Lecture Notes: Conservation of Energy in Fluid Dynamics

Overview

• Review of concepts from the last video.
• Discussion of fluid dynamics involving an oddly shaped pipe.

Key Concepts

Variables and Notations

• V1: Input velocity of fluid.
• P1: Pressure on the left-hand side (input pressure).
• A1: Area of the input hole.
• V2, P2, A2: Corresponding variables for the output side.

Conservation of Energy

• The law states that the energy entering the system must equal the energy leaving the system (conservation of joules).
• Set up an equation based on the law of conservation of energy.

Energy Components

• Work Input: Pressure times the volume of mass over time, divided by density.
  • Equivalent to potential energy, typically denoted as mgh.
• Kinetic Energy Input: Mass of the fluid volume times its velocity squared divided by 2.
• Output Work: Similar calculations apply to the output side.
  • Considers a longer cylinder due to increased velocity.

Bernoulli’s Equation

• Setup of Bernoulli's equation which describes the conservation of energy in fluid flow.
• Equation: [ P1 + \rho g h1 + \frac{\rho v1^2}{2} = P2 + \rho g h2 + \frac{\rho v2^2}{2} ]
• Simplification by removing common terms (mass) and manipulating variables.

Implications of Bernoulli’s Equation

• If height (h) is constant:
  • Increase in velocity results in a decrease in pressure.
  • If pressure increases, velocity decreases.
• Application in understanding phenomena like flight.

Practical Problem: Fluid Ejection from a Cup

Scenario

• Cup filled with fluid, covered, and contains a vacuum.
• Hole poked at a height h below the surface.

Objective

• Determine output velocity (V2) of the fluid as a function of height (h).

Given Conditions

• Area of hole (A2) is 1/1000th of the surface area (A1).

Applying Bernoulli’s Equation

• Input Conditions:
  • Pressure at the surface is zero (vacuum).
  • Height input (h1) is h.
  • Velocity input (V1): derived from continuity equation.
    • [ V1 \times A1 = V2 \times \frac{A1}{1000} ]
    • [ V1 = \frac{V2}{1000} ]
• Output Conditions:
  • Need to calculate pressure at the hole (P2) and consider the setup for the next video.

Continuation

• The explanation was paused due to time constraints and will proceed in the next session.