Archimedes number#

Named after: Archimedes (c. 287-c. 212 BCE).

$$\text{Ar} \stackrel{\text{def}}{=} \frac{g L^{3} \rho (\rho_s - \rho_f)}{\mu^{2}} \sim \frac{\text{buoyancy}}{\text{viscous resistance}}$$

Measures buoyant forcing on a particle or bubble relative to viscous resistance. It indicates the settling or rising regime set by density contrast.

Name	Symbol	SI units	Dimension
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
solid-fluid density difference	$\rho_s - \rho_f$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$

Particle settling

Range	Regime	Description
0 – 1	viscous settling	Viscous resistance dominates buoyant motion. Particles or bubbles settle slowly in creeping flow.
1 – 1000	transitional settling	Buoyancy, viscosity, and inertia are all important. Wake effects and shape sensitivity may appear.
1000 – ∞	inertial settling	Buoyant forcing is large compared with viscous resistance. Inertial wakes and unsteady paths become important.

Name	Symbol	SI units	Dimension
gravitational acceleration	\(g\)	\(\mathrm{m}\,\mathrm{s}^{-2}\)	\(\text L\,\text T^{-2}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
solid-fluid density difference	\(\rho_s - \rho_f\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
dynamic viscosity	\(\mu\)	\(\mathrm{Pa}\,\mathrm{s}\)	\(\text L^{-1}\,\text M\,\text T^{-1}\)