10.1.1. Lec 30: The Navier-Stokes Equation III

Good morning all of you. Today we are going to discuss on nebustic locations and its approximation for simple flow problems. We will discuss that part. As we discussed in the last class how we can derive the basic Bernoulli's equations from Navier-Stokes equations that what we did it Navier-Stokes equations from that we derived Euler equations then we have derived Bernoulli's equations.

Today, the lecture introduces some fundamental concepts related to flow problems, specifically focusing on nebustic locations and their approximations. The instructor recaps the derivation of Bernoulli's equations from the Navier-Stokes equations, illustrating the foundational relationships between these equations in fluid mechanics.

Imagine a river flowing smoothly. The principles discussed in this lecture, like Bernoulli's equation, help us understand how the speed and height of the water can change depending on the shape of the riverbed, similar to how air flows around an airplane wing.

So that components also today I will recap it then we will go through these today topics which mostly I am following it the book of F M White fluid mechanics books for today lectures. So you can look at the similar line of derivations in F M White book.

The instructor emphasizes the importance of earlier lessons, indicating that today's lecture will build upon those concepts. F. M. White's Fluid Mechanics book serves as the primary reference for today's discussion.

Think of this as building a house: the previous classes laid the foundation, and today’s lecture will help define the structure. Just like a builder refers to blueprints, students can refer to White's book as a guide to solidify their understanding.

We will discuss more details. very interesting things, generations of rotationality. And then we will have a look at a simple solutions for incompressible viscous flow between a fixed and a moving plate.

The lecture will cover the interesting concept of rotationality generation in fluids. This section focuses on incompressible viscous flow, particularly examining how viscous fluids behave between a stationary plate and a moving plate, which is relevant in many engineering applications.

Consider how honey moves between two surfaces—one fixed and one moving. The flow characteristics you'll learn today can help explain the sticky behavior of honey and how it gradually flows when agitated.

So basically let us start with Navier-Stokes equations which is long back about 200 years back. So the equations what we derive in last two classes we will go more detail about that. Before going that let me I just write down the basics equations is the Euler equations okay.

The Navier-Stokes equations, fundamental to fluid mechanics, have a rich history of development spanning about 200 years. The instructor introduces the Euler equations as a foundational component, relevant for understanding non-viscous, incompressible flows.

Think of the Navier-Stokes equations as the rules of a game. Just as teams need to understand the rules to play effectively, engineers must grasp these equations to predict fluid behavior in various scenarios.

If you look it when I talk about Euler equations you can understand it. It is for incompressibles and non-viscous or fixed or less flow. So this is the reasons we can apply it that means if you consider flow past a tall building okay a high rise buildings okay flow past tall building.

The validity of Euler equations is specified for incompressible and non-viscous flows. An example is given where the flow of air past tall buildings can be modeled using Euler's equations under certain conditions, emphasizing their application in real-life fluid dynamics.

Picture a breeze flowing around a tall skyscraper. Understanding how air flows in this scenario can help architects design buildings that withstand wind pressure, similar to how Euler equations aid engineers in predicting fluid motion.

But newer to structures as the vorticity is there, the turbulence behaviors are there as well as it has the flow separations that is what is not valid for this external flow.

While Euler's equations apply broadly, they falter near structures where flow separation and turbulence occur. In such regions, real-world factors complicate the simple model provided by Euler's equations.

Consider a car driving through a heavy rain; the water droplets separate from the smooth flow as they hit the car’s surface. In fluid dynamics, as in this example, exploring complex behavior near surfaces or objects reveals greater challenges in prediction.

The basic idea comes that can you write these equations with a single scalar value okay that is the simple idea comes is that instead of looking at 3 scalar velocity component why we cannot write it with a simple a scalar functions that is what is the velocity potential function.

A significant concept highlighted is that instead of dealing with three velocity components, it is possible to simplify the process by utilizing a scalar function known as the velocity potential function. This reduces complexity in analysis.

Imagine using a GPS coordinate instead of listing every step of a journey. A single number can summarize location, just like the velocity potential function simplifies the understanding of fluid motion.

Thus if I define it the velocity is a gradient of phi, phi is a velocity potential function at which condition thus this is what justified it was. If you look at the very basic components if I velocity potentials. v equal to grade phi is the gradient of a scalar components will be justified okay or the reverse is also true that when you have v is equal to 0 okay.

The discussion delineates the conditions for applying the velocity potential function, emphasizing that it is valid under irrotational flow conditions. This ability to simplify depends greatly on the flow's rotational characteristics.

Consider a still pond. The water moves smoothly (irrotational) when nothing disturbs it. When a rock is thrown in (causing rotation), the situation changes—just as fluid mechanics relies on understanding these conditions.

So we need to draw the streamlines and the potential lines to show it that because just you interpreted the gradient of the velocity potential functions indicates as the velocity field that is very interesting part of here.

To visualize fluid flow, the concepts of streamlines and velocity potential lines are discussed. They reveal the relationship between the velocity field and fluid movement, enhancing our understanding of flow characteristics.

Think of the lines drawn on a weather map indicating wind patterns. Just as those lines illustrate airflow, streamlines reveal how fluid moves, helping meteorologists predict weather changes.

Now if you look at very interesting part which I can relate it with very basic equations what you know it is the Newton's first law of motions is you know it...

The lecture connects Newton's laws of motion to the concepts of fluid rotationality. It emphasizes the importance of understanding these laws’ implications in fluid behavior and how they relate to irrotational and rotational flows.

Think about a toy spinning on a table. It maintains its motion (just like a fluid flow remains irrotational) until an external force (your hand) acts on it, demonstrating Newton's law in both solids and fluids.

The fluid initially irrotational may become rotational only if there are the four conditions...

The section outlines four specific conditions under which an initially irrotational flow can transform into a rotational flow. Understanding these conditions is crucial in predicting fluid behavior in various applications.

Imagine butter melting in a hot pan. The surface remains smooth until stirred (introducing vorticity). Similarly, fluids remain irrotational until specific conditions introduce rotation.

We cannot use the Bernoulli's equations when you have the flow changes for rotationalities and the viscous components dominate...

The lecture concludes with the idea that traditional equations like Bernoulli's may not always be appropriate in scenarios with rotational flow and increasing viscosity. Identifying the right conditions for using fundamental equations is key to accurate analysis.

Similar to using a calculator for simple math but needing a computer for complex equations, different fluid scenarios require different analytical approaches to yield accurate results.

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