Which law describes the relationship between flow and the pressure differential, viscosity, and length?

Poiseuille's law specifically describes the relationship between flow rate, pressure differential, viscosity, and the length of a vessel. It is primarily applicable to laminar flow in a cylindrical tube and can be expressed by the formula:

[ Q = \frac{ΔP \cdot π \cdot r^4}{8 \cdot η \cdot L} ]

In this equation, ( Q ) represents the flow rate, ( ΔP ) denotes the pressure differential across the length of the tube, ( r ) is the radius of the tube, ( η ) is the viscosity of the fluid, and ( L ) is the length of the tube. This relationship emphasizes that flow is directly proportional to the pressure differential and the fourth power of the radius, while inversely proportional to the viscosity and length of the tube.

Understanding this law is critical in the context of sonography and fluid dynamics, particularly when interpreting Doppler ultrasound studies, where blood flow in vessels is influenced by those very factors.