The overflow rate says how fast a particle must fall. Stokes' law says how fast it can: $v_s = \dfrac{g\,(\rho_s - \rho)\,d^{2}}{18\,\mu}$ — read aloud v-s equals g, rho-s minus rho, d squared, over eighteen mu.

Every letter, in words. $v_s$ is the terminal settling velocity in metres per second — terminal because drag grows with speed until it balances weight, after which the particle falls steadily and stops accelerating. $g$ is 9.81 m/s². $\rho_s$ — rho-sub-s — is the particle's density in kg/m³, and $\rho$ — plain rho — is the water's, 1000 kg/m³. Their difference is what drives the fall, because the water buoys the particle up: subscript s is the solid, no subscript is the fluid, and confusing the two is the commonest error in the relation. $d$ is the particle diameter in metres, and $\mu$ — mu — is the water's dynamic viscosity, 0.001 Pa·s at 20 °C. The 18 comes out of the Stokes drag law and is not adjustable.

Now the prize, and it is the reason this lesson exists: $d$ is squared. Double the size of a floc particle and it falls four times as fast. Grow it ten times and it falls a hundred times as fast. That single exponent is the whole commercial argument for coagulation — alum does not make the solids heavier, it makes them bigger, and the square does the rest.

Run the contrast once and it stays with you. A 100 µm sand grain at 2650 kg/m³ falls at about 9 mm/s — nearly 800 m/d, which no clarifier could ever outrun, and that is why grit chambers work at all. A raw 2 µm clay particle falls thousands of times slower and would need weeks in a tank. Coagulate it into a 200 µm floc and it settles in minutes, even though the floc is barely denser than the water it sits in.

Know the edge. Stokes assumes a sphere in laminar fall, valid to about Re = 1 — roughly 100 µm for sand in water. Above that the wake goes turbulent and Stokes over-predicts. And flocs are not spheres, are not rigid, and shear apart in a strong current, which is why flocculation basins are stirred gently and more gently still toward the outlet.