STUDY #22 · 2026 · IN OBSERVATION

Viscous Fingering

A model-driven visual study of an interface that must pay for its corners.

MOVING IMAGE — THE SLOW FLUID HOLDS THE PEN σ 0.3 · p_in 8 · noise 0.09 · seed 8 · const-p · 512² · crude

WHAT IS THIS

Squeeze a runny fluid into a viscous one between two glass plates and the front between them breaks into fingers: any part that leads feels a steeper pressure gradient and advances faster, while surface tension charges a toll for every corner, forbidding details finer than one selected wavelength. This is the Saffman–Taylor instability, first studied as the reason waterflooded oil fields leave oil behind.

The model is one-sided Hele-Shaw flow: a Laplace pressure field in the viscous fluid, a curvature toll on the moving boundary, nothing else. This study runs it on the GPU — the pressure relaxes in a ping-pong texture while the interface advances through it — in real time.

the blossom — the hero's final state, the ledger of the negotiation σ 0.3 · p_in 8 · R0 12 · noise 0.09 · seed 8 · step ~3050 · crude

Motif Saffman–Taylor instability / Hele-Shaw cell / tip-splitting / one selected length

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation The pressure field of an undisturbed disc matches the exact logarithmic solution to 0.4%, and mass is conserved to machine precision — yet the same engine, given one grain of noise, refuses to stay round. Each ripple wavelength has a maximum toll it can afford: the measured tension where mode n stops growing follows the parameter-free prediction σ_c = (nΓ−1)ṘR²/(Γn(n²−1)) across a nearly threefold range, and a mode pushed beyond its price decays on cue. The web engine agrees with the Python engine to 0.9% in growth rate — two implementations, one negotiation.

Reference P. G. Saffman & G. I. Taylor, "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid," Proceedings of the Royal Society of London A, vol.245, 312-329 (1958); L. Paterson, "Radial fingering in a Hele Shaw cell," Journal of Fluid Mechanics, vol.113, 513-529 (1981).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

SAME FAMILY, A THIRD AUTHOR

Noise writes DLA. The lattice writes the snowflake. A balance of forces writes the fingering.

Study #09 — DLA Study #22 — Viscous Fingering

What writes the form chance — random walkers frozen where they first touch a balance of forces — drive against surface tension, deterministic

The characteristic length none — scale-free branching at every scale one selected wavelength λ* — every petal, split and groove written at that scale

The role of noise the builder — noise writes every branch the casting — physics picks the wavelength, noise picks whose finger wins

The limit between them the destination — lace with no length of its own σ → 0 — the censor retires and the flower grows toward DLA (Paterson 1984)

PARAMETERS EXPLORED

param meaning effect on the image

σ surface tension — the toll on curvature sets the single intrinsic length λ* ∝ √(σ/V): lower the toll and the petals grow teeth, raise it and the flower becomes a few slow lobes

p_in driving pressure of the invader sets the front speed V — pushing harder shrinks λ*, so the fingers get finer and faster; only the ratio σ/V decides the form

noise seed defects on the starting circle the physics picks the wavelength, the noise picks whose finger wins — determinism makes the symmetry, noise the individuality

R0 / protocol initial radius and drive protocol (const-p / const-Q) sets how many fingers appear first and how the invasion ages — the film runs one uncut const-p invasion

palette the imaging stage crude (oil and brine) / petrol / glycerine (dyed lab cell) / ember — the same physics photographed on different stages

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

One-sided radial Hele-Shaw flow: a harmonic pressure field in the viscous fluid, a curvature toll on the moving boundary, nothing else. The whole flower is written by these four lines.

u = -\frac{b^2}{12\mu}\,\nabla p \;\equiv\; -\lambda\,\nabla p, \qquad \nabla^2 p = 0

Darcy flow in the thin gap: the depth-averaged velocity follows the pressure gradient, and incompressibility makes the pressure harmonic in the viscous fluid. The invader is effectively inviscid — one uniform pressure pushes from inside.

p\big|_{\partial} = p_{\mathrm{in}} - \sigma\,\kappa, \qquad v_n = -\lambda\,\frac{\partial p}{\partial n}

The moving boundary: Young-Laplace charges the toll σκ for every corner, and the front advances at the Darcy velocity. These two lines are the entire negotiation.

\omega_n = (n-1)\,\frac{\dot{R}}{R} \;-\; \lambda\sigma\,\frac{n(n^2-1)}{R^3}

The radial dispersion relation, re-derived for this study (it matches Paterson 1981). The drive term is parameter-free — geometry and kinematics only; the toll term censors the high modes. Its zero is each mode's critical tension — the price list.

\lambda^{*} = 2\pi\sqrt{3\lambda\sigma/V}

The fastest-growing wavelength — the pattern's single intrinsic length. Petal width, tip splits and grooves are all written at this scale; lowering the toll or pushing harder makes the handwriting finer.

A visual interpretation of the model, not a claim of scientific precision. The historic Saffman-Taylor single-finger 1/2-width limit needs a long narrow channel and was not reached in this square domain — recorded as unresolved.

