Hagen-Poiseuille flow describes a viscous fluid's steady, incompressible flow through a cylindrical tube with a constant radius R. This flow profile is often applied to understand fluid transport in narrow channels, such as capillaries. It serves as a foundational example of laminar flow. In this model, cylindrical coordinates (r,θ,z) are used to describe the radial (r), angular (θ), and axial (z) dimensions within the tube. For Hagen-Poiseuille flow, the velocity profile is purely axial,...

A streamline represents the trajectory that is always tangent to the fluid's velocity vector at any given point. The velocity of a fluid particle is always directed along the streamline, ensuring the particle continuously follows the streamline's path. Streamlines are particularly useful for visualizing the overall direction of flow in a fluid system, and they provide an instantaneous representation of the flow's velocity field. In steady flow, where conditions do not change over time,...