Poiseuille’s law of resistance, also known as Poiseuille’s equation, 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.

The law is based on the observation that the flow of fluid in a pipe is impeded by the resistance of the pipe walls to deformation and by the friction between the fluid and the walls. The pressure difference is the driving force that overcomes this resistance, causing the fluid to flow. The viscosity of the fluid and the diameter of the pipe both play a role in determining the resistance to flow.

Mathematically, Poiseuille’s law of resistance 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 an important concept in the field of fluid mechanics and has wide-ranging applications in areas such as cardiovascular physiology, biomechanics, and fluid transport in pipelines and other engineering systems.