# 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.