NavierStokes

Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of viscous fluid substances, such as liquids and gases. Named after Claude-Louis Navier and George Gabriel Stokes, these equations are fundamental in fluid dynamics, providing a mathematical framework for analyzing how fluids flow under various conditions.

The equations express the conservation of momentum and mass within a fluid system. The momentum conservation equation accounts for the velocity field's acceleration, pressure gradient forces, viscous forces, and external forces like gravity, while the continuity equation ensures mass conservation. Together, they are used to model a wide range of fluid behaviors, from laminar and turbulent flows to complex phenomena like turbulence, boundary layers, and flow separation.

In their general form, the Navier-Stokes equations involve variables including velocity, pressure, density, and viscosity. Solutions to these equations can be analytical in simplified cases or, more commonly, obtained through numerical methods such as computational fluid dynamics (CFD) for complex real-world problems.

The Navier-Stokes equations are pivotal in engineering, meteorology, oceanography, astrophysics, and biomedical sciences. Despite their long history of study, predicting turbulent flow remains challenging, and questions regarding the existence and smoothness of solutions in three dimensions are listed among the Millennium Prize Problems by the Clay Mathematics Institute.

Overall, the Navier-Stokes equations are key to understanding the physics of fluids and continue to be a subject of active research aimed at solving one of mathematics' most significant open problems.