NavierStokesSystem

NavierStokesSystem generally refers to the mathematical framework formed by the Navier–Stokes equations, which describe the motion of viscous, Newtonian fluids. In many contexts the term encompasses both incompressible and compressible formulations and includes appropriate boundary and initial conditions. The system is fundamental in fluid dynamics and underpins analyses in engineering, physics, and geophysics.

For incompressible flow with constant density, the governing equations consist of the continuity equation and the momentum equation. The continuity equation is ∇·u = 0, expressing mass conservation, while the momentum equation is ∂u/∂t + (u·∇)u = -∇p/ρ + ν∇²u + f, where u is the velocity field, p the pressure, ρ the density, ν the kinematic viscosity, and f represents body forces. The viscosity and density determine the fluid's behavior, with the Reynolds number Uz = UL/ν serving as a dimensionless indicator of inertial versus viscous effects. Boundary conditions and initial conditions complete the system on a given domain.

Key features of the NavierStokesSystem include nonlinearity and coupling between velocity and pressure, which complicate analysis and computation. Simplified forms include the Stokes equations for very slow, viscous-dominated flows and the full compressible Navier–Stokes equations for variable density and energy transport. Mathematical questions, notably the existence and smoothness of solutions in three dimensions, remain open and are central to the Millennium Prize Problems. Numerical methods, such as finite elements and projection schemes, are widely used for simulations in engineering and scientific applications.