Archimedes number

Physical key figure
Surname	Archimedes number
Formula symbol	${\displaystyle {\mathit {Ar}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Ar}}={\frac {\Delta \rho gL^{3}}{\rho \nu ^{2}}}}$
${\displaystyle \Delta \rho }$ Density difference between the body and the fluid ${\displaystyle g}$ Acceleration due to gravity ${\displaystyle L}$ characteristic length of the body ${\displaystyle \rho }$ Density of the fluid ${\displaystyle \nu }$ kinematic viscosity
Named after	Archimedes
scope of application	Buoyancy of bodies

The Archimedes number ( symbol :) is a dimensionless number , named after the ancient scholar Archimedes . It can be interpreted as the ratio of lift force to friction force and is defined as ${\displaystyle {\mathit {Ar}}}$

${\displaystyle {\begin{aligned}{\mathit {Ar}}&={\frac {\Delta \rho \,g\,L^{3}}{\rho \,\nu ^{2}}}=\left(1-{\frac {\rho _{\mathrm {K} }}{\rho }}\right)\cdot {\frac {g\,L^{3}}{\nu ^{2}}}\\&={\frac {\rho \,\Delta \rho \,g\,L^{3}}{\eta ^{2}}}\end{aligned}}}$.

The incoming sizes are

• the difference between the density of the body and the density of the fluid ${\displaystyle \Delta \rho =\rho _{\mathrm {K} }-\rho }$ ${\displaystyle \rho _{\mathrm {K} }}$ ${\displaystyle \rho }$
• the acceleration of gravity , on earth ${\displaystyle g\approx 9{,}81\,\mathrm {\frac {m}{s^{2}}} }$
• the volume calculated from the characteristic length of the body ${\displaystyle L}$ ${\displaystyle L^{3}}$
• the kinematic viscosity of the fluid , which differs from the dynamic viscosity by the factor . ${\displaystyle \nu }$ ${\displaystyle \eta =\rho \cdot \nu }$ ${\displaystyle \rho }$

Other definition

An alternative definition of the Archimedes number, which can be interpreted as the ratio of buoyancy force to inertia force or between free and forced convection , is identical to the definition of the Richardson number and reads:

${\displaystyle {\mathit {Ar}}={\frac {\Delta T\,g\,L\,\beta }{{u_{\infty }}^{2}}}={\frac {\mathit {Gr}}{{\mathit {Re}}^{2}}}}$.

It is

• ${\displaystyle \beta }$the isobaric expansion coefficient
• ${\displaystyle \Delta T=T_{\infty }-T_{\text{Wand}}}$ the driving temperature difference
• ${\displaystyle u_{\infty }}$ the ambient speed
• ${\displaystyle {\mathit {Gr}}}$: Grass yard number
• ${\displaystyle {\mathit {Re}}}$: Reynolds number .

Individual evidence

1. ↑ Repetition of technical thermodynamics : Achim Dittmann, Teubner study books, mechanical engineering ISBN 3-519-06354-9
2. ↑ Hanel, Bernd M., Raumlufströmung, Müller Verlag Heidelberg, 1994 pp. 31 + 72
3. ↑ VDI 6019 sheet 1, Beuth Verlag Berlin, 2006 p. 37 ff