# Hagen–Poiseuille Law

on . Posted in Fluid Dynamics

Hagen–Poiseuille law, also called Hagen–Poiseuille equation, Poiseuille equation, or Poiseuille law, gives the pressure loss in a fluid flowing through a long cylindrical pipe.  It describes the laminar flow of a Newtonian fluid through a long, straight, cylindrical pipe of constant cross-section.  It relates the pressure drop across the pipe to the flow rate and the properties of the fluid and pipe.

The law assumes laminar flow, which is characterized by smooth, ordered flow patterns and is observed at low Reynolds numbers.  The equation also assumes that the fluid is incompressible, the flow is steady-state, and the pipe is long and straight with a constant cross-sectional area.  The Hagen-Poiseuille law is commonly used in engineering and fluid dynamics to calculate pressure drop and flow rate in pipes, as well as to determine the properties of fluids from pressure flow rate data.

## Hagen–Poiseuille Law formula

$$\large{ \Delta p = \frac { 8 \; \mu \; l \; Q }{ \pi \; r^4 } }$$     (Hagen–Poiseuille Law)

$$\large{ \mu = \frac { \pi \; r^4 \; \Delta p }{ 8 \; l \; Q } }$$

$$\large{ l = \frac { \pi \; r^4 \; \Delta p }{ 8 \; \mu \; Q } }$$

$$\large{ Q = \frac { \pi \; r^4 \; \Delta p }{ 8 \; \mu \; l } }$$

$$\large{ r = \left( \frac { \pi \; \Delta p }{ 8 \; \mu \; l \; Q } \right)^{ \frac{1}{4} } }$$

Symbol English Metric
$$\large{ \Delta p }$$ = pressure loss $$\large{\frac{lbf}{in^2}}$$  $$\large{Pa}$$
$$\large{ \mu }$$  (Greek symbol mu) = dynamic viscosity of the fluid $$\large{\frac{lbf-sec}{ft^2}}$$ $$\large{Pa-s}$$
$$\large{ l }$$ = length of the pipe $$\large{ft}$$ $$\large{m}$$
$$\large{ Q }$$ = volumetric flow rate of the fluid $$\large{\frac{ft^3}{sec}}$$ $$\large{\frac{m^3}{s}}$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = pipe inside radius $$\large{in}$$ $$\large{mm}$$