Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant.

The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation:

P + ½ρv² + ρgh = constant

where P is the absolute pressure, ρ is the fluid density, v is the velocity of the fluid, h is the height above some reference point, and g is the acceleration due to gravity. If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Bernoulli’s equation is a form of the conservation of energy principle. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

The general form of Bernoulli’s equation has three terms in it, and it is broadly applicable. To understand it better, we will look at a number of specific situations that simplify and illustrate its use and meaning including static fluids and constant depth.

Let us first consider the very simple situation where the fluid is static—that is, v₁ = v₂ = 0. In that case, we get

P₂ = P₁ + ρgh₁

This equation tells us that, in static fluids, pressure increases with depth.

Another important situation is one in which the fluid moves but its depth is constant—that is, h₁ = h₂. Under that condition, Bernoulli’s equation becomes

P₁ + ½ρv₁² = P₂ + ½ρv₂²

Situations in which fluid flows at a constant depth are so important that this equation is often called Bernoulli’s principle. It is Bernoulli’s equation for fluids at constant depth. As we have just discussed, pressure drops as speed increases in a moving fluid.

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Key Points

• Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant.

• Bernoulli’s equation is a form of the conservation of energy principle. and a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

• In static fluids, pressure increases with depth: P₂ = P₁ + ρgh₁

• Bernoulli’s principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid’s potential energy

• For fluids at a constant depth, pressure drops as speed increases in a moving fluid: P₁ + ½ρv₁² = P₂ + ½ρv₂²

Key Terms

Fluid density: The average density of a substance or object is defined as its mass per unit volume, ρ = m/ V.

Bernoulli’s equation: The equation resulting from applying conservation of energy to an incompressible frictionless fluid: P + ½ρv² + ρgh = constant, through the fluid.

Bernoulli’s principle: Bernoulli’s equation applied at constant depth: P₁ + ½ρv₁² = P₂ + ½ρv₂²