The Flow of the Universe

The Navier-Stokes equations describe the motion of viscous fluid substances. These balance equations arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term and a pressure term.

The Equations

The incompressible Navier-Stokes equations can be written as:

Let's break down the terms:

1. Unsteady acceleration ($\frac{\partial \mathbf{u}}{\partial t}$): The rate of change of velocity with respect to time.
2. Convective acceleration ($(\mathbf{u} \cdot \nabla) \mathbf{u}$): The acceleration of a fluid particle due to its motion through a varying velocity field.
3. Pressure gradient ($-\frac{1}{\rho} \nabla p$): The force due to differences in pressure.
4. Viscosity ($\nu \nabla^2 \mathbf{u}$): The internal friction of the fluid.
5. Body forces ($\mathbf{f}$): External forces like gravity.

A Millennium Prize Problem

Despite their physical importance, we still don't know if smooth solutions always exist in three dimensions. This makes the Navier-Stokes existence and smoothness problem one of the seven Millennium Prize Problems in mathematics.

Computational Fluid Dynamics (CFD)

Because analytical solutions are rare, we often use computers to solve these equations numerically. Here is a conceptual example of a simulation loop:

function simulateFluid(grid, dt) {
  // 1. Add forces
  applyForces(grid, dt);
  
  // 2. Advect (move the fluid along itself)
  advect(grid, dt);
  
  // 3. Diffuse (viscosity)
  diffuse(grid, dt);
  
  // 4. Project (ensure mass conservation/incompressibility)
  project(grid);
  
  return grid;
}

Understanding these equations is crucial for everything from designing airplanes to predicting the weather.