Understanding fluid flow behavior through pipes is critical in fluid mechanics, especially in applications like oil transportation through pipelines. Hagen-Poiseuille's law provides an exact solution derived from the Navier-Stokes equations for steady, incompressible, and laminar flow within a circular pipe. Hagen-Poiseuille's law helps determine the necessary pressure drop across a pipeline section by determining parameters like pipe length, radius, oil viscosity, and the desired volumetric flow rate.

Hagen-Poiseuille's law describes the pressure drop (ΔP) required to sustain a specific flow rate in a circular pipe under laminar conditions. The law is expressed as:

Where:

1. μ is the dynamic viscosity of the oil,
2. L is the length of the pipeline
3. Q is the desired volumetric flow rate, and
4. R is the pipe's inner radius.

This relationship highlights the role of pipe radius in determining pressure drop. Because the radius is raised to the fourth power, minor changes to the pipe's diameter can profoundly affect the pressure required to maintain a given flow rate.

Ensuring laminar flow requires calculating the Reynolds number (Re), which assesses whether the fluid motion will be smooth (laminar) or chaotic (turbulent). The Reynolds number is given by:

Where:

1. ⍴ is the density of the oil,
2. v is the mean velocity of the oil,
3. D is the pipe diameter (twice the radius), and
4. μ is the dynamic viscosity.

Flow is considered laminar for Reynolds numbers below 2100, making Hagen-Poiseuille's law applicable. If the Reynolds number exceeds this threshold, adjustments may be necessary, such as reducing the flow rate or increasing pipe diameter to maintain laminar conditions.

Hagen-Poiseuille's law can determine the optimal pipe dimensions to meet specific flow requirements and pressure constraints, whether in water distribution, wastewater management, or oil transport. Selecting the correct pipe radius ensures efficient fluid movement, minimizes energy loss, and reduces operational costs, enhancing infrastructure systems' overall performance.