Unit 8: Fluids

Introduction

The motion of fluids is governed by conservation laws, including the continuity equation and Bernoulli's equation, which describe how mass and energy are preserved in fluid systems. These principles are essential for understanding real-world applications such as pipe flow, flight, and hydrodynamics.

Key Concepts

• Conservation of Mass (Continuity Equation): \[ A_1 v_1 = A_2 v_2 \] where:
  • A = cross-sectional area (m²)
  • v = velocity of the fluid (m/s)
  This equation states that for an incompressible fluid, the volume flow rate remains constant.
• Bernoulli's Principle (Energy Conservation in Fluids): \[ P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant} \] where:
  • P = pressure (Pa)
  • ρ = fluid density (kg/m³)
  • v = velocity (m/s)
  • h = height (m)
  This principle states that the total energy (pressure energy, kinetic energy, and gravitational potential energy) is conserved in a fluid.
• Applications of Bernoulli's Equation:
  • Faster-moving fluids have lower pressure (e.g., airplane wings create lift).
  • Fluids speed up when moving through narrow passages (Venturi effect).

Mathematical Routines

• Use A₁v₁ = A₂v₂ to analyze changes in flow speed.
• Apply Bernoulli's equation to determine pressure differences.
• Calculate fluid velocity from height differences using energy conservation.

Creating Representations

• Diagrams: Illustrate streamlines and pressure variations in different pipe sections.
• Graphs: Plot pressure vs. velocity to visualize Bernoulli's principle.

Scientific Questioning & Argumentation

• Why does increasing fluid speed lower pressure in a system?
• How does the continuity equation explain why water speeds up in a narrow pipe?

Use flow diagrams and Bernoulli’s equation to support your answers.

Practice Activities

Summary & Exam Preparation Tips

• The continuity equation ensures mass conservation in fluid flow.
• Bernoulli's equation relates pressure, velocity, and height in a moving fluid.
• Faster-moving fluids lower pressure, explaining aerodynamic lift.
• Flow rate remains constant for incompressible fluids.