Lecture: An Overview of the Navier-Stokes Equations

Introduction

• Key Question: Can we predict how a lake flows over time given the velocity and pressure at every point in the lake?
• Central Concept: Navier-Stokes equations, which describe the behavior of any fluid (water, air, honey, etc.)
• Significance: Part of the Millennium Prize Problems, with a prize of one million dollars for solving it.

Importance of Navier-Stokes Equations

• Applications:
  • Weather forecasting
  • Modeling airplanes and rockets
  • Predicting water currents
• Mathematical Challenges: Despite extensive usage, there's still much unknown.

Assumptions for Simplification

• Newtonian Fluid:
  • Shear stress rate has no effect on viscosity
  • Example: Ketchup (higher shear rate decreases viscosity)
• Incompressible Fluid:
  • No considerable volume change under pressure
• Isothermal Fluid:
  • No heat loss or gain during flow

The Navier-Stokes Equations

• Mass Conservation: Indicates that mass is conserved within the fluid.
  • Uses the divergence of a velocity vector field
• Newtonian Physics:
  • Derived from Newton's second law (sum of forces = mass × acceleration)
  • For Fluid Molecules:
    • Replace mass with density (mass/volume = density)
    • Acceleration is the derivative of velocity vector field
    • Internal Forces:
      • Pressure Gradient: Fluid moves from high to low pressure (e.g., drinking from a straw)
      • Friction/Viscosity: Resistance within fluid molecules (water vs. honey)
    • External Forces: Often simplified to gravity (denoted as ρg)

Mathematical Expression

• First Equation: Divergence of velocity vector field = 0 (mass conservation)
• Second Equation: Rewritten Newton's second law for fluids
• Terms:
  • Density (ρ): Mass of the fluid
  • Velocity Vector Field (u): Describes fluid's movement
  • Pressure Gradient (∇p): Change in pressure
  • Friction/Viscosity Term: Describes internal resistance
  • External Force (f): Typically gravity

The Millennium Prize Problem

• Challenge: Prove one of four statements regarding the existence and smoothness of solutions
• Smoothness: Solution must be differentiable
• Chaos Theory: Small changes in initial conditions lead to large differences in outcomes
  • Example: Weather prediction limits (~7 days)
  • Example: Predicting turbulence in airplanes

Conclusion

• Despite its complexity, Navier-Stokes equations are crucial in many practical applications and remain a significant mathematical challenge.