Formula: Poiseuille's Equation Volumetric Flow Rate Length Viscosity Radius Pressure
$$Q ~=~ \frac{ \pi \, R^4 \, \left( \mathit{\Pi}_1 ~-~ \mathit{\Pi}_2 \right) }{ 8 \eta \, L }$$
$$Q ~=~ \frac{ \pi \, R^4 \, \left( \mathit{\Pi}_1 ~-~ \mathit{\Pi}_2 \right) }{ 8 \eta \, L }$$
$$L ~=~ \frac{ \pi \, R^4 \, \left( \mathit{\Pi}_1 ~-~ \mathit{\Pi}_2 \right) }{ 8 \eta \, Q }$$
$$\eta ~=~ \frac{ \pi \, R^4 \, \left( \mathit{\Pi}_1 ~-~ \mathit{\Pi}_2 \right) }{ 8 L \, Q }$$
$$R ~=~ \left( \frac{ 8\eta \, L \, Q }{ \pi \, \left( \mathit{\Pi}_1 ~-~ \mathit{\Pi}_2 \right) } \right)^{1/4}$$
$$\mathit{\Pi}_1 ~=~ \frac{ 8\eta \, L \, Q }{ \pi \, R^4 } ~+~ \mathit{\Pi}_2$$
$$\mathit{\Pi}_2 ~=~ \mathit{\Pi}_1 ~-~ \frac{ 8\eta \, L \, Q }{ \pi \, R^4 }$$
Volumetric Flow Rate
$$ Q $$ Unit $$ \frac{\mathrm{m}^3}{\mathrm s} $$
Volumetric Flow Rate (Volume velocity) is the volume of a fluid (e.g. water) that passes through a cross-sectional area of a pipe per second. The volume flow is due to a pressure difference \( \Delta \mathit{\Pi} = \mathit{\Pi}_1 - \mathit{\Pi}_2 \) between the ends of the pipe.
Length
$$ L $$ Unit $$ \mathrm{m} $$
Length of a pipe through which a fluid moves.
Dynamic viscosity
$$ \eta $$ Unit $$ \frac{\mathrm{kg} \cdot \mathrm{m}}{ \mathrm s } $$
Viscosity \( \eta \) (pronounced: Eta) is a property of a fluid (liquid or gas) and describes how difficult it is to move a body through this fluid.
| Liquid | Viscosity \( \eta \) |
|---|---|
| Olive oil | \( 108 \cdot 10^{-3} \, \frac{\mathrm{kg}}{ \mathrm{m}\cdot \mathrm{s} } \) |
| Honey | \( 10\,000 \cdot 10^{-3} \, \frac{\mathrm{kg}}{ \mathrm{m}\cdot \mathrm{s} } \) |
| Glycerin | \( 1500 \cdot 10^{-3} \, \frac{\mathrm{kg}}{ \mathrm{m}\cdot \mathrm{s} } \) |
| Water | \( 1.008 \cdot 10^{-3} \, \frac{\mathrm{kg}}{ \mathrm{m}\cdot \mathrm{s} } \) |
| Tar | \( 100\,000 \cdot 10^{-3} \, \frac{\mathrm{kg}}{ \mathrm{m}\cdot \mathrm{s} } \) |
Radius
$$ R $$ Unit $$ \mathrm{m} $$
Radius of the pipe through which a fluid moves. If you double the radius, then the flow rate becomes 16 times larger!
Pressure at one end
$$ \mathit{\Pi}_1 $$ Unit $$ \mathrm{Pa} = \frac{ \mathrm{N} }{ \mathrm{m}^2 } $$
Pressure at one end of the pipe. \(\Pi\) is pronounced as "Pi".
Pressure at the other end
$$ \mathit{\Pi}_2 $$ Unit $$ \mathrm{Pa} = \frac{ \mathrm{N} }{ \mathrm{m}^2 } $$
Pressure at the other end of the pipe.