The Navier-Stokes theorem is a fundamental equation in fluid mechanics that describes the motion of a fluid. It is named after Claude-Louis Navier and George Gabriel Stokes, who independently derived the equation in the 19th century.
The Navier-Stokes theorem consists of two equations: the continuity equation and the Navier-Stokes equation. The continuity equation describes the conservation of mass in a fluid, while the Navier-Stokes equation describes the conservation of momentum.
The Navier-Stokes theorem has numerous applications in various fields such as aerodynamics, weather forecasting, oceanography, and even in the design of everyday objects like pipes, pumps, and turbines. It is also used in the simulation of fluid flow in computer graphics and video games.
The Navier-Stokes equations are highly complex and nonlinear, making them difficult to solve analytically. Even with the use of numerical methods, solving these equations accurately can be computationally intensive and time-consuming. Additionally, the equations are sensitive to initial and boundary conditions, making it challenging to model real-world scenarios accurately.
There are a few known exact solutions to the Navier-Stokes equations, but they are limited to simplified scenarios. In most cases, numerical methods are used to approximate solutions to these equations. These methods involve dividing the fluid into small cells and solving the equations at each of these cells, using mathematical techniques such as finite difference or finite element methods.
