The Navier–Stokes equations are a set of fundamental mathematical equations that describe the motion of viscous fluid substances, such as liquids and gases. They are derived from Newton’s second law of motion, applied to fluid elements, and account for forces like pressure, viscous friction, and external body forces (e.g., gravity).
These equations express how the velocity, pressure, density, and viscosity of a fluid evolve over time and space. They are used to predict the behavior of fluids in both laminar (smooth) and turbulent (chaotic) flow conditions.
Key Features:
- They include conservation of mass (continuity equation) and momentum.
- Applicable to a wide range of fluids—air, water, oil, plasma, etc.
- Solving them exactly is extremely challenging, especially for turbulent flows.
Examples:
- Weather forecasting models use them to simulate atmospheric dynamics.
- Aerodynamics relies on them to design vehicles, aircraft, and wind turbines.
- Blood flow modeling and oil pipeline design also use these equations extensively.
The Navier–Stokes equations are central to fluid dynamics, but a general solution for all conditions remains one of the great unsolved problems in mathematics, recognized as a Millennium Prize Problem by the Clay Mathematics Institute.