Poiseuille’s law is a relationship that describes the flow of fluid through a cylindrical pipe or vessel. It states that the flow rate of fluid through a pipe is directly proportional to the pressure difference across the pipe and inversely proportional to the fluid’s viscosity and the radius of the pipe to the fourth power, and the length of the pipe.

Mathematically, Poiseuille’s law can be expressed as:

$$Q = \frac{\Delta P\pi R^4}{8\eta L}$$

where Q is the flow rate, r is the radius of the pipe, ΔP is the pressure difference across the pipe, η is the viscosity of the fluid, and L is the length of the pipe.

Poiseuille’s law is useful for understanding the factors that affect fluid flow in pipes and can be used to calculate the flow rate in a variety of applications, such as blood flow through blood vessels, air flow through airways, and fluid flow in pipelines.