Stokes equation

This article refers to the Stokes equation. For further regularities, rules and theorems established by the physicist and mathematician Sir George Gabriel Stokes, see Stokes' laws .

Streamlines around a sinking sphere in a liquid; The buoyancy force is denoted here with and the gravitational force with . ${\displaystyle F_{d}}$ ${\displaystyle F_{g}}$

The Stokes equation , which is based on Stokes' law , is used to calculate the sedimentation velocity of spherical bodies in a liquid or a gas. For non-spherical bodies, instead of the particle radius , its halved equivalent diameter is used as a rough approximation . ${\displaystyle r}$

The Stokes equation is valid for slow sedimentation with Reynolds numbers smaller than one ( laminar flow ). This is the case when the inertia of the fluid is insignificant for the flow caused by the sinking body. With a higher Reynolds number, the formation of eddies must also be taken into account ( turbulent flow ), then the u. G. Formula no longer.

It follows from the approach ${\displaystyle F_{\text{Reibung}}=F_{\text{Gewicht}}-F_{\text{Auftrieb}}}$

${\displaystyle F_{\text{Reibung}}=6\;\pi \;r\;\eta \;v_{p}\!}$ ( Stokes friction ) and

${\displaystyle F_{\text{Auftrieb}}=\rho _{f}\;V_{p}\;g}$ ( static buoyancy )

${\displaystyle F_{\text{Gewicht}}=\rho _{p}\;V_{p}\;g}$ ( Gravity )

the constant rate of descent

${\displaystyle v_{p}={\frac {2}{9}}\cdot {\frac {r^{2}\;g\;(\rho _{p}-\rho _{f})}{\eta }}}$

The individual symbols stand for the following quantities :

• ${\displaystyle v_{p}}$ - sedimentation rate
• ${\displaystyle r}$ - radius of the sinking object
• ${\displaystyle V_{p}}$- volume of the particle (for balls: ) ${\displaystyle V_{k}={\frac {4}{3}}\pi \,r^{3}}$
• ${\displaystyle g}$- gravitational acceleration
• ${\displaystyle \rho _{p}}$- density of the particle
• ${\displaystyle \rho _{f}}$ - density of the fluid
• ${\displaystyle \eta }$- dynamic viscosity of the fluid.