Nearly all fluid flow is governed by the Navier-Stokes equations. This is impressive considering the diversity of all possible flowfields. The Navier-Stokes equations lose relevance for non-continuum flows such as rarefied gasses. Here, the molecules behave more independently of each other and the theory of gas kinetics applies.

Navier-Stokes is a system of nonlinear, second-order, inhomogeneous partial differential equations as shown below in vector form, without source/sink ter5ms. The five vector equations describe the transport of mass, momentum (x,y, and z), and energy. The description of chemical reactions requires additional equations. Despite their broad applicability, the Navier-Stokes equations are unsolvable in closed form without gross simplifications. Only the most simplified, usually one-dimensional, forms can be solved in closed form or with simple numerical analysis. This realization set the stage for traditional fluid mechanics analysis and design based on one-dimensional and experimentally-derived empirical methods.

Over the last 25 years, the rise of Computational Fluid Dynamics (CFD), which is capable of addressing the full equations, has led to a revolution in the analysis and design of fluid dynamic components. Even with CFD, some simplifications to the Navier-Stokes equations are usually made to reduce computing time.