Lecture on Bernoulli's Equation

Introduction

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• Bernoulli's Equation: Key formula in physics and engineering for fluid flow analysis.
• Describes relationships: pressure, velocity, elevation in fluid dynamics.
• Applications: Aircraft lift, liquid drainage, etc.

Bernoulli's Equation

• Published by Daniel Bernoulli in 1738.
• Sum of three pressures along a streamline remains constant:
  1. Static Pressure (P): Pressure of fluid.
  2. Dynamic Pressure: Function of fluid density (ρ) and velocity (v), representing kinetic energy per unit volume.
  3. Hydrostatic Pressure: Pressure due to gravity (g) and elevation (H).
• Forms of the equation: Pressure, Head, Energy.
• Concept: Conservation of energy in fluid flow.

Streamline and Application Example

• Streamline: Path traced by fluid particle, tangent to velocity vector.
• Pipe flow with diameter change:
  • Bernoulli's equation helps determine pressure changes.
  • Continuity equation applies for incompressible fluids (conservation of mass):
    • Mass flow rate = fluid density × area × velocity.
  • Smaller cross-sectional area increases velocity, decreases pressure.

Bernoulli's Principle

• Horizontal flow: Increased velocity decreases pressure.
• Applications:
  • Airfoil (Lift): Faster fluid over airfoil creates low pressure, generating lift.
  • Bunsen Burners: Air drawn in due to low pressure from high gas velocity.
  • Flow Measurement Devices: Pitot-static tube (measures airspeed in aircraft), Venturi meter (flow rate through pipe).

Flow Measurement Devices

• Pitot-static tube:
  • Measures stagnation and static pressure to determine fluid velocity.
• Venturi meter:
  • Measures pressure drop across converging pipe section for flow rate.
  • No moving parts, reliable measurement.

Practical Example: Beer Keg

• Calculate beer draining speed using Bernoulli's equation.
• Assumptions: Atmospheric pressure, negligible fluid velocity at point 1.

Limitations of Bernoulli's Equation

• Assumptions for applicability:
  • Laminar, steady flow.
  • Inviscid flow (negligible viscosity).
  • Incompressible fluid (valid for liquids, not high-velocity gases).
• Adapted versions exist for unsteady, compressible flows.

Conclusion

• Recognizing and applying Bernoulli's principle is a valuable skill.
• Extended examples available on Nebula.
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