Poiseuilleelvek

Poiseuille flow, also known as Hagen-Poiseuille flow, is a steady, laminar flow of a viscous fluid through a long, cylindrical pipe of constant cross-section. It is named after Jean Louis Marie Poiseuille, who first described the flow in 1840. The flow is characterized by a parabolic velocity profile, where the velocity is highest at the center of the pipe and decreases towards the walls.

The flow is governed by the Navier-Stokes equations, which describe the motion of fluid substances. In the case of Poiseuille flow, the equations simplify significantly due to the steady, laminar nature of the flow and the constant cross-section of the pipe. The resulting velocity profile is given by the formula:

v(r) = (ΔP * R^2) / (4 * η * L) * (1 - (r/R)^2)

where v(r) is the velocity at a distance r from the center of the pipe, ΔP is the pressure difference between the two ends of the pipe, R is the radius of the pipe, η is the dynamic viscosity of the fluid, and L is the length of the pipe.

Poiseuille flow is of great importance in various fields, including physiology, where it is used to model blood flow in capillaries, and engineering, where it is used in the design of fluid transport systems. The flow is also used in the study of microfluidics, where the small scales involved make Poiseuille flow a dominant factor.