Snow Globe

Why do the flakes sink so slowly?

A flake is heavier than the liquid, so gravity pulls it down with the net weight F = (ρp − ρf) V g (weight minus buoyancy). As soon as it moves the liquid pushes back. For slow, smooth flow around a small object the push is the Stokes drag, proportional to speed: Fd = c · μ · d · v.

For a round bead c = 3π, which gives the classic vt = (ρp − ρf) g d² / (18 μ). For a thin flat flake falling flat-on the exact result for a disc is Fd = 8 μ d v (c = 8), and the flake here is a disc of thickness d/8, so vt = π(ρp − ρf) g d² / (256 μ). Compare the factors: 1/18 = 0.0556 for the bead and π/256 = 0.0123 for the flake, so the flat flake sinks about 4.5 times slower than a bead of the same width. The sinking speed grows with d² and is inversely proportional to the viscosity μ; the graph on the right shows both curves.

When Stokes is not enough

The Reynolds number Re = ρf v d / μ compares inertia with viscosity. For Re well below 1 the Stokes law is excellent. In water-like liquids with big flakes Re reaches 10 to 100 and the drag grows faster than the speed; the page multiplies the drag by the Schiller-Naumann factor f(Re) = 1 + 0.15 Re0.687 and solves for the speed. The green dashed line in the third graph is the plain Stokes law; the blue line is what the page uses.

How quickly does a flake react?

Newton's law for a flake is (m + ½ mf) dv/dt = F − c μ d f(Re) (v − u) where u is the liquid velocity and mf is the liquid displaced (the ½ is the added mass). The speed relaxes exponentially to u + ws with the response time τ = (m + ½ mf) / (c μ d f), usually milliseconds. The page uses the exact exponential step, so it is stable for any time step. The Stokes number St = τ U / l compares τ with the eddy time l/U: for St << 1 flakes ride the swirls, for St > 1 they keep going straight.

The shaken liquid

The liquid velocity is a random, divergence-free swirl built from 8 waves: u = ∇ × (w(r) ψ) + w(r) Ω × x with ψ = Σ (A/k) sin(k · x + φ) e, e ⊥ k, and w = 1 − (r/R)6 so the flow dies away at the glass without flowing through it. Its rms speed U decays because big eddies break up and viscosity rubs them out: dU/dt = −C U² / l − ν (π/R)² U with l = R, C = 0.2. This has the closed form U(t) = b U0 e−bt / (b + a U0(1 − e−bt)) with a = C/l, b = ν π² / R², drawn dashed in the second graph. A twirl adds solid-body rotation that slows with the Ekman time R / √(ν Ω).

Resting and lifting

Flakes that reach the platform stop. They lift off again when the local liquid speed exceeds about three times the Shields friction speed u*c = √(0.05 (ρp − ρf) g d / ρf), so bigger or heavier flakes are harder to stir up. A shake also kicks about half of the resting flakes upwards.

Honest simplifications

• Flakes are points with a drag law; they do not collide with each other or tumble, and wall effects on the drag (Faxén) and Brownian motion are ignored.
• The liquid flow is a prescribed random field with the right symmetry and decay law, not a solution of the Navier-Stokes equations. It is incompressible and does not flow through the glass, but its details are not those of a real shake.
• Flakes are drawn about 1.7 times larger than real so you can see them (untick “Draw flakes larger” in the Layers to see true size). The model does not stop flakes from passing through the little house, so flakes that land inside the house are hidden.
• Real snow globe liquids are water and glycerine mixes (about 1 to 1000 mPa·s at room temperature); the presets use values in that range.

Model checks

The page tests itself: (1) one flake integrated in still liquid is compared with an independent force-balance root; (2) the stepped shake turbulence is compared with the closed form; (3) the finite-difference divergence of the flow field and the flow through the glass are checked to be zero. See “Model checks” in the panel.

• Press Shake! (or tap the globe, or press Space) and watch the swirl die away in the second graph.
• Switch between Flat flakes and Round beads: the beads sink several times faster at the same width.
• Pick Thick syrup and then Glitter in water: the sinking speed changes by a factor of 100 while the flake stays the same.
• Double the Flake size and read the sinking speed: it goes up about four times (Stokes) or a bit less when Re is large.
• Choose Moon gravity or turn the Flake density below the liquid density to make the flakes rise.
• Turn on Flow arrows and Trace a flake to see how one flake follows the swirl.