October 9, 2026
Three Forces and a Terminal Velocity
A particle released in a still fluid feels three forces: gravity pulling it down, buoyancy pushing it up, and drag opposing its motion. Drag grows with speed, so the particle accelerates only until drag plus buoyancy exactly balance gravity. From then on it falls at a constant speed, its terminal settling velocity. In practice small particles reach it almost instantly, so terminal velocity is what matters for sizing settling equipment.
Everything about the problem rests on the drag term, and drag depends on the particle Reynolds number, Re_p = ρ_f·v·d/μ. At very low Re_p, viscous forces dominate and the flow creeps smoothly around the sphere. That is the regime George Stokes solved exactly in 1851, while studying how air resistance affects pendulums.
Stokes' Result and the Diameter-Squared Rule
Stokes showed that in creeping flow the drag on a sphere is F = 3πμdv. Setting that equal to gravity minus buoyancy gives the terminal velocity v_t = g·d²·(ρ_p − ρ_f)/(18μ). The striking feature is the d². Halve the particle diameter and it settles four times more slowly; reduce it tenfold and it settles a hundred times more slowly.
That one exponent explains a lot of process design. Coarse sand drops out of water in seconds, while clay-sized particles of a few micrometres can stay suspended for hours. To get fines out, plants either make particles bigger by adding flocculants that bind them into loose clusters, or make gravity bigger by spinning the suspension in a centrifuge or hydrocyclone, where centrifugal acceleration can be hundreds or thousands of times g.
Checking the Regime Before Trusting the Answer
Stokes' law only holds when the particle Reynolds number is below roughly 1. The standard procedure is to calculate v_t with Stokes' law, then compute Re_p with that velocity and check. If it comes out well above 1, the answer is wrong, and Stokes' law overpredicts the velocity, because real drag has grown beyond the creeping-flow value.
Beyond the Stokes range lies an intermediate regime where the drag coefficient falls gradually with Re_p, and then the Newton regime, roughly 1000 < Re_p < 2 × 10⁵, where the drag coefficient levels off near 0.44 and terminal velocity grows only with the square root of diameter. Between regimes, terminal velocity normally has to be found by iteration. Haider and Levenspiel's 1989 correlation gives it explicitly for spheres and for non-spherical particles, using sphericity to account for shape.
Crowded Suspensions and Thickener Design
Stokes' law describes a single particle in an infinite fluid. In a concentrated slurry, each falling particle displaces fluid that has to flow back up past its neighbours, so everything settles more slowly. Richardson and Zaki's 1954 experiments captured this hindered settling with a simple expression, v = v_t·ε^n, where ε is the voidage, the volume fraction of fluid. At low Reynolds number the exponent n is about 4.65, so even a modest solids fraction cuts the settling speed substantially.
Continuous thickeners, the large shallow tanks used in mineral processing and water treatment, have to be sized for this hindered behaviour rather than for a single particle. Kynch's 1952 theory of sedimentation showed how to extract the needed settling flux from a single batch settling test in a graduated cylinder, by following the interface between clear liquid and slurry over time. That approach still forms the basis of thickener area calculations.
References
- Stokes, G. G. (1851). On the effect of the internal friction of fluids on the motion of pendulums. Transactions of the Cambridge Philosophical Society, 9(II), 8–106.
- Haider, A., & Levenspiel, O. (1989). Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technology, 58(1), 63–70. doi:10.1016/0032-5910(89)80008-7
- Richardson, J. F., & Zaki, W. N. (1954). Sedimentation and fluidisation: Part I. Transactions of the Institution of Chemical Engineers, 32, 35–53. Reprinted in Chemical Engineering Research and Design, 75 (1997), S82–S100. doi:10.1016/S0263-8762(97)80006-8
- Kynch, G. J. (1952). A theory of sedimentation. Transactions of the Faraday Society, 48, 166–176. doi:10.1039/TF9524800166
- McCabe, W. L., Smith, J. C., & Harriott, P. (2005). Unit Operations of Chemical Engineering (7th ed.), Chapter 7. New York: McGraw-Hill.