Hagen–Poiseuille equation

The Hagen–Poiseuille equation gives the volume flow rate of a viscous fluid in steady laminar flow through a cylindrical tube. It relates flow to pressure drop, tube radius and length, and fluid viscosity.

Poiseuille flow: Steady laminar flow of a Newtonian fluid through a long, straight circular pipe, with a parabolic velocity profile. The equation quantifies this idealized flow regime.
Dynamic viscosity: A fluid property measuring resistance to shear deformation and flow. Flow rate is inversely proportional to this property.
Vascular resistance: The opposition to blood flow through the systemic or pulmonary circulation. The equation models how vessel radius, length, and blood viscosity affect this resistance.
Turbulent flow: Irregular fluid motion characterized by fluctuating velocities and enhanced mixing. The equation does not describe the extra momentum transport of turbulence.
Navier–Stokes equations: The equations governing momentum and mass conservation in flowing fluids. The equation follows from simplifying these fluid-motion equations for a pipe.
Pressure gradient: The spatial rate of change of pressure in a fluid. The pressure drop along the tube drives the flow.
Microfluidics: The science and technology of controlling small volumes of fluid in channels typically measured in micrometres. Its radius dependence helps predict flow through small channels.
Non-Newtonian fluid: A fluid whose viscosity varies with shear rate or whose stress is not proportional to deformation rate. Its changing rheology violates the equation's constant-viscosity assumption.
No-slip condition: The boundary condition that a viscous fluid's velocity matches a solid surface's velocity at their interface. Zero fluid speed at the tube wall helps produce the parabolic profile.
Volumetric flow rate: The volume of fluid passing a cross-section per unit time. This is the quantity the equation predicts.

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