Fluid Dynamics Simulator

What Are the Navier-Stokes Equations?

The Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous fluid substances. First developed by Claude-Louis Navier and George Gabriel Stokes in the 19th century, these equations form the foundation of computational fluid dynamics (CFD) — one of the most important fields in modern engineering and physics. They remain one of the seven Millennium Prize Problems in mathematics.

How This Simulator Works

This tool implements Jos Stam's "Stable Fluids" algorithm (1999), widely used in VFX and game dev. Each frame performs four operations:

1. Add Forces — Your mouse input is added to the velocity field
2. Diffuse — Viscosity equalizes neighboring velocities
3. Advect — The velocity field moves itself and dye through back-tracing
4. Project — Pressure solve ensures divergence-free (mass-conserving) flow

The simulation runs on a 128×128 grid with Gauss-Seidel relaxation. RGB dye channels are tracked independently, allowing natural color mixing. Runs at 60+ FPS entirely client-side.

Real-World Applications

• Aerospace — Airflow optimization reducing fuel consumption by up to 15%
• Weather — Atmospheric fluid models predict storms days in advance
• Medical — Blood flow simulation for heart valve design
• Film & VFX — Realistic water, fire, and smoke effects
• Games — Real-time fluid effects in modern titles

Frequently Asked Questions