8. Fluid Mechanics

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.

The Navier-Stokes equations describe how fluids move and are essential for understanding fluid mechanics. They consist of four equations that represent mass conservation and linear momentum. For incompressible Newtonian fluids, the dependent variables are the velocity components (u, v, and w) and pressure. Solving these equations enables us to determine the velocity and pressure fields in fluid flows.

Think of the Navier-Stokes equations as the rules of a game for fluids. Just like players need to follow certain rules to play effectively, fluids follow these equations to move and interact. Whether you're studying how a river flows or how air moves over an airplane wing, these rules apply.

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.

When using the Navier-Stokes equations, specific assumptions are made to simplify the equations, especially for various types of fluid flows. These assumptions may include treating the fluid as incompressible and Newtonian, where the density remains constant, and the viscosity does not change significantly. Understanding these assumptions is crucial for applying the equations correctly.

Imagine trying to bake a cake. If you want to make a chocolate cake, you assume you have the right ingredients—flour, sugar, eggs, and chocolate. If you decide to add other ingredients unpredictably, the final product may not turn out as expected. Similarly, if fluid mechanics practitioners don't adhere to these key assumptions, the results of their calculations may not be reliable.

If you look at this Navier-Stokes equations which is in Cartesian coordinates if I can write in vector forms which is easy to remember it is del cross dot v divergence of velocity vectors is 0 for incompressible fluid flow and that is the equations of continuity equations.

The continuity equation for incompressible flow states that the divergence of the velocity field is zero. This means that fluid does not compress, and the liquid's density remains constant. The equation captures the principle of mass conservation, emphasizing that what flows into a volume must flow out if there is no accumulation of fluid.

Consider a garden hose full of water. If you cover the end of the hose, the pressure builds up, and no water can escape. Similarly, in a system governed by the continuity equation, if fluid cannot compress or disappear, what goes in must come out, just like the water in a hose.

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.

The linear momentum equations describe how the momentum of a fluid changes due to different forces, like pressure, gravity, and viscous forces. In vector form, this relationship helps us understand how these forces interact and influence the overall motion of the fluid. It’s crucial for predicting behavior in systems where multiple forces act on the fluid.

Think of riding a bicycle. Your acceleration changes when you push harder (increasing force) or go uphill (gravitational force). Similarly, the linear momentum equations analyze the different forces acting on a fluid to determine how it moves—just like you must adjust your pedaling to maintain speed on different terrains.

So, all these terms I have to look it which are having a dominancy part. How I can simplify these equations for examples if I consider the mu is very very close to 0.

Identifying which terms in the Navier-Stokes equations have a dominant effect is crucial for simplification. If for instance, the viscosity (mu) is negligible (close to 0), we can ignore specific terms and reduce the equations to a simpler form. This approach allows us to focus on the most impactful forces and interactions while simplifying the analysis and computations.

Imagine trying to read a book in a noisy cafe. If you focus only on the conversations closest to you and ignore background sounds, you can understand the plot more easily. This simplifies the reading experience, just as ignoring less relevant forces in fluid mechanics helps simplify complex equations for analysis.

If I read it again, the equations that I have the linear momentum equations I just again writing for you linear momentum equations. Vector forms what we get it the rho d v d t this is the accelerations component minus rho g del phi plus the love as operators of v. Laplandian operators of V.

When specific terms become negligible or zero, the Navier-Stokes equations can be transformed into Euler's equations, which apply under conditions where viscous forces are minor. Euler's equations represent a more straightforward relationship among the fluid variables, allowing for easier calculation of motion in fluid dynamics.

Think of Euler's equations as a simplified version of a travel guide where only the most critical landmarks are shown. While detailed maps (like the Navier-Stokes equations) provide extensive information, choosing to focus solely on essential routes makes travel faster and less complicated.

Now, if you will go for the next part which is the boundary conditions which I just introduced to you that first conditions is no slip boundary condition.

Boundary conditions define the behavior of a fluid at its interface with surfaces, such as walls or free surfaces. The no-slip condition states that a fluid in contact with a stationary surface will have zero velocity relative to that surface. This requirement is essential for accurately modeling fluid dynamics and understanding how fluids interact with their environment.

Imagine sliding your hand across a smooth glass surface. The glass remains still, and so does the layer of fluid at the surface of the glass. Just as you don’t slide past the glass, the fluid has no relative velocity at the boundary; it sticks to the glass. This no-slip condition is a foundational concept in fluid mechanics.

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