# Dynamic or Kinematic viscosity

Utilising the formula $$\eta = \frac { \left( 2\left( { p }_{ s }-{ p }_{ l } \right) g{ r }^{ 2 } \right) }{ 9v }$$

Where $$p_s$$ is density of a sphere with radius $$r$$ traveling at velocity $$v$$ through a liquid $$l$$. I would like to know whether this equation is utilised to find kinematic or dynamic viscosity.

Thanks in advanced for the help.

It is quite easy to figure out using dimensional analysis. The SI unit of the dynamic viscosity commonly denoted $$\eta$$ or $$\mu$$ is the pascal-second (also called poiseuille named after Jean-Léonard-Marie Poiseuille). Therefore its dimension is

$$\left[\eta\right] = \mathsf{M}\mathsf{L}^{-1}\mathsf{T}^{-1}.$$

Since the kinematic viscosity $$\nu$$ is a diffusivity coefficient

$$\left[\nu\right] = \mathsf{L}^2\mathsf{T}^{-1}.$$

$$\eta \propto \frac{(\rho_\mathrm{s}-\rho_\mathrm{l})gr^2}{v} \Longrightarrow \left[\eta\right] = \frac{[\rho][g][r]^2}{[v]} = \frac{(\mathsf{M}\mathsf{L}^{-3})(\mathsf{L}\mathsf{T}^{-2})(\mathsf{L})^2} {\mathsf{L}\mathsf{T}^{-1}} = \mathsf{M}\mathsf{L}^{-1}\mathsf{T}^{-1}$$