Archimedes number

In viscous fluid dynamics, the Archimedes number (Ar), is a dimensionless number used to determine the motion of fluids due to density differences, named after the ancient Greek scientist and mathematician Archimedes.

It is the ratio of gravitational forces to viscous forces^[1] and has the form:^[2]

${\displaystyle {\begin{aligned}\mathrm {Ar} &={\frac {gL^{3}{\frac {\rho -\rho _{\ell }}{\rho _{\ell }}}}{\nu ^{2}}}\\&={\frac {gL^{3}\rho _{\ell }(\rho -\rho _{\ell })}{\mu ^{2}}}\\\end{aligned}}}$

where:

• ${\displaystyle g}$ is the local external field (for example gravitational acceleration), m/s²,
• ${\displaystyle L}$ is the characteristic length of body, m.
• ${\displaystyle {\frac {\rho -\rho _{\ell }}{\rho _{\ell }}}}$ is the submerged specific gravity,
• ${\displaystyle \rho _{\ell }}$ is the density of the fluid, kg/m³,
• ${\displaystyle \rho }$ is the density of the body, kg/m³,
• ${\displaystyle \nu ={\frac {\mu }{\rho _{\ell }}}}$ is the kinematic viscosity, m²/s,
• ${\displaystyle \mu }$ is the dynamic viscosity, Pa·s,