Stokes' Law

A special case of the Navier-Stokes Equation can be used to determine the frictional force (called the drag force) arising from the viscosity of a fluid on a spherical body.

Definition §

Drag force on a spherical body in a Fluid

$$F_d=6\pi \cdot \eta \cdot r\cdot v⟺{\vec{F}}_d=6\pi \cdot \eta \cdot r\cdot -\vec{v}$$

Note

• $F_d$ = Drag force onto a spherical body. Being a frictional force, it opposes the direction of velocity, i.e. $-\vec{v}$ (in $\text{N}$)
• $\eta$ = Coefficient of Viscosity of the fluid (in ${\text{Ns/m}}^2\text{ or Pa}\cdot \text{s}$)
• $r$ = Radius of spherical body (in $\text{m}$)
• $\vec{v}$ = Velocity (in $\text{m/s}$)

Implications §

Can be solved for $v$:

$$v=\frac{2}{9}\frac{{\rho }_{obj}-{\rho }_{fluid}}{\eta }g{r}^2⟹v\propto r^2$$

#todo full derivation!