NavierStokesGleichung

Navier-Stokes-Gleichung, often called the Navier-Stokes equations, describe the motion of viscous, Newtonian fluids. They arise from conservation of mass and momentum and combine fluid inertia with viscous stresses.

In the incompressible case with constant density, the equations are ρ(∂u/∂t + (u·∇)u) = -∇p + μΔu + f, together with ∇·u = 0. Here u is the velocity field, p is pressure, μ is dynamic viscosity, ρ is density, and f represents external body forces such as gravity. For compressible fluids the momentum equation includes the viscous stress tensor τ and is completed by an equation of state.

The system is solved with appropriate boundary conditions, commonly no-slip u=0 on solid boundaries, and an initial velocity field u(x,0). It is a nonlinear, coupled set of partial differential equations in space and time.

Mathematically, the three-dimensional, time-dependent incompressible problem remains a central open question regarding global existence and smoothness; Leray’s weak solutions exist, but regularity in general is unresolved. In three dimensions, this issue is one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute.

In practice, numerical methods such as finite elements, finite volumes, and spectral methods are used, and turbulence is modeled via approaches like Reynolds-averaged Navier-Stokes (RANS) or Large Eddy Simulation (LES).

Applications span aerodynamics, weather and climate modeling, oceanography, engineering, and biomechanics. The name honors Claude-Louis Navier and George Gabriel Stokes; the German term Navier-Stokes-Gleichung is commonly used in German-language texts.