3.6. Problem Solving for 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 shift our focus from laminar flow in pipes to laminar flow between parallel plates. Similar principles apply; however, the arrangement of the system is different. The fixed parallel plates establish a controlled environment allowing students to explore how fluids behave under shear between stationary boundaries.

Imagine squeezing a tube of toothpaste. If you hold the bottom of the tube still and push down on the top, the toothpaste flows out in a way that resembles how fluid moves between fixed plates. The toothpaste in the middle is similar to the fluid moving slower where it is closest to the surface.

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 segment introduces how we visualize the spacing, referred to as 't', between the plates, and establishes the flow direction. By dissecting the system into infinitesimally small elements (delta x and delta y), we can analyze the forces acting due to pressure and shear stresses. The balance of these forces will be critical in deriving meaningful equations for flow between the plates.

Picture two flat surfaces, like a sandwich, being pressed together. The pressure between the slices creates a scenario similar to the way forces act across the plates, showing how the 'thickness' influences movement and pressure distribution between them.

So, with this in background, we start to write the force or we do the force balance analysis along the flow direction. So, for your convenience I have made the free body diagram that is there, I have just reproduced it in the top corner. You see, P delta y, that is, force per unit length minus this one here and this again is in the negative direction tau delta x.

In this chunk, we perform a force balance analysis by identifying all acting forces. The pressure force acts in one direction while the shear force acts in the opposite direction. This balance of forces is fundamental to finding the relationship governing the flow characteristics, particularly the pressure gradient.

Think of a game of tug-of-war where one side pulls with force derived from pressure and the other side exerts a resisting force via friction (shear). The equilibrium between these forces will determine who wins – in this case, the 'winner' tells us how the fluid will flow!

Now, if we divide both sides by delta x into delta y, that is, area, what we are going to get is, so, this becomes P delta x delta y and this also become P delta x into delta y whole square. So, this will get cancelled here, here. So, if you do these cancellations what we are actually left with is dP dx here and d tau dy, rest of the things will get cancelled. So, this is one important equation that we are going to get.

Upon performing the mathematical manipulations, we arrive at a simplified equation that relates the pressure gradient (dP/dx) and the rate of change of shear stress (dτ/dy). This equation forms the basis for predicting how changes in pressure affect flow rates within the confines of the parallel plates.

Imagine a flow of cars (fluid) on a road (the plate space). If there is a speed limit enforced (pressure gradient) and traffic flow (shear), understanding how one influences the other can help city planners optimize traffic flow – much like how we optimize fluid flow using dP/dx and dτ/dy.

Now, because here it is double integral, so we will integrate it twice. So, we will get, because we have assumed y, you see, y is in this direction. So, this is the equation we get. Now, the boundary conditions are u at y equal to 0 if the plate is fixed. So, due to no slip condition the velocity will be 0 also at the thickness at the other end after the distance t.

This part discusses the application of boundary conditions, specifically the no-slip condition (fluids in contact with a solid surface don't move relative to that surface). By integrating the relationship we previously derived and applying these conditions, we can arrive at the velocity profile of the fluid between the plates, which, notably, takes on a parabolic shape.

Think of icing on a cake. The frosting spreads out more in the middle and less at the edges (the plates), representing how fluid velocity behaves where it reaches a maximum at the center and diminishes near the surfaces due to the no-slip condition.

Now, repeating the same procedure of average velocity, maximum velocity and discharge. The average velocity is again given by integral 0 to t u dy / t... This means the maximum velocity again is at the center line, which is y is equal to t / 2.

In this segment, we calculate the average velocity and maximum velocity of fluid flow between parallel plates. The average is derived through integration of the velocity profile, and it distinctly shows that the maximum velocity occurs at the midpoint of the gap between the plates. This discussion also leads to how we determine the overall discharge.

If you've ever watched a fountain, the water shoots highest at the center but slows as it splashes around. This animation is akin to how velocities vary in fluid flow between the two plates, showcasing a peak in the midsection.

--