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Fluid Mechanics Simulator

Terminal Settling Velocity Simulator — Sphere Drag and Reynolds Regimes

Iteratively compute the terminal velocity of a spherical particle settling under gravity. Vary diameter, density and viscosity to compare the Stokes, intermediate and Newton drag regimes.

Parameters
Particle diameter D
mm
Particle density ρ_p
kg/m³
Fluid density ρ_f
kg/m³
Fluid viscosity μ
Pa·s

Gravity g = 9.81 m/s². Iteration up to 20 steps with convergence |ΔU/U|<1e−4.

Results
—
Terminal velocity U_t
—
Terminal Reynolds number Re_t
—
Drag coefficient C_D
—
Flow regime
Re ― U_t plot (theory curves at several diameters)

Horizontal axis = Reynolds number (log) / Vertical = U_t (log) / Gray dots = D=0.01,0.1,1,10mm theory / Yellow dot = current (Re, U_t) / Dashed lines = regime boundaries Re=0.1, 1000

Theory & Key Formulas

For a sphere settling under gravity, the terminal velocity is reached when the effective weight is balanced by fluid drag.

General force balance (gravity − buoyancy = drag):

$$U_t = \sqrt{\dfrac{4\,g\,D\,(\rho_p-\rho_f)}{3\,\rho_f\,C_D}}, \qquad Re=\dfrac{\rho_f\,U_t\,D}{\mu}$$

Drag coefficient by Reynolds regime:

$$C_D = \begin{cases} 24/Re & (Re<0.1)\\ \dfrac{24}{Re}\,(1+0.15\,Re^{0.687}) & (0.1\le Re<1000)\\ 0.44 & (1000\le Re<2\times10^5) \end{cases}$$

In the Stokes regime an explicit solution exists:

$$U_t^{\text{Stokes}} = \dfrac{g\,D^2(\rho_p-\rho_f)}{18\,\mu}$$

In the intermediate regime C_D depends on U_t, so we iterate Re→C_D→U_t until convergence.

What is the Terminal Settling Velocity Simulator?

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When I drop sand into a glass of water it speeds up at first and then seems to fall at a steady pace. What is happening physically?
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Roughly speaking, gravity (minus buoyancy) accelerates it, but as it speeds up the fluid drag grows. Once the two are equal there is no more acceleration and the speed is constant — that constant value is the terminal settling velocity U_t. Press the "Sand / water" preset above and you will see a 1mm sand grain settles in water at about 0.14 m/s.
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Does the formula change for very small versus very large particles? The chart on the right has a kink in it.
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Good catch. The flow around the sphere is characterized by the Reynolds number Re=ρUD/μ. In the slow Stokes regime (Re<0.1) viscosity dominates, C_D=24/Re, and you can solve directly: $U_t=gD^2\Delta\rho/(18\mu)$, which scales as D squared. In the Newton regime (Re>1000), C_D≈0.44 is nearly constant and $U_t \propto \sqrt{D}$ — only the square root.
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My sand grain landed at Re=138, in the intermediate range. How is that solved?
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That is the interesting part. In the intermediate regime C_D depends on U_t, but solving for U_t needs C_D — a chicken-and-egg loop. So we iterate: start with the Stokes guess, compute Re, compute C_D, update U_t, repeat. This tool runs at most 20 iterations until |ΔU/U|<1e−4. Try the "Steel ball / oil" preset to see what a 5mm steel sphere does in viscous oil — increasing the viscosity drops Re and pulls the operating point back into the Stokes regime.

Frequently Asked Questions

This tool assumes a sphere. Real powders and sand grains are irregular and have larger drag coefficients than spheres at the same Re. In practice we apply a sphericity correction, use an equivalent volume or velocity diameter, or switch to a non-spherical correlation such as Haider-Levenspiel. Treat the value here as an upper bound and add a safety margin in design.
This tool assumes infinite dilution (a single particle in the fluid). At higher concentrations the upward return flow and inter-particle interactions reduce the apparent settling speed — known as hindered settling. The classical correction is Richardson-Zaki: U_int = U_t·(1−φ)^n, where φ is the particle volume fraction and n depends on Re. Hindered settling dominates the design of thickeners and clarifiers.
When ρ_p < ρ_f the particle rises. This tool reports the speed magnitude and the regime (e.g. an oil bubble or polymer fragment rising in water). Note that bubbles often deform under surface tension and do not stay spherical at moderate Re, requiring different models such as Hadamard-Rybczynski or the Mendelson wave theory.
Above Re ≈ 1000 a turbulent wake with separated vortices forms behind the sphere, and pressure drag dominates. Viscous stresses contribute relatively little, and the flow pattern becomes nearly Re-independent, giving C_D ≈ 0.44. Above Re ≈ 2×10^5 the boundary layer transitions to turbulent, the separation point shifts rearward, and C_D drops sharply — the famous "drag crisis".

Real-world applications

Sedimentation tanks for water and wastewater: The settling velocity of suspended solids sets the upper limit on the surface loading rate (flow per tank area). Engineers compute U_t in the Stokes regime, then size the tank from the target removal diameter. In practice coagulation is added so flocs grow larger and settle faster.

Cyclone separators and centrifugal classification: In a cyclone, the centrifugal field replaces g with ω²r and powders are sorted by the resulting terminal velocity differences. Cyclones are ubiquitous in powder processes and air cleaners; their design hinges on the C_D-Re relationship of the particles.

Atmospheric and rainfall physics: Raindrop terminal velocity grows with size — about 4 m/s for a 1mm drop and about 9 m/s for a 5mm drop (lower than the spherical theory because of deformation). This is the basis of rainfall intensity estimation and of algorithms that infer drop-size distribution from radar reflectivity.

Chemical processes and fluidized beds: In a fluidized-bed catalytic reactor, gas faster than U_t entrains the particles. Stable operation requires the gas velocity to lie between U_mf (minimum fluidization velocity) and U_t. The terminal velocity from this tool serves as the upper bound for entrainment onset.

Common misconceptions and pitfalls

The most common misconception is to assume "settling velocity scales linearly with diameter". In fact U_t scales as D² in the Stokes regime, as √D in the Newton regime, and somewhere in between in the intermediate regime. Sweep the diameter slider from 0.001 mm to 100 mm and watch U_t and Re. A factor of 10 in D changes U_t by very different amounts depending on regime. Designers who assume a linear relationship will be far off, especially for grains larger than fine sand.

Another common mistake is using the Stokes formula across all sizes. Stokes is only valid for Re<0.1. With the default values (sand 1 mm in water) the Stokes estimate gives Re=818 — well outside its range — and U_t≈0.82 m/s, about six times the real value. In the intermediate and Newton regimes you must update C_D from Re iteratively. This tool does it automatically; just keep an eye on the regime card to know which formula is in effect.

Finally, do not assume "terminal velocity is reached instantly". A particle starting from rest needs a finite time and distance to reach 99% of U_t — milliseconds for micron-scale Stokes particles, a few seconds for centimeter-scale Newton particles. In sedimentation tanks and short drop tubes the acceleration distance can be non-negligible, and a head-start correction is needed for short residence times.

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