SELECTED STILLS — 6

the blossom — the hero's final state in the waterflooding stage σ 0.3 · p_in 8 · R0 12 · noise 0.09 · seed 8 · step ~3050 · crude

the same moment in petrol — the stage changes, the negotiation does not σ 0.3 · p_in 8 · R0 12 · noise 0.09 · seed 8 · step ~3050 · petrol

the same moment in glycerine — the dyed table-top experiment σ 0.3 · p_in 8 · R0 12 · noise 0.09 · seed 8 · step ~3050 · glycerine

slow lobes — the toll raised threefold, the flower reduced to a few slow petals σ 0.9 · p_in 6.5 · R0 16 · noise 0.07 · seed 7 · step 2600 · crude

toll near zero — the censorship lifted, every petal grows teeth σ 0.05 · p_in 12 · R0 12 · noise 0.1 · seed 5 · step 2600 · ember

first ripples — the moment the wavelength is selected σ 0.3 · p_in 8 · R0 12 · noise 0.09 · seed 8 · step 760 · crude

PROCESS — PARAMETER SWEEPS

The exploration as a morphology plane — surface tension against driving pressure, sixteen runs. The same shape repeats along the iso-σ/p_in diagonals, because only the selected length λ* ∝ √(σ/V) decides the form: the "one selected length" hypothesis made visible. Down the main diagonal, the whole range from a single slow lobe to a crowd of fine fingers.

the σ × drive morphology sheet — the same form along every iso-ratio diagonal σ 0.6→0.075 (rows) × p_in 2.5→20 (cols) · N 384 · R0 16 · noise 0.05 · seed 3 · const-p

SIGNATURE — THE PRICE LIST

Every ripple wavelength has a price it can no longer pay.

An interface pushed into a slower fluid wants to ripple, and every ripple is a bid. The drive pays each mode a wage fixed by geometry and kinematics alone — no adjustable constants — while surface tension charges for the corners, hardest on the finest ripples. Written out, the ledger becomes a dispersion relation, and each mode's growth rate crosses zero at its own critical tension σ_c: a price list for ripples.

The list was checked on the study's own engine. The measured tension where mode n stops growing follows the parameter-free prediction σ_c = (nΓ−1)ṘR²/(Γn(n²−1)) across a nearly threefold range (n 8–14, linear fits R² ≥ 0.994), and a mode pushed beyond its price — n 16 held at 1.4 σ_c — decays on cue. The check was built on σ_c deliberately: the zero crossing is an invariant, unmoved by the instrument's multiplicative response. And the other side of the ledger balances too — mass is conserved to machine precision (1.6 × 10⁻¹⁶).

the price list — each mode's growth rate crossing zero at its own critical tension; the dotted lines are the parameter-free prediction ω(σ) zero crossings · n 8–14 · linear fits R² 0.994–0.999 · mean deviation 8.0% · Γ = finite-wall factor

COLOUR = THE WATERFLOODING STAGE

The hero palette is the phenomenon's original stage: waterflooding an oil reservoir. The dark ground is crude oil; the pale invader is brine; the amber glow around the flower is the pressure field itself — the oil that is about to yield. The faint rings inside the invader are arrival-time isochrones: the pattern is the receipt of its own invasion. In the ember variation the toll is near zero and every petal grows teeth — the same physics photographed at a different price.

Form is physics throughout: petal width is the selected wavelength λ*, tips split when they grow wider than the toll allows, and the grooves are where the slow fluid held its ground.

the pressure made visible — the amber halo is the oil about to yield, the rings are isobars of the negotiation σ 0.9 · p_in 6.5 · R0 16 · noise 0.07 · seed 7 · step 2600 · crude

All palettes are artistic approximations of literature imagery, not measurements.

REFERENCES

P. G. Saffman, G. I. Taylor. "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid." Proceedings of the Royal Society of London. Series A, vol.245, no.1242, 312-329 (1958). https://doi.org/10.1098/rspa.1958.0085
L. Paterson. "Radial fingering in a Hele Shaw cell." Journal of Fluid Mechanics, vol.113, 513-529 (1981).
H. S. Hele-Shaw. "The flow of water." Nature, vol.58, 34-36 (1898).
L. Paterson. "Diffusion-Limited Aggregation and Two-Fluid Displacements in Porous Media." Physical Review Letters, vol.52, no.18, 1621-1624 (1984).
G. M. Homsy. "Viscous fingering in porous media." Annual Review of Fluid Mechanics, vol.19, 271-311 (1987).
D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Shraiman, C. Tang. "Viscous flows in two dimensions." Reviews of Modern Physics, vol.58, no.4, 977-999 (1986).
J. W. McLean, P. G. Saffman. "The effect of surface tension on the shape of fingers in a Hele-Shaw cell." Journal of Fluid Mechanics, vol.102, 455-469 (1981).

INTERACTIVE STUDY

A thin layer of viscous fluid, invaded from the centre. The pressure field decides who advances; surface tension charges every corner. surface tension and injection act live mid-invasion — the toll can be raised while the fingers grow — and noise re-injects the flower with new flaws. It is a deliberately simplified instrument, capped at a 160² lattice with a few curated knobs and no export, separate from the full engine used to author the finished works.

SIMPLIFIED INSTRUMENTHELE-SHAW · CURVATURE TOLL · σ LIVE

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.