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Bernoulli's principle is a statement in fluid dynamics that describes how the speed of a moving fluid relates to its pressure. In its most common form for incompressible, non-viscous, steady flow along a streamline:

• Where the fluid flows faster, its static pressure is lower.
• Where the fluid flows slower, its static pressure is higher.

Key points to understand:

• Conservation of energy: In a horizontal flow (no height change), the Bernoulli equation can be written as:
  P + ½ρv² = constant
  where P is the static pressure, ρ is the fluid density, and v is the flow speed.
  This expresses that the sum of pressure energy and kinetic energy per unit volume remains constant along a streamline.

• Interpretations:

  • If the speed (v) increases, the pressure (P) must decrease to keep the total energy constant.
  • If the speed decreases, the pressure increases.

Common examples and applications:

• Air over an airplane wing: The wing is shaped so air travels faster over the top surface than below. Faster flow on top lowers pressure, creating lift that helps the airplane rise.
• Venturi effect: In a constricted section of pipe, velocity increases and pressure decreases. This principle is used in carburetors, atomizers, and some flow meters.
• Chimney effect and sprinkler systems: Differences in air flow speed create pressure differences that drive movement of air or fluids.
• Baseball with a seam: Spin can affect local airflow and pressure around the ball, influencing its trajectory (though this is often a combination of Bernoulli and other effects).

Important caveats and common misconceptions:

• Bernoulli’s principle applies along a streamline for idealized flows (incompressible, non-viscous, steady). Real fluids have viscosity, turbulence, and energy losses, so the simple P + ½ρv² constant form is an idealization.
• It does not by itself explain lift entirely; a complete explanation of wing lift also involves Newton’s third law and the change in momentum of air, circulation, and the angle of attack. Bernoulli’s principle is a part of the broader fluid dynamics picture.
• Pressure differences can also arise from height differences (hydrostatic pressure) and from viscosity and flow separation; these factors mean real-world situations require careful analysis beyond the simplest version.

A simple takeaway: In a flowing fluid, speeding up the flow tends to lower the pressure, and slowing the flow tends to raise the pressure, when you can treat the flow as steady and along a streamline.