4. Detailed Derivation Process

The main objective of this module is going to be able to derive Navier Stokes equation from scratch, so how to start from the beginning and derive the Navier Stokes equation.

In this part of the lecture, the professor emphasizes that the goal is to derive the Navier-Stokes equation in a methodical manner. This equation is foundational in fluid mechanics, modeling the motion of viscous fluid substances. Starting from the basic principles, students will learn how to systematically approach the derivation, understanding each step along the way.

Think of deriving the Navier-Stokes equation like cooking a complex recipe. Just as a recipe requires you to follow step-by-step instructions to achieve the final dish, deriving the equation involves following mathematical rules and principles to arrive at a crucial formula about fluid motion.

Our difference from the regular classes to this one is going to be that I will be teaching it by hand, we will take the help of slides as little as possible.

The professor intends to teach the derivation by writing on a board rather than relying heavily on slides. This tactile approach often aids understanding, as writing by hand allows for a more interactive and detailed exploration of each step in the derivation process.

Consider how learning a new instrument works. Often, hands-on practice and guidance from a tutor can be more beneficial than watching videos. The professor's choice to use the board signifies a preference for a hands-on approach to learning.

Some general points; one is a fluid is a substance that deforms continuously under the action of a shear force, this means, it cannot resist shear. A solid however, can resist shear and remain at rest.

The distinction between fluids and solids is crucial in understanding fluid dynamics. Fluids, unlike solids, cannot maintain a fixed shape when subjected to shear forces. This fundamental difference affects their behavior and how we model them with equations like the Navier-Stokes.

Imagine trying to push a block of ice across a table (solid) versus trying to push a cup of water across the table (fluid). You'll find that the solid resists your push and maintains its shape, while the water flows and changes shape, illustrating the concept of shear.

First is kinematic property, that is, velocity, acceleration, vorticity, rate of strain, angular velocity etc.

Kinematic properties describe the motion of fluid elements in terms of velocity and acceleration without direct consideration of the forces causing the motion. Understanding these properties is essential for analyzing fluid flow characteristics effectively.

Think about driving a car. Kinematic concepts refer to how fast you're going (velocity) and how quickly you can speed up (acceleration). These were fundamental concepts before understanding what forces are needed to make the car accelerate or decelerate.

So the idea of this particular lecture today is that we are going to write what actually material derivatives are.

The material derivative combines local and convective changes in a fluid property as observed from the viewpoint of a moving fluid particle. It allows us to describe how properties, like velocity or temperature, change not just in space, but over time as well.

Imagine you're riding a roller coaster. As you move, you experience changes in your surroundings and your feelings — the material derivative captures these experiences from your moving perspective, combining the changing landscape (position) and your changing thrill (time) into one understanding.

A fluid element can undergo the following 4 types of motion or deformation: translation, rotation, extensional strain (dilation), and shear strain.

These types of motion speak to how different fluid particles interact and change shape under different conditions. Understanding these motions helps in visualizing how fluids behave under various forces and influences, which is vital for deriving fundamental equations.

Think of how dough behaves when being kneaded. It moves (translation), spins (rotation), stretches (extensional strain), and squishes (shear strain). Each of these movements illustrates the various motions that fluid elements undergo.

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