Archimedes Number (Buoyancy against Viscosity)

The Archimedes number is built the way it is for one reason: it contains no velocity. Everything in $Ar = gL^3\rho(\rho_s-\rho)/\mu^2$ is known before anything moves — the particle size, the two densities, the fluid viscosity and the gravitational field. That is what makes it the right entry point to settling and fluidisation.

To see why that matters, try the problem without it. The terminal velocity of a particle satisfies a balance between submerged weight and drag, and drag depends on the drag coefficient, which depends on the particle Reynolds number, which depends on the velocity you are trying to find. You iterate. Recast the same balance in terms of $Ar$ and the velocity drops out of the known side entirely: correlations are written as $Re_p = f(Ar)$ and solved directly. $Ar$ is, in fact, $C_D Re_p^2 \times 3/4$ — the one combination of drag coefficient and Reynolds number in which the velocity cancels.

The regimes follow the usual pattern and carry the usual caveat. Below $Ar \approx 30$ the settling is Stokesian: drag is purely viscous, $Re_p < 1$, and the terminal velocity is $Ar\,\mu/(18\rho L)$, which is Stokes' law in disguise. Above roughly $10^5$ drag has gone fully inertial, $C_D$ flattens near 0.44, and terminal velocity scales with $\sqrt{L}$ rather than $L^2$. Between them is the intermediate regime where every published correlation is a fit rather than a derivation. Nothing happens at 30. The data are smooth and the boundaries are conventions drawn for convenience.

Names are a hazard here. Some texts call this identical group the Galileo number; others define Galileo without the $(\rho_s-\rho)/\rho$ factor, so $Ga = gL^3\rho^2/\mu^2$ and $Ar = Ga\,(\rho_s-\rho)/\rho$. Check which one a chart means before reading a value off it. And a heat-transfer Archimedes number $gL\Delta\rho/(\rho v^2)$ also exists, which is the Richardson number under another name — a different group with the same name, in the same field.

Three honesty notes, and the middle one is the largest source of error in practice.

The cube on $L$ means the answer hangs entirely on a particle size that, for any real material, is a distribution rather than a number. A $d_{50}$ uncertain by a factor of two carries a factor of eight into $Ar$. Quoting the group to four figures from a sieve analysis is false precision of an impressive order.

The correlations $Ar$ feeds were fitted to single, isolated, roughly spherical particles. Real beds hinder each other's settling — the Richardson–Zaki correction can halve a velocity at modest concentrations — and real grains are angular, which changes both the drag and the orientation they fall in. Both effects are larger than the precision of the number.

And $\mu^2$ makes viscosity the most leveraged input on the page. Water at 5 °C is about 1.5 times as viscous as at 20 °C, which more than halves $Ar$ and can push a settling problem from the intermediate regime back into the Stokes one. Any clarifier that performs well in summer and poorly in winter is telling you about that square.

Gravity is a variable here for a practical reason. A centrifuge is a settling device with $g$ replaced by $\omega^2 r$, and a hydrocyclone does the same trick with the flow itself. $Ar$ scales linearly with the field but with the cube of particle size, so spinning at 1000 g does for a fine particle roughly what growing it tenfold would do — which is a good way to see both what a centrifuge buys and why it is never a substitute for flocculation.