Ink in Water

Fluid model

Different parts of a drop move at different velocities, stretching the dye into sheets and filaments. Molecular diffusion gradually erases the concentration gradients across these structures.

The model follows dissolved dye in three dimensions. Each concentration c_i is scaled to equal one in the initial ink solution. The fractional density change is b = ∑_i β_ic_i, summed over all inks, giving density ρ = ρ₀(1 + b). Here ρ₀ is water density and β_i is each ink’s fractional density contrast. The Boussinesq approximation retains these small density differences only in buoyancy.

Here u is velocity, t is time, π is pressure divided by water density after subtracting hydrostatic pressure, and e_y is the upward unit vector. The equations conserve fluid volume, evolve velocity under pressure, viscosity and buoyancy, and transport each ink through the shared flow.

Kinematic viscosity is ν = 10⁻⁶ m² s⁻¹, dye diffusivity is D = 10⁻⁹ m² s⁻¹, and gravity is g = 9.81 m s⁻².

The solver transports velocity, applies viscosity and buoyancy, then solves for pressure to keep the flow nearly incompressible. Shared fluxes between neighbouring grid cells conserve the amount of each ink.

Initial currents combine sixteen sinusoidal modes with seeded directions, phases and polarisations. Each mode is divergence-free; pressure projection adjusts the sampled flow to the selected boundaries. This initial motion subsequently evolves under the momentum equation. The seed also determines each drop’s shape perturbation and orientation. Drops have the same nominal radius; Reset repeats the initial conditions.

Speed changes the requested rate of fixed time steps; hardware may limit the achieved rate. Currents sets the initial water motion and restarts the experiment; zero starts at rest. Up to five inks can share the flow. Adding or removing ink also restarts it; density and colour edits apply immediately. Positive density contrast drives sinking, negative values drive rising, and zero makes ink neutrally buoyant. Fine uses smaller cells and more tracer samples. View rotates around the full domain.

Boundary conditions

The cuboid, cylinder and sphere are sealed containers. Water cannot cross their walls and can slide along them without friction (free-slip). Dye has zero flux through a wall. Curved containers are approximated by stair-step surfaces made from grid cells; the velocity, dye and reflected tracers use these same walls.

The flat 3-torus identifies opposite cuboid faces in all three directions. It has no boundary. The displayed box is a repeating cell: fields satisfy periodic boundary conditions, and ink crossing one face re-enters through the opposite face. Its buoyancy uses b − 〈b〉, where 〈b〉 is the volume mean, removing uniform acceleration of the whole fluid.

Colour

Grid concentrations determine buoyancy. The image uses finer tracer samples, each carrying a fixed amount of one ink. Away from walls, a tracer’s position X follows

u_h is the interpolated grid velocity. Brownian motion W represents molecular diffusion: each step Δt adds an independent Gaussian displacement with variance 2DΔt per coordinate.

Normalised Gaussian kernels spread the samples into concentration integrated along viewing rays. Each ink contributes its absorption coefficient times this integrated amount. Beer–Lambert absorption converts their sum into transmitted colour. Picker colours set illustrative RGB transmittances for a 2 mm reference column, independently of the dynamics.

Limitations

The fluid grid has 112 × 168 × 112 cells at Standard resolution and 160 × 240 × 160 at Fine. Flow advances in fixed 0.01 s steps; dye transport uses smaller substeps when needed for stability.

Fine optical filaments do not alter the coarser concentration field driving buoyancy. Grid spacing, numerical diffusion, tracer sampling and optical smoothing limit the detail; interpolated velocities near stair-step corners introduce further error. The model omits pigment settling, surface chemistry, variable viscosity, refraction and light scattering.

References

Young, Y.-N., et al. (2001). On the miscible Rayleigh–Taylor instability: two and three dimensions. Journal of Fluid Mechanics 447, 377–408.

Arecchi, F. T., et al. (1996). Fragmentation of a drop as it falls in a lighter miscible fluid. Physical Review E 54, 424–429.

Meunier, P., & Villermaux, E. (2022). The diffuselet concept for scalar mixing. Journal of Fluid Mechanics 951, A33.