2.2. Flow Over a Smooth Flat Plate

Now, we consider the flow over a smooth flat plate, like this. So, there is a flow which is coming with a speed U, there is a flat plate and at any distance x, there is a section 1 - 1. I will remove this, but just to mark, this is the section 1 – 1 and, that is, located at a distance x, from the leading edge.

In this chunk, we begin discussing the scenario of flow over a smooth flat plate. A flat plate is placed in a fluid (like water or air) with a uniform speed, denoted as U. At a specific point on the plate, referred to as the leading edge, we can measure distances along the plate. This setup is crucial for understanding how fluid behaves when it makes contact with surfaces.

Imagine a flat soup ladle moving through a pot of soup. The ladle is like the flat plate, and as it moves through the soup, the fluid sticks to the surface of the ladle, just like fluid sticks to the flat plate when they are in motion.

Now, at section 1 - 1, we consider an elemental strip of thickness dy. Suppose, if b is the width of the plate, that is, the width, then the area of the strip can be given by, b into dy. The strip will have the height dy and the width is b.

We focus on a small strip of thickness dy located at a distance y from the plate's surface. To understand the flow characteristics at this section, we calculate the area of the strip (height dy and width b), which is b * dy. This area is important because it helps us determine the mass flow rate across this strip as the fluid moves past it.

Consider a thin slice of bread (the strip) being smeared with butter (the fluid). The thicker the butter applied, the more you will have on that slice, similar to how mass flux changes based on velocity and area.

Therefore, the reduction in the mass flux through the elemental strip, compared to the uniform velocity profile will be the difference of the mass fluxes. This is due to uniform, that is, in the boundary layer, or if we take ρ outside, b and dy outside, it becomes ρ into U minus u into bdy.

In this chunk, we calculate the difference in mass flux between the elemental strip and the conditions if the flow were uniform (i.e., how it would behave without a boundary layer). The mass flux normally would be ρ U, but because of the boundary layer formation (u < U), there is a reduction, represented by ρ (U - u). By rearranging, we can express the total reduction in mass flux as an integral across the strip's thickness dy.

Picture a garden hose with a nozzle. When you partially cover the end of the hose, the flow of water (mass flux) decreases significantly compared to when the nozzle is fully open. This is similar to how mass flow reduces due to the boundary layer effects on the flat plate.

Now, the total reduction in mass flux through BC, will be, we have considered and its thickness of dy. So, what we are going to do? We are going to simply integrate over this length. And let us say, this is so, let me, so, this distance is delta. So, the integration goes from 0 to delta and ρ U - u b dy.

Here, we take the previously defined mass flow reduction and integrate it from the surface of the plate (0) to the thickness of the boundary layer (delta). This integral gives us a cumulative measure of mass flux loss affecting the overall momentum transfer due to viscous flow against the plate.

Think of it like pouring sand through a cone. If the cone narrows, the sand flow slows. By integrating the rate of flow at each point from the cone's tip to the top, we can determine how much sand has been lost in the process.

Thus, for an incompressible fluid, we obtain, delta dash is equal to ... and this is the equation for the displacement thickness.

We derive the relation for displacement thickness (delta dash), which is a measure of how much the streamline outside the boundary layer has been pushed away due to the viscous effects at the plate. The derivation leads to a formulation used for calculating delta dash in real-world applications.

Consider a swimmer pushing through water. The water gets displaced, creating a wake as they move. The distance the water is pushed aside from its original position gives an idea similar to displacement thickness, representing how far the streamline has shifted.

Now, proceeding, what is the momentum thickness theta? So, momentum thickness theta is the loss of momentum flux in the boundary layer, as compared to that of the potential flow.

The momentum thickness (theta) refers to the reduction in momentum flux experienced in the boundary layer compared to what it would be under potential or ideal flow conditions. Understanding theta helps assess efficiency losses in fluid systems.

Imagine a train moving through a tunnel. The flow of air in the tunnel is like the ideal flow, while the air moving around and through every small crack (boundary layer) creates additional resistance, much like how momentum is lost.

Now, we have to make note of some important point, that is, the boundary layer theory is based on the fact that the boundary layer is thin.

A key assumption in boundary layer theory is that the boundary layer is thin compared to the overall flow dimensions. This concept underpins many calculations and simplifications used in fluid mechanics, indicating that x (distance from leading edge) is significantly larger than thickness measures like delta, delta dash, and theta.

Think of a stick being waved through a stream of water. The thickest part of the water’s movement is around the stick (boundary layer), while further away, the water flows freely. The stick's influence is limited to just a thin layer near its surface.

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