## A Pedagogical Derivation of the Navier-Stokes Equation

**Authors:** Armando V.D.B. Assis

This brief paper is part of my research on the origins of turbulence. Since the derivations of the Navier-Stokes equation are frequently cumbersome, I would like to provide this pedagogical derivation (I hope), discussing the properties of the continuum fluids under a heuristical approach, viz., we provide a heuristical derivation of the so-called Navier-Stokes equation. We turn out to be concerned with the physical insight regarding the system under consideration, a system of continuum. Derivations of the Navier-Stokes equation are, frequently, pedagogically cumbersome, loosing the main heuristics one should grasp under the transition to the continuum. This transition turns out to naturally encapsulate neglected degrees of freedom due to the intrinsically thermodynamic domain. This pedagogical derivation discusses the properties of the continuum fluids and the relation to the taken limit encapsulating the continuum hypothesis, which turns out to raise the question of lack of validity over extremely distorted subdomains, once a grown rarefied subdomain may not provide sufficient large statistics to a smooth description via its center of mass, which is the main hypothesis of the infinitesimal limit process for the local description under the continuum hypothesis. Such transient, albeit not presented here, once it would change the characteristic of this paper to the research one connected to the important question of unicity of the (3+1)-dimensional Navier-Stokes differential equation, is to be pointed out, once it provides ansatz for research on the unicity of description of fluids by the Navier-Stokes equation.

**Comments:** 13 pages. English.

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### Submission history

[v1] 2011-12-20 17:51:50

[v2] 2012-01-23 22:23:59

[v3] 2012-01-29 15:09:21

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