3. Laminar Flow Between Parallel Plates

Now we have done the laminar flow through the circular pipe. Now we are going to see the laminar flow between the parallel plates. So, this is a parallel plate. So, whatever assumptions that we have made for laminar flowing pipes are valid here also. In this case both the plates are fixed plate and this is also fixed plate. As you can observe, this is x direction and this is y direction.

In this chunk, we introduce the concept of laminar flow occurring between parallel plates. Laminar flow is characterized by smooth, orderly fluid motion where layers of fluid slide past one another without mixing. The scenario under discussion involves two fixed parallel plates, where fluid flows between them. By recognizing that the assumptions applied to circular pipe flow are also valid here, students understand continuity in fluid mechanics principles.

Imagine sliding your hand across a smooth surface, like a sheet of ice. As you glide, the layers of air right above the ice move together without causing any disturbance to each other, similar to how fluid flows smoothly between the fixed plates.

The thick the distance between the plate is given by t, the flow is in this direction positive x axis and what we have assumed is an element having the thickness this length is delta x and this is delta y this is the direction. And as we have written all the forces in a similar fashion as we did it for the pipe flow.

This chunk focuses on the force analysis within the flow between the plates emphasizing the force balance in the x-direction. The thickness of the fluid layer between the plates is defined as ‘t’, and a fluid element is conceptualized with lengths ‘delta x’ and ‘delta y’. By analyzing forces acting on this element, including pressure forces and shear forces, students can visualize the dynamic behavior of the fluid under laminar flow conditions.

Think of a thin layer of honey between two flat sheets. As you attempt to pull one sheet away from the other, you can feel the resistance due to the honey's viscosity. This resistance mimics the shear forces and pressure balance discussed in the flow analysis between parallel plates.

So, now we have established that flow is laminar, therefore, Q is going to be the formula that we have derived — Q = - (pi / 8 mu) (dP/dx) R^4.

In this part, we acknowledge that since the flow is laminar, we can utilize the derived equation for volumetric flow rate (Q). Specifically, we reintroduce the equation that relates flow rate with pressure gradient and other parameters. This relationship allows us to model and predict the flow characteristics of the fluid passing between the plates.

If you consider a garden hose with varying water flow based on how much you restrict its opening, similar dynamics take place here. The flow rate of water depends on the hose diameter, the viscosity of the water, and how much you compress the hose, which parallels how pressure gradient influences laminar flow rates.

Now, because here it is double integral, so we will integrate it twice. So, we will get. This is the equation that we get after we put in this, this is equation 8...

In this chunk, we delve into the mathematical process required to find the velocity distribution within the fluid. By applying boundary conditions (like the no-slip condition at the fixed plates), we perform integration to derive the velocity profile equation. This mathematical foundation is crucial for understanding fluid behavior in laminar flows.

Consider the way syrup flows when you pour it over pancakes. The syrup sticks to the edges of the plate (the no-slip condition), and how it moves from the top towards the bottom displays a smooth gradient in velocity. The principles here reflect the same flow dynamics governed by the equations we are examining.

Here, also the velocity distribution is parabolic, you see, ty - y square, parabolic in nature. This is how it looks.

This section focuses on the velocity profile resulting from laminar flow between parallel plates. The established velocity distribution is parabolic, meaning that the fluid velocity is highest at the center and decreases towards the plates. This parabola shape is characteristic of laminar flows and essential to understand how different layers of fluid experience varying velocities.

Imagine a crowded highway where cars tend to drive faster in the middle lanes compared to those closer to the edge. In the same way, the fluid particles in the center of the flow between the plates travel faster than those close to the surface of the plates.

--