve for) the pressure, velocity, and height. The equation assumes the fluid is incompressible (constant density), the flow is steady (not changing with time), there is no friction (inviscid flow), and the two points are on the same streamline. While real fluids have viscosity, Bernoulli's equation gives excellent approximations for many engineering problems.
The key insight of Bernoulli's principle is that where fluid velocity increases, pressure decreases, and vice versa. This explains how airplane wings generate lift (faster air over the curved top creates lower pressure), how carburetors mix fuel with air (fast air in a narrow throat creates low pressure that draws fuel in), and how a shower curtain gets sucked inward (fast-moving water creates low pressure inside the shower).
Bernoulli's Equation Formula
Bernoulli's Equation:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Head Form:
P/(ρg) + v²/(2g) + h = constant
(pressure head + velocity head + elevation head)
Energy per Unit Volume:
P + ½ρv² + ρgh = constant
Torricelli's Theorem (tank draining):
v = √(2gh)
Venturi Effect:
v₂ = v₁ × (A₁/A₂)
P₁ - P₂ = ½ρ(v₂² - v₁²)
Example Calculation
Water (ρ = 1000 kg/m³) flows through a horizontal pipe. At point 1, P₁ = 101325 Pa and v₁ = 2 m/s. At point 2, v₂ = 4 m/s. Both points are at the same height. Find P₂:
Given: P₁ = 101325 Pa, v₁ = 2 m/s, v₂ = 4 m/s
h₁ = h₂ = 0 (horizontal), ρ = 1000 kg/m³
P₁ + ½ρv₁² = P₂ + ½ρv₂² (h terms cancel)
P₂ = P₁ + ½ρ(v₁² - v₂²)
P₂ = 101325 + ½×1000×(4 - 16)
P₂ = 101325 + 500×(-12)
P₂ = 101325 - 6000 = 95325 Pa
Pressure decreased by 6000 Pa as velocity increased.
This demonstrates Bernoulli's principle: faster flow → lower pressure.
Bernoulli Equation Reference Table
| Scenario | P₁ (Pa) | v₁ (m/s) | v₂ (m/s) | P₂ |
|---|---|---|---|---|
| Water pipe narrowing | 101325 | 2 | 4 | 95325 Pa |
| Venturi meter | 200000 | 1 | 3 | 196000 Pa |
| Airplane wing (air) | 101325 | 60 | 70 | 100540 Pa |
| Water tower (10m) | 101325 | 0 | 0 | 199425 Pa |
| Fire hose nozzle | 400000 | 2 | 20 | 202325 Pa |
| Pitot tube | 101325 | 0 | 50 | 99800 Pa |
Frequently Asked Questions
What is Bernoulli's equation?
Bernoulli's equation states that for an ideal fluid flowing along a streamline, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant. Mathematically: P + ½ρv² + ρgh = constant. It was published by Daniel Bernoulli in 1738 and is derived from the conservation of energy applied to fluid flow. It is one of the most widely used equations in fluid mechanics and aerodynamics.
What are the assumptions of Bernoulli's equation?
Bernoulli's equation assumes: (1) the fluid is incompressible (constant density), (2) the flow is steady (not changing with time), (3) the fluid is inviscid (no friction or viscosity losses), and (4) the equation is applied along a single streamline. Real fluids violate these assumptions to varying degrees, but the equation still provides useful approximations for many engineering applications, especially for low-viscosity fluids at moderate speeds.
How does Bernoulli's principle explain airplane lift?
An airplane wing (airfoil) is shaped so air flows faster over the curved upper surface than the flatter lower surface. According to Bernoulli's principle, faster flow means lower pressure. The pressure difference between the lower surface (higher pressure) and upper surface (lower pressure) creates an upward net force — lift. However, this is a simplified explanation; the full picture also involves the angle of attack and Newton's third law (deflecting air downward).
What is the Venturi effect?
The Venturi effect is the reduction in fluid pressure that occurs when a fluid flows through a constricted section of pipe. By the continuity equation (A₁v₁ = A₂v₂), fluid speeds up in the narrow section. By Bernoulli's principle, this increased velocity causes decreased pressure. Venturi meters measure flow rate by measuring this pressure drop. Carburetors, aspirators, and paint sprayers all exploit the Venturi effect.
What is Torricelli's theorem?
Torricelli's theorem is a special case of Bernoulli's equation for fluid draining from a tank through a small hole. If the tank is open to the atmosphere and the hole is at depth h below the surface, the exit velocity is v = √(2gh) — the same speed as an object falling from height h. This assumes the tank is large (surface velocity ≈ 0) and there are no losses. It is used to calculate drain times and design water features.
When does Bernoulli's equation not apply?
Bernoulli's equation fails when: (1) the fluid is compressible (high-speed gas flows above Mach 0.3), (2) there is significant viscous friction (long pipes, very slow flows), (3) the flow is unsteady (water hammer, pulsating flows), (4) there is energy addition or removal (pumps, turbines), or (5) you compare points on different streamlines in rotational flow. For these cases, use the full Navier-Stokes equations or modified Bernoulli with loss terms.
Practical Applications
Bernoulli's equation is used extensively in engineering design. HVAC engineers use it to size ductwork and calculate pressure drops. Civil engineers apply it to design water distribution systems and dam spillways. Aerospace engineers use it for preliminary wing design and wind tunnel analysis. Medical devices like nebulizers and venturi masks use the Venturi effect to entrain air or medication. Even the curve of a baseball pitch can be partially explained by Bernoulli's principle acting on the spinning ball.