The Navier–Stokes Equations are the fundamental mathematical equations governing fluid motion. They express conservation of mass, momentum, and energy in viscous fluids. These equations describe how velocity, pressure, and density evolve in space and time. Navier–Stokes equations apply to a wide range of flows, from slow laminar motion to complex turbulence. Despite their broad applicability, exact solutions are known only for simple cases. Numerical simulation is often required for realistic flows. Understanding these equations is central to fluid mechanics, aerodynamics, and hydrodynamics. The mathematical complexity of the Navier–Stokes equations makes them one of the most challenging problems in physics and applied mathematics.
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