NavierStokeslikningen

NavierStokeslikningen, commonly referred to as the Navier–Stokes equation, is a fundamental description of fluid motion. It comprises a set of nonlinear partial differential equations that govern the velocity field and pressure of a viscous fluid. In the incompressible Newtonian case, the equations are: continuity equation ∇·u = 0 and the momentum equation ∂u/∂t + (u·∇)u = -∇p/ρ + ν∇^2 u + f, where u(x,t) is the velocity field, p is the pressure, ρ is the density, ν is the kinematic viscosity, and f represents external body forces such as gravity. For compressible flows, the momentum equation is combined with an energy equation and a state equation.

The Navier–Stokes equations express conservation of mass and momentum and are used to model phenomena ranging from laminar pipe flow to turbulent atmospheric dynamics. The Reynolds number Re = UL/ν, based on a characteristic velocity U and length L, characterizes the relative importance of inertial to viscous forces and helps predict flow regimes.

Boundary and initial conditions are required to solve the equations: an initial velocity field u(x,0) = u0(x) and boundary conditions such as no-slip on solid boundaries (u = 0) or slip conditions on membranes.

Mathematically, the Navier–Stokes problem raises important questions about existence and smoothness of solutions. In three dimensions, global existence of smooth solutions from smooth data is an open problem; weak solutions exist (Leray), and the issue is one of the Clay Mathematics Institute Millennium Prize Problems.

Applications span engineering, meteorology, oceanography, and biomechanics, underscoring the central role of the Navier–Stokes equations in fluid dynamics.