3.2. Force Balance Analysis

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 initial part, we begin our force balance analysis by laying out the forces acting on a fluid element in laminar flow. A free body diagram helps visualize these forces: the pressure force acting in the positive direction and the shear stress acting in the opposite direction. The pressure force per unit length is represented as P delta y, while the force due to shear stress is represented as tau delta x. This setup is crucial for deriving further relationships in fluid mechanics.

Think of a thick book resting on a smooth table. If you push the book (applying a force), the frictional force (which acts in the opposite direction) resists your push. Similarly, in fluid flow, the pressure inside a fluid pushes it forward, while shear stress (like the friction from the table) opposes the motion.

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.

By dividing both sides of the force balance equation by the area (delta x times delta y), we simplify the equation. This helps us isolate the differentials – the change in pressure (dP) with respect to x and the change in shear stress (d tau) with respect to y. This mathematical manipulation is crucial as it leads us to understand how these forces interact within a fluid element as it flows.

Imagine you have a stretched rubber band. The tension in the rubber band would represent pressure, while the stretching forces along the length symbolize shear stress. If you cut the rubber band into small segments, examining tension over each segment helps us understand how the entire band behaves under stretching, similar to analyzing forces in fluid flow.

Now, this is again a laminar flow assumption and also we will also assume that the mu the viscosity is constant. So, this is the equation that we get after we put in this, this is equation 8.

Under the assumption that flow is laminar and viscosity remains constant, we can express the relationship obtained into an equation (Equation 8). This equation relates the pressure gradient to shear stress and, crucially, outlines that the behavior of fluid flow can be described mathematically. This is fundamental in fluid mechanics as it provides a direct connection between forces acting on a fluid and the resulting motion.

Consider a slow-moving stream of honey. It's thick and flows smoothly. The constant thickness (viscosity) helps us predict how the honey will flow in different conditions. Similarly, the equivalent equation in our analysis helps predict fluid behavior under varying pressures.

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.

The equation derived from our force balance needs to be integrated to consider the cumulative effect of pressure and shear stress along the flow direction (y). By performing double integration, we account for variations in both pressure and shear stress across the height of the fluid element. This integration will yield velocity profiles and help in understanding the flow behavior more completely.

Think of filling a two-dimensional container with water. As you pour the water in (the primary force), the water distributes itself throughout the container bottom, creating a varied depth profile. Similarly, double integration in our analysis allows us to capture the varying effects of forces in the flow across a depth.

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.

When setting up our equations, we must apply boundary conditions to account for real-world scenarios. The no-slip condition dictates that the fluid at the boundary (in contact with a fixed plate) has zero velocity. Similarly, at the opposing boundary (thickness t), the velocity remains zero. These conditions are crucial for ensuring that the equations reflect actual behavior of fluids in confined spaces.

Imagine a car that comes to a stop at a red light. The tires touching the asphalt (fixed plate) aren't moving. This situation affects the whole car's behavior (similar to fluid behavior), where the velocity gradient starts to form as you go away from the stopping point towards the fast-moving center of the road.

This is also something that you can try to remember because it is quite simple. Here, also the velocity distribution is parabolic, you see, ty - y square, parabolic in nature.

The resulting equation for fluid velocity profile shows a parabolic distribution, which is typical for laminar flow between parallel plates. This implies that fluid flows fastest at the centerline and slowest at the boundaries due to viscous effects. Understanding this profile is key in applications where laminar flow is critical, such as in designing certain types of piping systems.

Think of a slide in a water park. The water flows faster in the middle (the centerline), mimicking the parabolic shape of the velocity profile, while the edges (where it contacts the slide) slow down due to friction. Recognizing this helps in predicting where water moves fastest and planning better for flow systems.

--