[CFD | ENG] Introduction to the momentum conservation equation and the Navier-Stokes equations

In this video, we will mathematically derive the momentum conservation equations for fluids, which are fundamental equations for simulating fluid flows. We will start by explaining Newton's 2nd Law, which is the law under which fluid dynamics is based. Then, we will explain basic mathematical concepts such as the material derivative, which is used to describe the spatial acceleration of fluids.

We then will proceed to unify these ideas to derive both the integral and differential form of the momentum conservation equation. We will also expressed these equations in Cartesian coordinates.

Moreover, we are going to give a brief introduction to Newton's Law of Viscosity, and other important concepts, such as stress tensor, deformation tensor, extensional and shear deformations. This will lead us to obtain the Navier-Stokes equations, which are the most famous and most important equations used in CFD. We will conclude with an analogy between these equations and other conservation equations (energy and mass conservation with chemical reaction).

Timestamps
00:00 Newton's 2nd Law for solid bodies
03:05 The material derivative
09:47 The material derivative in Cartesian coordinates
12:13 Definition of concepts (momentum, stresses)
22:28 Derivation of the momentum conservation equation for fluids
27:27 Momentum equation in Cartesian coordinates
29:16 Newton's Law of Viscosity (simplified version)
38:40 Newtonian fluids
44:10 Introduction to the Navier-Stokes equations
46:20 The Navier-Stokes equations for incompressible fluids in Cartesian coordinates
48:56 Similarity with other conservation equations

This video is part of the course called "Introduction to CFD" which I am offering for free in the channel. The presentation called "Introduction to continuity equation (mass conservation)", which is the previous video, can be found here:    • [CFD | ENG] Introduction to continuit...  

The next video called "Simple shear-flow for a Newtonian fluid" can be found here:    • [CFD | ENG] Simple shear flow for a N...  

I encourage you to watch all the videos of the course. Enjoy the content.

Links of interest
Personal webpage where you can read more about my research related to fluid dynamics and other popular-science articles: https://hugocastillocom.wordpress.com