7.3.1. Application of Newton's Laws

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.

This chunk introduces the focus of the lecture: the Navier-Stokes equations, which are crucial for understanding fluid dynamics. It emphasizes that these equations form the foundation for computational fluid dynamics (CFD), a field used to solve complex fluid flow problems. The significance of understanding how these equations are derived and applied lays the groundwork for further discussion on fluid mechanics principles.

Think of the Navier-Stokes equations like the underlying rules of a game of basketball. Just as players need to understand and follow the rules to play effectively, engineers and scientists must understand these equations to predict how fluids behave in various situations, such as in air over airplane wings or water flowing through pipes.

The derivations part of Navier-Stokes equations that what today I will cover it. The most of the part what I am covering it, it is comes from Sinzel Cimbala book, the derivations part. If you are not following the line by line the derivations, I just encourage you to go through the Sinzel Cimbala book. If you look it for vector notation and related to the vectors, operations in fluid mechanics, the best course which is available in MIT OpenCourseWare which on fluid mechanics you can get it.

The lecture underscores the importance of studying the Navier-Stokes equations, highlighting their derivation and the resources available for a better understanding, specifically mentioning textbooks and courses. This indicates that mastering fluid mechanics involves looking at various sources to grasp these complex ideas. The reference to historical context serves to illustrate that these equations represent a significant milestone achieved by physicists Claude-Louis Navier and mathematician George Stokes, who independently developed the equations.

Imagine learning a new sport by watching videos from various coaches (different textbooks) and practicing (solving equations) based on their guidance. Just as it's important to gather different perspectives to improve your game, understanding the Navier-Stokes equations requires consulting multiple resources and practicing their applications.

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.

This section introduces the distinction between Newtonian and non-Newtonian fluids, critical for understanding how different types of fluids respond to forces. Newtonian fluids, like water, have a linear relationship between shear stress and shear rate, while non-Newtonian fluids, like ketchup or blood, do not follow this linearity. This concept is important when applying the Navier-Stokes equations, as they are primarily derived based on Newtonian fluid behavior.

Consider how different liquids pour. Water (a Newtonian fluid) flows smoothly and consistently regardless of how hard you shake the container. On the other hand, ketchup (a non-Newtonian fluid) won't come out of the bottle despite shaking; it requires a specific amount of force to flow. This variability illustrates the differences in behavior that need to be accounted for in fluid dynamics calculations.

Let us go to the next slides as 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.

Here, we transition to discussing Cauchy's equations, which form the basis for understanding momentum in fluid mechanics. The equation presented outlines how changes in fluid momentum relate to density, velocity, gravitational forces, and the stress tensor. This chunk emphasizes the significance of these components in analyzing fluid behavior, particularly how they interact under varying conditions.

Think of a crowded subway train; as the train accelerates, people (fluid particles) have to adjust their positions based on the movement (force acting on them) and the interactions with one another (stress tensor). Just like the balance of momentum among passengers, fluid dynamics also depends on these forces acting upon the fluid's particles.

The same concept we are talking about but we are talking about a control volume. That is the difference the control value acted by at the surface as a surface tension as it has a body forces which is the gravity components what we are considering it.

This chunk focuses on the application of Newton's laws within a defined control volume in fluid mechanics. A control volume is an imaginary box used for analyzing fluid behavior under the influence of various forces—gravity, pressure, and surface tension. By looking at these forces acting on a control volume, we gain insights into how fluids respond to external influences.

Imagine a swimming pool and break it down into sections (control volumes). When someone jumps into the pool (an external force), the water in that control volume must push against the edges of the pool. The water's behavior is influenced by the forces acting across these boundaries (surface tension, gravity), mirroring how we analyze fluids in a control volume.

--