8.1.1. Navier-Stokes Equation part 2

Good morning. Let us start today's class on Navier-Stokes equations. In the last class, we have derived the Navier-Stokes equations, which is the four equations, mass conservation equations and the linear momentum equations. As we have the four equations, as well as we have four dependent variables like velocity field u, v, w, and the pressures for incompressible flow. So, solving that equations you can get the solutions velocity field and the pressure field for incompressible Newtonian fluids.

In this section, the professor introduces the Navier-Stokes equations, which are vital for understanding fluid mechanics. These equations consist of four critical equations: mass conservation and linear momentum equations. The variables involved include the velocity components in three dimensions (u, v, and w) and pressure. For incompressible Newtonian fluids, solving these equations helps determine the velocity and pressure fields of the flow system.

Think of the Navier-Stokes equations like the rules for playing a complex board game. Just as players must follow certain rules to achieve a goal, fluid particles must obey these equations to accurately depict their behavior in motion, such as how water flows through a pipe.

Please always have a remember what are the assumptions we have when you are deriving basic fluid equations today as we are discussing about Navier-Stokes equations. Today I will talk about the Navier-Stokes equations, how we do the approximations of Navier-Stokes equations for a simplified fluid flow with giving a series of illustrations that how we can simplify these Navier-Stokes equations.

This section emphasizes the importance of understanding the assumptions behind the Navier-Stokes equations before proceeding with their applications. The continuity equation, which ensures mass conservation, must be considered. The professor mentions that approximations will be discussed to simplify the equations for fluid flow analysis, which is essential for practical engineering applications.

Consider a busy highway full of cars. The continuity equation is like managing traffic flow: if one lane gets congested (mass doesn't flow through), adjustments must be made in other lanes to maintain overall traffic balance, similar to how mass conservation maintains flow in fluids.

If you look at the next the linear momentum equations as we have derived this is linear momentum equations in vector forms. So, in vector forms it is comes like rho So, it is the accelerations part is equal to is rho g grade p pressures into mu Laplace operators of velocity field.

This chunk discusses linear momentum equations presented in vector form. The equations reflect the balance of forces acting in a fluid. The quantity 'rho' refers to density, while 'g' represents gravitational influence, and 'p' is pressure. The Laplacian operator applies to the velocity field, indicating how velocity gradients influence the flow. Understanding these equations helps in analyzing how fluids respond to applied forces such as pressure and gravity.

Imagine a soccer ball on a field, where the ball (fluid) experiences forces when kicked (pressure) and pulled (gravity). The linear momentum equations would describe how the ball reacts to these forces, just as they describe the behavior of fluids in various conditions.

We have considered the isothermal case the mu is a constant, we have considered incompressible flow where the rho is a constant. So, all the three assumptions are giving us equations which expanded in the x directions.

In this part, the professor summarizes key assumptions made when working with fluid flow. These include treating the dynamic viscosity (mu) as constant, assuming incompressible flow (constant density, rho), and looking at specific directions while expanding the equations. These assumptions are crucial, as they simplify complex fluid dynamics, making it easier to apply the Navier-Stokes equations in practical scenarios.

These assumptions can be compared to simplifying a recipe by assuming perfect kitchen conditions — using constant temperature and quantity of ingredients makes the resulting dish predictable, just as the assumptions in fluid flow make calculations straightforward and manageable.

If I can say that this is becomes 0 then it is for me it is easy for integrating these terms because these are easy to integrate it along this field they are.

The section discusses how certain conditions, which essentially cause specific terms in the equations to equal zero, make it easier to integrate and solve the equations. By identifying conditions where viscous effects are negligible, the complexity of the equations can be reduced, allowing for more intuitive models like the Euler equations to be used instead of the full Navier-Stokes.

Consider a ski slope: if there’s no friction between the skis and the snow (like neglecting viscosity), the skier can glide down effortlessly without complex calculations. This is analogous to simplifying fluid equations and focusing only on major interactions, simplifying the problem greatly.

Now, if you look at these terms, if I look at this d v by d t, if I just expand it I will have a components is a local accelerations component which is with respect to time and I will have a components V dot grade.

This segment elaborates on the Euler equations, which arise under conditions where viscous effects are minimal or ignored. These equations help describe fluid motion without complexities introduced by viscosity and are particularly useful in theoretical scenarios, such as ideal flow conditions where the fluid behaves as a perfect fluid.

Imagine sailing on a calm sea where the wind is steady and there are no waves. The boat moves smoothly and predictably — this situation mimics ideal flow in fluids, where the complex factors like friction can be considered negligible.

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