2.3. Mass Flux Through Elemental Strip

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. So, this is the leading edge. The orientation is clear to you, I hope. Now, at section 1 - 1, we consider an elemental strip of thickness dy. Now, this dy so, this is the elemental thickness dy and it is located at a distance y from the plate.

In this chunk, we are setting the scenario for our analysis. We are looking at a flat plate with fluid flowing over it at a speed U. The important aspect here is understanding that at a certain distance (x) from the leading edge of the plate, we can analyze a small section (1 - 1). This section contains an 'elemental strip' of thickness denoted as dy. Essentially, dy represents an infinitesimally small vertical slice of the flow right next to the plate. Knowing these characteristics allows us to examine properties of the flow at a specific point.

Imagine you are observing a smooth river flowing over a flat rock. If you were to take a tiny slice of the water that flows directly against the rock’s surface, that slice would be akin to our 'elemental strip.' Just as we can look at how the water behaves in that small section, we can analyze the behavior of fluid in our elemental strip next to the plate.

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. Therefore, the area is simply bdy, here. Now, the mass flux through the elemental strip is given by, ρ u into dA. ρ u is the sort of a momentum, ρ into u, mass into velocity. Of course, we have not considered the volume, right now, but we multiply it with dA, to have the mass flux. So, dA if you put dy, this becomes ρ U b into dy.

In this segment, we calculate the mass flux through our elemental strip. The mass flux (mass flow per unit area) through this strip is found by multiplying the density (ρ) of the fluid with its velocity (u) and the area of the strip (dA = bdy). The width of the strip is b, and since we’re working with an infinitesimally small thickness dy, the area of our strip becomes bdy. Therefore, the mass flux expression simplifies to ρ U b dy, where U is the velocity of the flow.

Think of this in terms of water flowing through a hose. The hose is a uniform tube (like our plate), and when you look at a small segment of this hose, the amount of water flowing through it per second can be thought of as mass flux. If the water has a certain density and flows at a certain speed, you can calculate how much mass of water passes through that small section of hose in a given time.

Therefore, the reduction in the mass flux through the elemental strip, compared to the, you know, 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.

This chunk discusses how we determine the difference in mass flux between the ideal uniform flow scenario and the actual flow within the boundary layer. The resulting reduction in mass flux can be calculated by comparing the mass flux in a region where the flow is fully developed (U) vs. the boundary layer flow (u). By factoring out constants such as ρ, b, and dy, we find that the reduction simplifies to ρ (U - u) b dy, indicating how much the velocity defect reduces the mass flow.

Imagine a river flowing smoothly (uniform flow). If you drop a stick into it, the stick will float and move with the river’s speed. Now think about the surface of the river near the bank where the water slows down due to friction with the riverbed (boundary layer). The stick would appear to lag behind this slower-moving water; the difference in speed represents the reduction in mass flux, as the slower water has less influence on moving objects.

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, that is, the total reduction in mass flux through BC. I hope, this is clear, this is equation number 1.

Here, we're discussing how to quantify the total reduction in mass flux across the entire boundary layer by performing an integration. We set up to integrate our earlier expression of mass flux reduction from the point where the flow first touches the plate (y=0) to the boundary layer's outer edge (y=delta). This provides us with a complete picture of how mass flux changes as we move through the boundary layer.

Think of counting how many apples fall from a tree. Instead of just counting one apple at a time, you can gather apples from all the branches at once. By summing them all up—that's essentially what integration does for our problem. You're tallying the total effect (in this case, total mass flux reduction) across a range, rather than just at a single point.

Now, when the plate is displaced by delta dash, such that, the velocity at delta dash is equal to U, then the reduction in mass flux through the distance delta dash is going to be, very simple. It is going to be ρ U because the velocity there is, U delta dash into b. So, we have assumed, that the place, I mean, the plate is displaced by delta dash, such that, the velocity at delta dash is equal to the uniform velocity U, that was, outside the boundary layer, then the reduction in the mass flux through that distance is going to be ρ U delta dash b, because at the plate it was 0.

In this part, we are deriving the formula for displacement thickness. We assume the plate has been shifted such that at this new position (delta dash), the fluid velocity equals U. The mass flux reduction can then be represented as ρ U delta dash b. Notably, at the plate itself, the fluid velocity is 0. Thus, this simplification provides us with an essential relationship that helps us later quantify the effects of the boundary layer.

Think about pushing a board through water. As you push the board faster, the water beneath it gets displaced. If you displaced the board by a certain length and measured the speed of water flowing at that new point, that’s akin to our concept of displacement thickness—it quantifies the impact of moving the board on the surrounding water flow.

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