7. Poiseuille’s Law

So, the equation for fully developed laminar flow in pipe can be derived using 3 approaches. What are these 3 approaches? One is from Newton’s second law, which is applied directly. Second is from using the Navier-Stokes equation. The third one is from dimensional analysis.

In deriving the equation for fully developed laminar flow in a pipe, we can use three main approaches:

1. Newton’s Second Law: This law relates to the motion of fluids and can be directly applied to analyze the forces acting on the fluid within the pipe.
2. Navier-Stokes Equation: This equation provides a comprehensive mathematical description of fluid motion by taking into account viscosity and other factors affecting flow.
3. Dimensional Analysis: This approach helps us understand the relationships between different physical quantities through their dimensions, enabling us to derive important relationships in fluid dynamics.

Think of the three approaches like different strategies for reaching a destination. For instance, you could take the direct route (Newton’s second law), navigate the traffic patterns (Navier-Stokes equation), or use a map to understand the distance (dimensional analysis). Each method may lead to the same destination but offers unique insights into the journey.

So, this is the snapshot of a fluid element at time t, this one here. And in the fully developed laminar flow this is the velocity profile, as we have seen in our laminar and turbulent flow analysis. And this velocity is only a function of radial distance r.

In a pipe exhibiting fully developed laminar flow, we can represent the fluid as a small cylindrical element.

• The velocity of the fluid varies across the radius of the pipe but remains constant at any given radial position.
• This means that as you get closer to the wall of the pipe, the velocity of the fluid decreases to zero at the wall, due to the no-slip condition, while it is at its highest at the center of the pipe.

Imagine smooth, slow-moving traffic on a highway, where the cars near the edges (the walls) are moving slower than those in the center. Just like the cars, the fluid particles near the walls of the pipe lose speed due to friction, while those in the middle can move faster.

The shear stress τ varies linearly with r and is defined by the equation τ = -µ (du/dr), where µ is the dynamic viscosity of the fluid.

In laminar flow, the shear stress, which is the internal friction within the fluid, increases linearly as you move from the center of the pipe towards the walls. This relationship is expressed mathematically as τ = -µ (du/dr), which indicates how shear stress is directly related to the velocity gradient within the fluid.

• In essence, the steeper the gradient (or the change in velocity across a distance), the greater the shear stress experienced by the fluid.

Think about spreading butter on bread. If you push the knife harder (increasing the velocity gradient), it takes more force to spread smoothly (increasing shear stress). The butter close to the knife moves faster than the butter near the bread, similar to how fluid velocity changes across a pipe's diameter.

So, we have determined u as a function of r and the flow rate across the entire pipe will be integral of 2 pi r dr, you see, the area. If, we start integrating at a distance, we take an element of thickness dr, at a distance r, we can write, Q will be u integral dA and we integrate from r = 0 to D/2 that the half.

To derive Poiseuille’s Law, we need to calculate the flow rate (Q) through the pipe.

• We expressed the flow velocity (u) as a function of radial distance (r) and integrated this over the pipe's cross-sectional area, which gives us the total flow rate. The integral of the function involves considering the area of thin disks of fluid at varying radii within the pipe. The result derived from this integration leads us to the well-known formula for volumetric flow rate in laminar flow through a cylindrical pipe.

Consider how water flows out of a garden hose. If you cover a little of the opening with your thumb, the flow increases at the tip. The same concept applies in a pipe: the tighter the pipe or smaller the section area through which the fluid flows, the more controlled and increased the flow rate becomes.

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