Presets
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Parameters
Click either panel to place a drop (up to 6).
Model checks
The science
Random walk and Fick's law
Each ink particle takes independent random steps with variance 2D dt per axis. After time t the mean square distance from the start is ⟨r2⟩ = 4Dt in two dimensions. The same spreading obeys Fick's second law ∂c/∂t = D ∇2c, whose solution for a Gaussian drop of initial width σ0 is c(r,t) = M/(2π(σ02 + 2Dt)) exp(-r2/(2(σ02 + 2Dt))). The right panel integrates this PDE on a grid with an explicit finite-difference scheme; the left panel shows the particles.
Stokes-Einstein
A sphere of radius a in a fluid of viscosity η at temperature T diffuses with D = kBT/(6πηa). Water viscosity falls with temperature, η(T) = 2.414e-5 · 10247.8/(T - 140) Pa s (T in kelvin), so ink spreads faster in warm water; a thicker liquid (viscosity multiplier) slows it. A dye molecule of 0.5 nm has D of about 4e-10 m2/s in 20 C water.
Time compression
Real diffusion is slow: a dye drop spreads about 3.6 mm in an hour. The time-compression slider shows how many simulated seconds pass per real second.
Honest simplifications: still water with no convection, the Stokes-Einstein formula for a hard sphere applied to a dye molecule, a single non-interacting species, reflecting walls (the ⟨r2⟩ = 4Dt law holds only until the cloud reaches a wall), and the field panel shows the total concentration of all drops.
What to try
- Heat the water from 5 C to 70 C and watch D roughly quadruple.
- Raise the viscosity multiplier to mimic syrup.
- Use a 50 nm particle: D drops by a factor of 100 compared with a dye molecule.