In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances. These balance equations arise from applying Newton’s second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term – hence describing viscous flow. The main difference between them and the simpler Euler equations for inviscid flow is that Navier–Stokes equations also in the Froude limit (no external field) are not conservation equations, but rather a dissipative system. Navier–Stokes equations are useful because they describe the physics of many things of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things.
Coupled with Maxwell’s equations they can be used to model and study magnetohydrodynamics. The Navier–Stokes equations are also of great interest in a purely mathematical sense. This paper presents an overall view on the Navier-Stokes equations for the compressible flow of a viscous conducting medium in Eulerian frame according to various numerical methods. Strong and weak solutions are treated within coupled or decoupled references describing the space-time dependence. Initial boundary conditions are introduced and non linearity questions are discussed in terms of stochastic behaviour through the introduction of attractors and inertial manifold. Ultimately, the main steps of a calculus routine for the solutions search are presented. The Appendix hereby shows inherent theoretical issues, whereas CFD applications to aerodynamics focusing on the Navier Stokes equations are illustrated in the introductory section.