7.3.2. Deriving Momentum Equations

Good morning all of you. Today we are going to discuss about very interesting topics, the topics on Navier-Stokes equations. That is what the basic foundations of computational fluid dynamics and many of the complex fluid flow problems also we can solve it by using Navier-Stokes equations.

In this introduction, the speaker sets the stage for discussing the Navier-Stokes equations, which are essential in the field of fluid mechanics. These equations form the basis of computational fluid dynamics (CFD) and help in solving complex problems related to fluid flow. The Navier-Stokes equations are fundamental to understanding how fluids behave under various conditions, making them crucial for engineers and scientists.

Think of the Navier-Stokes equations as the rules of a game played by fluid particles. Just as players follow rules to move a ball—like passing or shooting—fluids follow mathematical equations to move and interact with their environment. Understanding these equations allows us to predict fluid behavior, just as knowing game rules allows players to anticipate actions in a game.

if you look go through its who is Navier and who is Stokes okay the way back the French physicist okay Claude-Louis Navier okay so it is almost 200 years back and also this George Stokes from as a mathematicians from England both have derived the Navier-Stokes equations independently.

The Navier-Stokes equations are named after two individuals, Claude-Louis Navier and George Stokes, who developed them over 200 years ago. Navier was a physicist from France, while Stokes was a mathematician from England. They independently derived these equations, contributing to the mathematical treatment of fluid dynamics. Their work has been validated and built upon, leading to the comprehensive theories we use today.

Imagine inventors who create a groundbreaking device, like a smartphone. One inventor might work on the software while another develops the hardware, and both are unaware of each other's discoveries. Their combined work ultimately leads to the final product that revolutionizes communication. Similarly, Navier and Stokes independently tackled fluid equations, and their separate contributions merged to form the foundation of fluid dynamics.

I will talk about start from a introductions part. Also, I will talk about revisit Newtonian fluid and non-Newtonian fluid part which is quite interesting to look at how we can use for this Navier-Stokes equations.

In fluid mechanics, fluids are classified as Newtonian or non-Newtonian based on their behavior when they flow. Newtonian fluids, like water, have a constant viscosity, meaning they flow consistently regardless of the applied stress. Non-Newtonian fluids, like ketchup, can behave differently; their viscosity can change based on the force applied, often flowing more easily under certain conditions or becoming thicker when stirred. Understanding these differences is critical when applying the Navier-Stokes equations since they assume a certain fluid behavior.

Consider syrup and water. When you pour water, it flows readily and consistently; this is like a Newtonian fluid. Now, think about syrup, which flows slowly and thickly at first but becomes more fluid-like when shaken—this shows its non-Newtonian behavior. Knowing how to handle different fluids in recipes is similar to how engineers must adjust their calculations for different fluid types when using the Navier-Stokes equations.

If you look it from Cauchy equations what we get it just in vector forms. In vector form if you remember it which is the linear momentum equations deriving with the components like rho by rho t rho v is velocity plus is equal to del dot product of rho v is equal to rho g is a vectors plus we have the del product or divergence of the stress tensor.

Cauchy's equations describe motion based on momentum conservation inside a fluid. Using vector notation, they represent the relationship between fluid density (rho), velocity (v), external forces such as gravity (g), and stress within the fluid. The equations consider changes over time (∂/∂t) and spatial variations, allowing us to derive the dynamic behavior of the fluid. This lays the groundwork for the momentum equations we derive later from the Navier-Stokes framework.

Think of a crowded highway where cars (representing fluid elements) are moving. The speed of each car (velocity) changes based on how many cars are on the road (density) and how much pressure there is from behind (stress). Cauchy's equations help us analyze how these factors influence traffic flow, just as Navier-Stokes equations help in analyzing fluid behavior.

If you go back very basic things, the the vectors divergence okay. If you look at that del and V the velocity divergence it is talking about the net volume outflow from a control volume.

In fluid dynamics, the term 'divergence' refers to a mathematical operation that measures how much a vector field spreads out from a point. Specifically, the divergence of velocity indicates how much fluid is exiting or entering a control volume. A positive divergence means fluid is flowing out, while a negative divergence means fluid is flowing in. Understanding this helps us quantify flow rates and mass conservation, which are critical for applying the Navier-Stokes equations.

Imagine a fire hose spraying water. The area where water flows out has a positive divergence because water is leaving that space, while the area where water is sucked back in, such as from a drain, has negative divergence. By measuring how much water exits versus what enters, you can better manage water flow, much like how engineers manage fluid flows in their designs.

So if you look at this Chezy's equations if you have simple derivations alternative form okay which more derivations you can just go through the derivations go through the Senzel and Cimbala book. I am not deriving one by one.

Chezy's equations are a form of the momentum equations applied to fluid flow. They stem from Newton's second law (force equals mass times acceleration) applied to a control volume within the fluid. This means calculating the forces acting on a small volume of fluid and relating it to changes in motion. This foundational understanding is necessary to derive the Navier-Stokes equations accurately.

Consider a skateboarder on a flat surface. The push of their legs (force) leads to acceleration, propelling them forward. In fluid dynamics, Chezy's equations analyze similar forces and movements in fluid flow, providing a mathematical way to understand how feeble pushes can lead to massive motions, like a river's flow.

If I try to interpret this physically this part nothing else that is what change in momentum okay. Here we have a linear momentum within the control volume with respect to time.

This chunk discusses the relationship between changes in momentum within a control volume in relation to time. It emphasizes understanding how momentum changes due to outside forces such as body forces (gravity, for example) and surface forces (shear stress from fluid viscosity). This sets the stage for expressing the Navier-Stokes equations and how they account for these various forces acting on fluids.

Think about how a swimmer pushes the water behind them to propel themselves forward. The force they exert changes the water's momentum, influenced by both their movements (surface force) and the water pushing against gravity (body force). The Navier-Stokes equations help model these interactions mathematically, allowing predictions of swimming techniques' effectiveness depending on the swimmer's style and technique.

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