The Cluster's Bernoulli's Principle

A page about Bernoulli's principle — the conservation of energy in fluid flow.

The Bernoulli equation

Bernoulli's principle relates the pressure, velocity, and height in a steady, incompressible, inviscid flow: P + (1/2) rho v^2 + rho g h = constant

The three terms represent:

• P: static pressure (energy per unit volume from pressure)
• (1/2) rho v^2: dynamic pressure (kinetic energy per unit volume)
• rho g h: hydrostatic pressure (potential energy per unit volume)

The sum is the total mechanical energy per unit volume along a streamline.

In the cluster, the edit Bernoulli equation gives an edit energy conservation.

The derivation

From the Euler equation (Navier-Stokes without viscosity): rho (dv/dt) = -nabla P - rho g nabla h For steady flow along a streamline: rho v (dv/ds) = -dP/ds - rho g (dh/ds) Integrating: P + (1/2) rho v^2 + rho g h = constant

The derivation requires:

• Steady flow (partial/partial t = 0)
• Incompressible fluid (rho = constant)
• Inviscid flow (eta = 0)
• Along a streamline

In the cluster, the edit derivation requires an edit streamline.

The applications

• Airplane wings: Faster flow over the top creates lower pressure (lift). Note: Bernoulli alone is insufficient — circulation and the Coanda effect also matter.
• Venturi effect: Constricted pipe -> higher velocity -> lower pressure. Used in carburetors, flow meters, aspirators.
• Garden hose: Pinching the nozzle increases velocity (and decreases pressure inside, which is why the hose feels the pressure).
• Sports: The Magnus effect on a spinning ball — different velocities on different sides create lift.
• Blood pressure: Narrowing of an artery increases flow velocity, decreasing lateral pressure (can be misread).

In the cluster, edit applications include:

• edit Airplane wings
• edit Venturi effect
• edit Garden hose
• edit Sports
• edit Blood pressure

The limitations

Bernoulli's equation is not valid when:

• Viscous effects are important (boundary layers, pipe flow with friction)
• The flow is unsteady
• The fluid is compressible (high Mach number)
• Heat transfer or chemical reactions occur

For compressible flow, the isentropic Bernoulli equation replaces (1/2) rho v^2 with integral dP / rho.

In the cluster, the edit limitations restrict an edit validity.

This principle

This page is about Bernoulli's principle. P + 1/2 rho v^2 + rho g h = constant. Along a streamline. Requires: steady, incompressible, inviscid. The principle is real.