The Bernoulli equation can be expressed as:
P+12 𝜌v2+ 𝜌𝑔ℎ = constant

Where:

  • 𝑃: Static pressure of the fluid (Pa)
  • 𝜌: Fluid density (kg/m33)
  • 𝑣: Flow velocity (m/s)
  • 𝑔: Acceleration due to gravity (9.81 m/s22)
  • ℎ: Elevation above a reference point (m)

Derivation of the Bernoulli Equation

The Bernoulli equation can be derived from the principle of conservation of energy.

  1. Work-Energy Principle: In fluid flow, the total mechanical energy per unit volume remains constant, assuming no energy is added or lost.
  2. Pressure Work: The work done by the pressure force over a small distance Δ𝑥is 𝑃Δ𝑉, where Δ𝑉is the volume of the fluid element.
  3. Kinetic Energy: The kinetic energy of a moving fluid is given by 12 𝜌v2Δ𝑉, where v is the velocity.
  4. Potential Energy: The potential energy due to gravity is 𝜌𝑔ℎΔ𝑉, where ℎ is the height above a reference level.

Applying Energy Conservation
For a fluid element moving from point 1 to point 2:

  • The work done by the fluid’s pressure difference is P1Δ𝑉−P2Δ𝑉.
  • The change in kinetic energy is 12v22Δ𝑉−12v12 Δ𝑉
  • The change in potential energy is 𝜌𝑔h2Δ𝑉−𝜌𝑔h1Δ𝑉.

Equating the total energy at point 1 to that at point 2:
P1+ 12v12+ 𝜌𝑔h1= P2+12v22+ 𝜌𝑔h2

This equation states that the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant.

Assumptions

  • The fluid is incompressible and non-viscous.
  • The flow is steady, meaning conditions at any given point do not change with time.
  • The flow is along a streamline, meaning there’s no energy exchange with other streamlines.
  • There are no energy losses due to friction, heat transfer, or other factors.

Applications
The Bernoulli equation is widely used in various engineering fields to analyze fluid flow. Some common applications include:

  • Calculating fluid flow speeds in pipes or open channels.
  • Understanding the behavior of aircraft wings (lift) and hydrofoils.
  • Designing and analyzing flow meters (venturi meters, orifice plates, pitot tubes).
  • It’s a foundational principle for understanding fluid dynamics, allowing engineers to predict how fluids behave in various conditions.