Falling Sphere Lab (2D)

A sphere falling through a fluid

Three forces act on a sphere of radius r and density rho_p in a fluid of density rho_f and viscosity mu: weight m g, buoyancy rho_f V g (Archimedes) and drag. The fluid that must be pushed aside adds an apparent mass 1/2 rho_f V, so (rho_p + rho_f/2) V dv/dt = (rho_p - rho_f) V g - 1/2 rho_f Cd pi r^2 v |v|. A sphere lighter than the fluid has a negative terminal speed and rises.

The drag coefficient and the Reynolds number

With Re = 2 r rho_f v / mu the drag coefficient follows the Schiller-Naumann fit Cd = 24/Re (1 + 0.15 Re^0.687) up to the Newton regime, where it levels off at about 0.44. For Re well below 1 this is Stokes drag F = 6 pi mu r v, giving the terminal speed v = 2 (rho_p - rho_f) g r^2 / (9 mu) (proportional to the square of the radius). For Re above about 1000 the drag is quadratic and v = sqrt(8 r g (rho_p - rho_f) / (3 Cd rho_f)) (proportional to the square root of the radius).

Tank walls

In a 10 cm tank the walls add drag. The Francis factor (1 - d/D)^2.25 / (1 - 0.475 d/D)^4 reduces the settling speed (it was derived for creeping flow and is applied here at all speeds as an estimate; switch it off to see the unbounded fluid).

Brownian motion

A micrometre-sized particle is also kicked by the thermal motion of the fluid molecules. Its diffusion coefficient is D = k T / (6 pi mu r) (Stokes-Einstein) and the mean squared displacement in a plane is 4 D t. For a 1 micron sphere in water the random motion in one second is a few microns, comparable with the distance it sinks, so sedimentation and diffusion compete.

Numerical checks on this page

(1) The integrated terminal speed of a small glass sphere in glycerol equals the Stokes formula. (2) A large steel ball in air reaches the quadratic-drag speed. (3) 4000 simulated random walkers reproduce the Einstein mean squared displacement.

Honest simplifications

The sphere falls on the tank axis with no rotation or lateral motion; the Basset history force and unsteady wake effects are ignored; fluid properties are fixed values at about 20 C; the wall factor is a creeping-flow formula; the Brownian path is shown for sphere A in a separate microscope inset and does not act on the sphere in the tank (it matters only below a few microns). Spheres that are too small to see are drawn magnified.

• Steel and glass ball in water: the denser ball falls faster; compare the strobe spacing.
• Honey: the Reynolds number is tiny, the motion is Stokes flow and the sphere reaches terminal speed almost instantly.
• Steel ball in air: no terminal speed within 85 cm; the drag is quadratic and Re is about 1e5.
• Same density, different size: the large sphere falls much faster (v grows with the square of r in the viscous regime).
• One micron particle: watch the Brownian inset and compare the random distance with the sinking distance.