1.2. Assumptions for Small Values of y

So, if the y is very small we can assume that tau is equal to tau not, where tau not is the shear stress at the pipe wall and can be assumed to be a constant. So, at the wall the shear stress is assumed to be constant and equal to tau not.

In fluid dynamics, particularly when dealing with pipe flows, shear stress (denoted as tau) can change based on the distance from the wall, represented as y. However, for small values of y (very near the wall), we simplify our assumptions by saying that the shear stress remains constant (tau = tau not). Here, tau not represents the shear stress at the very wall of the pipe. This simplification allows us to treat tau not as a constant value for calculations, making our analysis easier.

Imagine you are stirring a thick syrup. Right near the walls of the container where the syrup touches, the syrup has a slow movement (viscous effect). If you measure the speed of the syrup at a point very close to the wall, you can approximately consider its flow speed the same as at the wall itself because there isn't much room for it to move. Thus, we treat this 'wall speed' as a constant for those calculations.

And therefore, what we can say, if we substitute tau is equal to tau not in equation 16, we can obtain... du / dy, this quantity actually tau not can be written as rho. So, but the catch here is, what is the catch? We have considered small value of y. So, du / dy can be written as, 1 / kappa y and under root tau / rho is rho u *.

When we substitute tau with tau not in our equations (specifically in equation 16), we derive a relationship for the gradient of velocity (du/dy). Due to our assumption of small values of y, we can express this gradient as 1/(kappa * y). Here, kappa is a constant related to the geometry and flow properties. Additionally, we can express tau not using the density (rho) of the fluid and a velocity term (u*), linking the shear stress back to the dynamic behavior of the flow.

Think of the flow of a stream of water in a narrow channel. As you get closer to the banks (representing small y values), the speed of water changes, but if we assume certain conditions (like laminar flow), we can predict this change more easily using simpler relationships. By writing equations that involve these constants (like kappa and rho), we find patterns in how the speed of the water changes near the banks.

If you integrate the equation number 17, so, what we can get is, simple integration, it will get... Then using the boundary conditions, what are the boundary conditions? So, u at y is equal to R, where R is the radius of the pipe. We will get, u is equal to u max.

Once we have our equations established, we often need to integrate them to find velocity profiles across the flow. As we integrate equation 17, we then apply boundary conditions—certain known values at specific locations. For example, at the radius of the pipe (y = R), we assume that the velocity (u) reaches its maximum value (u max). This provides a solid point from which we can fully solve for the velocity distribution along the radius of the pipe.

Imagine you're trying to predict how the flow of honey in a pipe varies from the center to the edge. At the very edge (y equals radius), you know it moves the slowest, while at the center, where it bulges out, it moves the fastest. By integrating and applying these known positions, you can piece together an entire flow pattern just like connecting the dots on a graph to predict how quickly the honey flows.

So, if we use this boundary condition u at y is equal to R is u max, we can get u is equal to... if we substitute kappa as 0.4, we can get... and this is just simple manipulation...

By applying our boundary conditions, we can solve for unknown constants in our equations. Substituting values like kappa (0.4 here) allows us to derive a specific form of the velocity profile across the pipe. This results in a logarithmic relationship as opposed to the parabolic profile seen in laminar flow. This aspect is crucial in understanding how turbulent flows differ significantly from laminar ones.

Think of a water slide that is steep in the middle and shallow at the edges. The way water speeds up and slows down as it travels down the slide can be modeled with similar equations. When conditions change (like the slope), the way we describe how quickly the water travels (the velocity profile) also changes from smooth (like laminar flow) to more chaotic (like turbulent flow).

u max minus u is called the velocity defect or velocity defect law, this is velocity defect law.

The velocity defect law describes the difference between the maximum velocity in a flow (u max) and the actual fluid velocity (u) at any point within the flow. This difference is crucial for understanding how much fluid velocity 'defects' or falls behind the maximum due to turbulence and wall friction, impacting flow performance in various applications.

Imagine a runner on a track. The fastest speed recorded on a track is like u max, while each runner will run at their own speed, causing some to finish slower than the best time. The difference in their speeds (the defect) gives insight into how factors like fitness level (like wall friction) impact their performance—the same way flow conditions affect fluid behavior in a pipe.

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