10.4.3. Deriving Velocity Distributions

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.

So 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 velocity potential is a mathematical function that helps simplify the analysis of fluid flow problems. By using velocity potentials, we can reduce the complexity of the equations we need to solve. For example, the Euler equations govern the motion of fluid under certain conditions, and when dealing with potential flows, we concentrate on the velocity being potentially derived from a single scalar function, rather than multiple components. This reduction allows for easier calculations, especially in fluid mechanics involving irrotational flows, where we can apply these potentials effectively.

Imagine you are trying to navigate through a crowded room filled with people (the fluid). It could be complicated to measure the flow of each person (velocity components) individually. However, if you draw a map (the velocity potential function) that identifies the pathways and directions that will help most people move smoothly through the room (streamlines), you can better understand and predict how they will flow together through the crowd.

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...

To use velocity potentials effectively, specific conditions must be satisfied. Notably, it should be an irrotational flow, meaning there are no swirling movements within the fluid. Mathematically, we can represent the velocity (v) as the gradient of the potential function (phi). This means that to derive the velocity components in the fluid (u, v, w), we only need to calculate the derivatives of phi, rather than solve for each component separately. Therefore, if the flow is not irrotational, the concept of a velocity potential function does not apply.

Consider a calm lake's surface on a peaceful day—there are no ripples or waves, representing an irrotational flow. If someone tosses a small stone (introducing a disturbance), you start to see ripples spreading out in circles. This scenario illustrates that as long as the lake is calm, we can utilize potential functions to describe what happens, but as soon as the water becomes turbulent, the situation requires a more complex analysis, illustrating why we need irrotational conditions.

Let me have a very simple way look at this relationship the orthogonality relationship it is not a big idea that you take a phi functions okay it is a line let me a two dimensional functions okay. So if you have a consider is phi is a constant and this line okay that is what along this velocity potential functions...

Streamlines and equipotential lines provide visual tools to understand fluid flow. Streamlines represent the paths followed by fluid particles, while equipotential lines indicate points of constant potential function (phi). Importantly, where these two sets of lines intersect, they form right angles, which is a characteristic feature. This orthogonality means that we can effectively analyze flows; if we know the behavior along one set of lines, we can infer how it influences the other set, allowing for a deeper understanding of the fluid dynamics at play.

Think about riding a bicycle on a countryside road versus through a valley surrounded by mountains. The road (streamlines) will direct how you can move while the valleys (equipotential lines) show the heights you might pass. Where the road intersects with the valleys, imagine being at a turn; you'll need to change directions, akin to the right angles formed in fluid flow. Therefore, understanding where these lines meet and how they influence each other can help you navigate smoothly through the landscape.

Now coming back to doing some simple approximations using Navier-Stokes equations to get simple solutions of between a fixed and moving plate viscous flow if you have we called cute flow okay. Conditions is that I have a fixed plate okay. So means the velocity is 0...

In fluid mechanics, understanding how fluid behaves between fixed and moving surfaces is critical. By using the Navier-Stokes equations, we can derive the velocity distribution of fluid in this confined space. Certain assumptions, like neglecting gravity in horizontal flow, help simplify these complex equations into more manageable ordinary differential equations. By applying boundary conditions, such as knowing one plate is stationary and the other is moving, we can derive simple linear velocity profiles that represent how fluid flows under these constraints.

Consider the gap between two slices of bread in a sandwich—one slice held still while you spread butter on the moving top slice. The butter (fluid) moves between the slices, and you can imagine the flow becoming easier and smoother as the top slice moves. When determining how fast the butter flows (melting and spreading) based on your spreading force, we use approximations similar to those applied in fluid mechanics when analyzing the flow between surfaces.

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