How Zero-Mud Suminagashi Fluid Marbling Physics Work
Traditional Japanese Suminagashi ("floating ink") and Turkish Ebru paper marbling suspend surfactant-treated pigments on the surface of a thickened water bath called size (historically prepared from carrageenan moss or tragacanth gum). Because each pigment ring forms an immiscible monomolecular film driven by Marangoni surface tension, swirling the bath with a horsehair stylus or brass-toothed comb stretches the bands into paper-thin ribbons without the colors ever dissolving into muddy brown.
Most digital paint mixers blur neighboring RGB pixels together, which quickly turns vibrant pastels into gray sludge. Liquid Marble & Acrylic Pour Studio solves this using Inverse Area-Preserving Homeomorphism Mapping—evaluating exact analytical fluid deformations backward from each pixel to the original concentric ink rings so boundaries stay crisp and glossy regardless of how many times you rake the canvas.
How to Create Classic Marbling Patterns
- Pour Concentric Bullseye Rings (Suminagashi Base): Click + Pour Bullseye in the floating dock (or switch the active tool to Pour Bullseye Rings and tap anywhere on the tray). Each new ink drop radially pushes all existing paint outward while preserving its exact surface area.
- Draw Fine Tendrils with the Needle Stylus: Switch the tool selector to Fine Needle Stylus and drag through a bullseye ring to pull hearts, petals, and serpentine eddies.
- Rake Nonpareil & Peacock Feathers with the 7-Tine Comb: Select 7-Tine Comb Rake and sweep horizontally, then vertically across the tray. Click Auto Peacock (Load Example) to watch an automated multi-drop pour followed by opposing comb passes.
- Export High-Resolution Artwork or Enter Fullscreen: Click Save PNG to download your marbled texture, or click Fullscreen Landscape or Fullscreen Portrait to fill 100% of your screen.
Mathematical Formulation of Incompressible Surface Marbling
Following the exact mathematical marbling framework established by Lu, Jaffer, Jin, Zhao, and Mao (Mathematical Marbling, IEEE Computer Graphics & Applications), floating ink behaves as a 2D incompressible inviscid fluid film:
f_drop(P) = C + (P - C) × sqrt(1 + R² / ||P - C||²) [Forward Displacement]
f_drop⁻¹(P) = C + (P - C) × sqrt(1 - R² / ||P - C||²) [Exact Inverse Map for ||P - C|| ≥ R]
2. Oseen Viscous Stylus & Multi-Tine Comb Advection:
ΔP(d) = M · V_stroke × exp(-d⊥² / (2 · w²)) [where d⊥ is perpendicular distance to tine axis]
3. Rotational Vortex Shear Field:
θ(r) = ω_max × exp(-r² / (2 · R_vortex²))
Comparison of Traditional & Digital Marbling Operations
| Marbling Operation | Traditional Studio Tool | Mathematical Fluid Operator | Visual Pattern Produced |
|---|---|---|---|
| Bullseye Pour | Bamboo brush + pine-soot & surfactant ink | Radial area-conserving expansion ($\sqrt{1 + R^2/r^2}$) | Agate rings & concentric geological strata |
| Stylus Tine Drag | Single horsehair, bamboo needle, or awl | Gaussian transverse shear wake along stroke vector | Heart chains, leafy stems, and stone veins |
| Multi-Tine Comb Rake | Wooden or brass rake with evenly spaced pins | Periodic comb array $\sum_k \exp(-d_{\perp,k}^2 / 2w^2)$ | Nonpareil waves, chevrons, andacock feathers |
| Spiral Vortex | Circular stylus twirl in thickened size bath | Localized angular twist $\theta(r)$ around cursor center | Galactic spirals, roses, and shell whorls |
Worked Example: The 18-Layer Peacock Combing Sequence
When you click Auto Peacock (Load Example) in the bottom dock, the studio executes a classic three-stage Ebru sequence:
- Stage 1 (Concentric Pour): Three bullseye clusters of 6 alternating rings (
R = 46 pxeach) are poured across the bath. Under area conservation, 6 concentric drops of radius $46\text{ px}$ expand the outer ring radius to $\sqrt{6 \times 46^2} = \sqrt{12,696} \approx 112.7\text{ px}$. - Stage 2 (Horizontal Rake): A 7-tine comb with `36 px` tooth spacing sweeps from left to right, stretching the circular bands into parallel sinusoidal ribbons.
- Stage 3 (Vertical Counter-Rake): A second comb pass sweeps downward with offset tines, folding the ribbons into interlocking peacock arches while modulating the Web Audio low-pass filter from `180 Hz` up to `680 Hz`.