2.4. Total Reduction in Mass Flux

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. Whereas, for the uniform velocity profile, so, the uniform velocity profile that exists in a region below this, the mass flux is given by, ρ into U, because this is the full velocity U. Here, it is u as a function of distance y, this one, so, here is ρ U b into dy, if there was the flow was uniform, and there was no plate, for example, and if you would have considered the same elemental area of thickness dy.

In this chunk, we discuss how mass flux is calculated in a fluid flow scenario over a flat plate. The mass flux (which is the mass flow per unit area) through an elemental strip is represented as ρu into dA. Here, ρ is the density of fluid, and u is the velocity at that point. When we take a small area differential (dA) equal to b multiplied by dy (where b is the width of the plate and dy is the thickness of the elemental strip), the mass flux becomes ρU b dy for the overall flow with a uniform velocity profile. This comparison helps establish a base for understanding how this changes in the presence of boundary layers.

Think of a garden hose. When you turn the water on, the flow of water comes out uniformly. This uniform flow can be imagined as a mass flux where water is flowing at a certain density and speed through a consistent area. However, if you were to place a fence within the stream of water (representing the flat plate), the water would start to flow differently around the fence, altering how much water (or mass flux) reaches certain areas downstream, just like the velocity profile changes near the boundary layer.

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 reduction in the mass flux caused by the presence of the boundary layer. The mass flux in a uniform flow can be expressed as ρU b dy, while in a non-uniform flow (within the boundary layer), it is expressed as ρu b dy. The difference between these two mass fluxes shows us how much mass flux has been reduced due to the boundary layer effects. This reduction can be summarized simply as: the change in mass flux due to flow separation caused by viscosity, resulting in differing velocities across the boundary layer.

Imagine a large crowd of people walking on a smooth surface. If everyone is walking at the same speed, it's like the uniform flow with the maximum mass flux. Now, if there's a narrow passage ahead that a few people have to squeeze through slowly, the overall flow is interrupted. The number of people that can pass through that area is reduced, similar to how the mass flux is reduced when a boundary layer forms in a fluid flow around an object.

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.

To find the total reduction in mass flux over an area influenced by the boundary layer, we integrate the expression we derived for the reduction in mass flux (ρ(U - u)b dy) from the bottom of the boundary layer (y=0) to its upper limit (y=delta). This gives us a complete picture of how much total mass flux is affected over this height, leading to what we define in terms of equation number 1. This approach illustrates how differential elements contribute to the overall system dynamics in engineering fluid mechanics.

Think of an hourglass: as the sand (representing mass flux) flows from the upper bulb to the lower bulb through the narrow passage (akin to the boundary layer), the total amount of sand that passes through is a cumulative effect of how much sand is passing at each moment. If there were more grains stuck near the sides (like the viscous effects in our case), less sand would reach the bottom bulb compared to if it were flowing freely.

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 section, we consider the situation where the plate is displaced by a certain amount delta dash. When this happens, we assume the velocity at this new hypothetical layer is equal to U — the velocity of the uniform flow outside the boundary layer. The mass flux over this section can now be simply calculated as ρU delta dash b (where b is the width) because it accounts for the entirely uniform state. This illustrates how boundary interactions change our understanding of mass flux within a fluid flow system.

Picture a river in which a boat moves downstream (the plate). If the boat suddenly accelerates and matches the speed of the water around it, the amount of water it displaces as it moves becomes much more predictable — you could say it’s now flowing uniformly with the water. This mirrors the idea that displacing the plate leads to a uniform effect on mass flux, akin to how the water interacts with the boat moving through.

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