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.


1 Answer 1


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

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

Back to your equation:

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

Thus your equation provides a relation for the dynamic viscosity.


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