Bernoulli's equation

In fluid dynamics, Bernoulli's equation, derived by Daniel Bernoulli, describes the behavior of a fluid moving along a streamline.

v = fluid velocity along the streamline

g = acceleration due to gravity on Earth

y = elevation in the direction of gravity

P = pressure along the streamline

ρ = fluid density

These assumptions must be met for the equation to apply:

• Inviscid flow - Viscosity (internal friction) = 0
• Steady flow
• Incompressible flow - ρ is constant
• The equation applies along a streamline. It applies throughout the flow field for irrotational flow .

The decrease in pressure simultaneous with an increase in velocity, as predicted by the equation, is often called Bernoulli's principle.

The equation is named for Daniel Bernoulli although it was first presented in the above form by Leonhard Euler.

The equation can be derived by integrating the Euler equations, or applying the law of conservation of energy in two sections along a streamline, ignoring viscosity, compressibility, and thermal effects:

the work done by the forces in the fluid + decrease in potential energy = increase in kinetic energy

The work done by the forces;

F₁· s₁ - F₂·s₂ = p₁·A₁·v₁·Δt - p₂·A₂·v₂·Δt

+ the decrease of potential energy:

m·g·h₁ - m·g·h₂ = ρ·g·A₁·v₁·Δt·h₁ - ρ·g·A₂·v₂·Δt·h₂

= the increase in kinetic energy:

1/2·m·v₂² - 1/2·m·v₁² = 1/2·ρ·A₂·v₂·Δt·v₂² - 1/2·ρ·A₁·v₁·Δt·v₁²

gives;

p₁·A₁·v₁·Δt - p₂·A₂·v₂·Δt + ρ·g·A₁·v₁·Δt·h₁ - ρ·g·A₂·v₂·Δt·h₂ = 1/2·ρ·A₂·v₂·Δt·v₂² - 1/2·ρ·A₁·v₁·Δt·v₁²

or

ρ·A₁·v₁·Δt·v₁²/2 + ρ·g·A₁·v₁·Δt·h₁ + p₁·A₁·v₁·Δt = ρ·A₂·v₂·Δt·v₂²/2 + ρ·g·A₂·v₂·Δt·h₂ + p₂·A₂·v₂·Δt

division by Δt, ρ and A₁·v₁ (= rate of fluid flow = A₂·v₂ as the fluid is incompressible) gives;

v₁²/2 + g·h₁ + p₁/ρ = v₂²/2 + g·h₂ + p₂/ρ

or v²/2 + g·h + p/ρ = C (as stated in the first paragraph).

Further division by g gives;

v²/(2·g) + h + p/(ρ·g) = C

A free falling mass from a height h will reach a velocity v = √(h/(2g)), or h = v²/(2·g). The term v²/(2·g) is called the velocity head.
As the hydrostatic pressure or static head is defined as p = ρ·g·h or h = p/(ρ·g), the term p/(ρ·g) is also called the pressure head.