# What are the Navier-Stokes Equations?¶

The movement of fluid in the physical domain is driven by various properties. For the purpose of bringing the behavior of fluid flow to light and developing a mathematical model, those properties have to be defined precisely as to provide transition between the physical and the numerical domain. Velocity, pressure, temperature, density, and viscosity are the main properties that should be considered simultaneously when conducting a fluid flow examination. In accordance with the physical incidents such as combustion, multiphase flow, turbulent, mass transport, etc., those properties diversify enormously, which can be categorized into kinematic, transport, thermodynamic, and other miscellaneous properties\(^1\).

Thermo-fluid incidents directed by governing equations are based on the laws of conservation. The Navier-Stokes equations are the broadly applied mathematical model to examine changes on those properties during dynamic and/or thermal interactions. The equations are adjustable regarding the content of the problem and are expressed based on the principles of conservation of mass, momentum, and energy\(^1\):

**Conservation of Mass**: Continuity Equation**Conservation of Momentum**: Momentum Equation of Newton’s Second Law**Conservation of Energy**: First Law of Thermodynamics or Energy Equation

Although some sources specify the expression of Navier-Stokes equations merely for conservation of momentum, some of them also use all equations of conservation of the physical properties. Regarding the flow conditions, the Navier-Stokes equations are rearranged to provide affirmative solutions in which the complexity of the problem either increases or decreases. For instance, having a numerical case of turbulence according to the pre-calculated Reynolds number, an appropriate turbulent model has to be applied to obtain credible results.

## History¶

Despite the fact that motion of fluid is an exploratory topic for human beings, the evolution of mathematical models emerged at the end of 19\(^{th}\) century after the industrial revolution. The initial appropriate description of the viscous fluid motion had been indicated in the paper “Principia” by Sir Isaac Newton (1687) in which dynamic behavior of fluids under constant viscosity was investigated\(^1\). Later, Daniel Bernoulli (1738) and Leonhard Euler (1755) subsequently derived the equation of inviscid flow which is now expressed as Euler’s inviscid equations. Even though Claude-Louis Navier (1827), Augustin-Louis Cauchy (1828), Siméon Denis Poisson (1829), and Adhémar St.Venant (1843) had carried out studies to explore the mathematical model of fluid flow, they had overlooked the viscous (frictional) force. In 1845, Sir George Stokes had derived the equation of motion of a viscous flow by adding Newtonian viscous terms, thereby the Navier-Stokes Equations had been brought to their final form which has been used to generate numerical solutions for fluid flow ever since\(^{1,2}\).

## Resources¶

\(^1\): White, Frank (1991).Viscous Fluid Flow. 3rd Edition. McGraw-Hill Mechanical Engineering. ISBN-10: 0072402318.

\(^2\): Stokes, George (1851). “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums”. Transactions of the Cambridge Philosophical Society. 9: 8–106.

\(^3\): White, Frank (2002). Fluid Mechanics. 4th edition. McGraw-Hill Higher Education. ISBN: 0-07-228192-8.

\(^4\): Cebeci, T., Shao, J.P., Kafyeke, F., Laurendeau, E (2005). Computational Fluid Dynamics for Engineers. Horizon Publishing Inc. ISBN: 0-9766545-0-4.

\(^5\): http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdf

\(^6\): https://software.intel.com/en-us/articles/fluid-simulation-for-video-games-part-1

\(^7\): http://www.mathcces.rwth-aachen.de/2research/0mms/0gases/start

\(^8\): Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2001). Transport Phenomena, 2th edition. John Wiley & Sons. ISBN 0-471-41077-2.

\(^9\): https://en.wikipedia.org/wiki/Isaac_Newton

\(^{10}\): https://en.wikipedia.org/wiki/Claude-Louis_Navier#/media/File:Claude-Louis_Navier.jpg

\(^{11}\): https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet#/media/File:Ggstokes.jpg