3.4. Boundary Conditions and Velocity Distribution

Now, because here it is double integral, so we will integrate it twice. So, we will get $u = \frac{1}{\mu} \frac{dP}{dx} \frac{y^2}{2} + C_1 y + C_2.$

Because of no slip condition, the velocity will be 0 at both fixed plates. Therefore, $u(0) = 0$ and $u(t) = 0.$

Using these boundary conditions in equation 9, we get $C_2 = 0$ and $C_1 = -\frac{1}{\mu} \frac{dP}{dx} \frac{t}{2}.$ Therefore, the final equation will be $u = \frac{1}{\mu} \frac{dP}{dx} \left( \frac{y^2}{2} - \frac{ty}{2} \right).$

In this chunk, we discuss the boundary conditions which are critical for solving fluid flow problems. In laminar flow between two fixed plates, the flow profile is influenced by the no-slip condition, which states that the fluid at the surface of the plate has zero velocity relative to the plate. Therefore, at both ends where the plates are fixed (y=0 and y=t), the fluid velocity is zero. This leads to the determination of constants in our velocity distribution equation, giving us a parabolic profile for velocity distribution as the fluid flows between the plates.

Think of this scenario like honey flowing between two pieces of ice. The honey adheres to the ice (fixed plates), so at the surfaces of the ice, the honey is 'stuck' and doesn’t flow (zero velocity). In the middle, however, the honey moves faster, creating a smooth, parabolic curve in terms of flow velocity, just like our velocity equation shows.

Now, repeating the same procedure of average velocity, maximum velocity, and discharge. The average velocity is again given by $V_{avg} = \int_0^t u \, dy / t.$

In this chunk, we derive the average velocity of fluid flow between the plates. The average velocity $V_{avg}$ can be calculated by integrating the velocity distribution across the thickness of the fluid layer and then dividing by the total thickness. It highlights how the parabolic velocity profile means that the maximum velocity occurs at the centerline of the flow, typically at $y = \frac{t}{2}$. Knowing how to derive these values is essential in fluid mechanics, especially for applications involving flow between plates or in pipes.

Imagine a crowded hallway where people are moving. The average speed of the crowd would be determined not just by those walking slowly at the edges (edges of the plate) but also by the few who walk quickly down the center (maximum speed at the centerline). The average velocity tells us how fast, on average, people are moving in the hallway, and just like that, our fluid has a maximum speed in the center while slower near the edges.

So, if you divide $u_{max} / V_{avg}$, what we get is, actually, $u_{max} = 1.5 V_{avg}.$

In pipe flow, it was $u_{max} = 2 V_{avg}.$ Here, it is 1.5 times $V_{avg}$ and $Q = V_{avg} \times A.$

This chunk puts into context how the calculated values relate to fluid dynamics applications. In laminar flow between parallel plates, the ratio of maximum velocity to average velocity differs from that in pipe flow due to varying flow dynamics. Knowing these relationships helps engineers and scientists predict fluid behavior in various systems, like designing pipes or channels.

Think of two water slides: one straight (like our parallel plates) and one curved (like a pipe). On the straight slide, the speed at the centerline is 1.5 times compared to the overall average speed of people coming down the slide. But on the curved slide, the maximum speed is double the average as more people might speed up at the curves. Understanding these differences helps in planning how to use space efficiently and ensure safety for everyone.

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