Derivation of the Navier–Stokes equations

The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as the application and formulation for different families of fluids.

The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum, in other words is not made up of discrete particles but rather a continuous substance. Another necessary assumption is that all the fields of interest like pressure, flow velocity, density, and temperature are differentiable, weakly at least.

The equations are derived from the basic principles of continuity of mass, momentum, and energy. For that matter, sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be applied. This finite volume is denoted by ${\displaystyle \Omega }$ and its bounding surface ${\displaystyle \partial \Omega }$. The control volume can remain fixed in space or can move with the fluid.

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