1.8. Differential Form of the Equation

Now, equation number 2 can be written as simply you see, so this is pressure at x + dx - P x by dx, so it can be written as, dP dx and this can be written as, d r into tau dr. So, this is the differential form of the equation.

This portion discusses how to express the pressure difference in a pipe as a differential equation. The notation dP/dx indicates how pressure changes along the length of the pipe (x direction). The term 'tau dr' suggests that shear stress (tau) varies with the radius (r) of the pipe, and when multiplied by thickness (dr), it contributes to the overall flow behavior in the system.

Imagine water flowing through a garden hose. As you push on the hose at one end, the pressure changes at various points along the hose. Similarly, in our equation, we are analyzing how pressure changes over a small section of the hose and how that affects the flow.

Now, if we use the standard tau is equal to minus mu du dr, why do we do this? Actually this is an assumption for laminar flow. So, if we have a laminar flow we can assume shear stress as a function of minus, you know, as a function of du dr or in other terms tau is equal to minus mu du dr.

In this chunk, we see the introduction of the shear stress equation specific to laminar flow. Here, 'tau' (shear stress) is represented as a function of the fluid's dynamic viscosity (mu) multiplied by the gradient of velocity (du/dr). This means that in laminar flow, where the fluid moves in smooth layers, shear stress is directly proportional to how quickly the velocity changes with distance.

Think of spreading butter on bread. If you spread it slowly, the butter layers slide smoothly over one another, similar to laminar flow. However, if you spread it too forcefully, the butter may splatter, akin to turbulent flow. In laminar flow, this gradual change in butter thickness represents how shear stress builds up.

So, now what is dP dx? So, to find dP dx, we have to consider the force balance on fluid element of radius R. So, we consider a fluid element of radius R, it is quite simple, their pressure here is P. So, it becomes pi R square P, this force and this force is pi R square P + dP and the same thing is there is the shear stress acting here.

This text segment outlines how to calculate the pressure gradient (dP/dx) in a fluid element. A cylindrical element of fluid with radius R is analyzed, where the pressure exerted on the element can be described mathematically. The forces acting on this fluid element (pressure and shear) are balanced to derive the relationship necessary for understanding flow dynamics.

Imagine a balloon. When you press on one side, the pressure inside the balloon changes. In our analysis, we're looking at a small piece of that balloon and figuring out how the pressure from your fingers (dP) affects the whole structure, just like analyzing a fluid element in the pipe.

If we integrate this equation, equation 4, so, this is the equation 4, after obtaining this result here.

Integrating the equation derived from the fluid element balance allows us to relate shear stress, velocity, and pressure changes over the length of the fluid. This mathematical operation helps us find the velocity distribution across different radii in the pipe, which is crucial for analyzing fluid behavior under laminar conditions.

Consider a water slide. As you slide down, different parts of your body might go at varying speeds based on where they are on the slide. Similarly, by integrating our equation, we can understand how different layers of fluid move within a pipe - some layers move faster while others move slowly due to friction, just like the varying speeds on the slide.

So, if we utilize the above boundary conditions, equation 5 will take the following form. So, C 1 and C 2 will be eliminated and we will have u as a function of r will be given as ...

In this chunk, we implement boundary conditions to solve for constants in our equations. These conditions define specific parameters, such as no-slip condition at the wall (0 velocity at the wall) and symmetry at the centerline. Using these, we determine the velocity profile of the fluid within the pipe, illustrating how velocity varies with the distance from the center.

Think of a traffic jam in a narrow street. Cars closer to the edges (representing the pipe walls) move much slower compared to cars in the center of the street. This analogy helps you visualize how fluid velocity changes across a section of the pipe, where the speed of flow is highest at the center and gradually decreases towards the walls.

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