Frequently Asked Questions about Foam Drainage

What is a Plateau border in foam?

A Plateau border (or Plateau channel) is the liquid-filled region where three soap films meet along a line. Named after physicist Joseph Plateau, these channels form the primary drainage network in foam. The cross-section of a Plateau border is a concave triangle whose sides are circular arcs set by the local curvature of the soap films.

How does Laplace pressure drive foam drainage?

Laplace pressure arises from the curvature of a liquid interface: P = 2γκ, where γ is surface tension and κ is mean curvature. Smaller bubbles have higher internal pressure than larger ones, creating pressure gradients across Plateau borders. Liquid flows from high-pressure regions (small bubbles) toward low-pressure regions (large bubbles), causing drainage and eventual coarsening.

What is disjoining pressure and why does it matter?

Disjoining pressure (Π) is the excess pressure in a thin liquid film arising from van der Waals, electrostatic double-layer, and steric forces between the two film surfaces. When a soap film thins to nanometre-scale, disjoining pressure builds to oppose further thinning, temporarily stabilising the film. If the applied pressure exceeds the maximum disjoining pressure, the film ruptures — the so-called spinodal or critical thickness h_crit.

What causes foam coarsening?

Foam coarsening (Ostwald ripening in foam) occurs because gas diffuses through soap films from small, high-pressure bubbles into large, low-pressure ones. Over time small bubbles shrink and collapse while large bubbles grow. The driving force is again Laplace pressure — the same curvature that causes drainage also drives coarsening. In 2D, von Neumann's law states each n-sided bubble grows at a rate proportional to (n − 6).

What determines the speed of foam drainage?

Drainage speed depends on Plateau border radius r (larger channels drain faster ∝ r²), surface tension γ, liquid viscosity η, and gravity. The channel model gives a drainage velocity V ≈ (ρg r²)/(αη) for gravity-driven flow, and a pressure-driven term ∝ ∇P_Laplace for surface-tension-driven drainage. Surfactant mobility (rigid vs mobile surfaces) adds a factor of 2–3.

How thin can a soap film get before it ruptures?

Common black films thin to two distinct equilibrium thicknesses. The Common Black Film (CBF) stabilises at ~30–100 nm by electrostatic repulsion. The Newton Black Film (NBF) is only 4–5 nm thick — essentially two surfactant monolayers. Below this, any perturbation causes rupture. The critical thickness h_crit varies with surfactant type, ionic strength, and applied pressure.

What is the foam drainage equation (FDE)?

The foam drainage equation is ∂A/∂t + ∂/∂z[A²(A − ∂A/∂z)] = 0, where A is the dimensionless Plateau border cross-sectional area and z is dimensionless height. It balances gravity-driven downward flow against surface-tension-driven flow (the ∂A/∂z term). The FDE predicts a sharp drainage front — a solitary wave — that propagates downward through the foam.

Why do dry foams behave differently from wet foams?

Foam wetness is characterised by liquid fraction φ_l. Wet foams (φ_l > 0.05) have spherical-like bubbles with thick Plateau borders; flow is primarily viscous. Dry foams (φ_l < 0.01) have polyhedral bubbles separated by near-flat films; Plateau border radii are small, drainage is slow, and films become unstable. The transition occurs near the random-close-packing fraction φ_l ≈ 0.36.

How does surfactant concentration affect foam stability?

Above the critical micelle concentration (CMC), surfactant molecules pack densely at both film surfaces, maximising disjoining pressure and surface elasticity (Marangoni effect). This slows drainage and film thinning. Below CMC, films are more mobile and rupture faster. Polymeric surfactants (e.g., proteins) form viscoelastic interfacial layers that can arrest drainage almost completely.

What is von Neumann's law for 2D foam coarsening?

In a 2D foam, von Neumann's law states that the area A_n of a bubble with n sides changes at rate dA_n/dt = k(n − 6), where k is a constant proportional to gas permeability and surface tension. Six-sided bubbles are in neutral equilibrium. Bubbles with fewer than 6 sides shrink; those with more than 6 sides grow. This elegant rule emerges purely from the 120° angle constraint at vertices.

What real-world applications rely on understanding foam drainage?

Foam drainage physics is critical in food science (beer head, cappuccino foam), cosmetics (shaving cream, shampoo), fire suppression (AFFF foams), mineral flotation, enhanced oil recovery, and lightweight structural foams. Controlling drainage rate determines product shelf-life, fire-fighting efficacy, and the texture of food emulsions.

About this simulation

Written by MySimulator Team · Reviewed by MySimulator Editorial Review

Last updated: 5 July 2026

This simulation models foam drainage — the way liquid trapped between bubbles gradually flows downward through the network of Plateau border channels that form wherever three soap films meet. A 2D Voronoi diagram builds the bubble network from random seed points, each cell tracks its own water content and effective radius, and Laplace pressure differences between neighbouring bubbles (P = γ/r) drive flow along the shared borders until thin films rupture and bubbles coarsen together.

🔬 What it shows

A live Voronoi foam of polygonal "bubbles", each coloured by its water content — brighter, more saturated cells are wetter. Liquid flows between neighbouring cells according to their Laplace pressure difference (Pi = γ/ri), gravity slowly drains liquid toward the bottom of the screen, and when a shared film thins below the critical thickness hcrit the two bubbles rupture and merge, which is exactly how real foam coarsens over time.

🎮 How to use

Drag Surface tension γ, Drainage rate κ and Critical film hcrit to change how fast liquid moves and how readily films rupture, or use Bubble count to rebuild the foam with more or fewer cells. Click or tap anywhere on the canvas to inject extra liquid into the nearest bubble and watch it drain into its neighbours. Pause/Resume and Reset are available in the HUD, and the ⓘ info button opens a modal with the underlying algorithm and key equations.

💡 Did you know?

Joseph Plateau worked out the geometric rules that govern soap-film networks — three films always meet at 120°, four Plateau borders always meet at the tetrahedral angle of 109.47° — despite having gone blind decades before finishing this work. The same rules shape everything from a pint's head to the packing of biological cells.

Frequently asked questions

What exactly is being drawn on screen?

Each polygon is a bubble in a 2D Voronoi foam generated from randomly placed seed points. Its fill colour encodes water content (darker and less saturated when dry, brighter cyan/blue when wet), the border thickness also grows with water content to suggest a thicker film, and small glowing dots at each vertex mark the Plateau borders where three films meet.

How does the simulation move liquid between bubbles?

Every bubble gets a Laplace pressure P = γ/r from its surface tension γ and effective radius r — smaller bubbles have higher pressure. For every pair of neighbouring bubbles the flow is Q = κ·(P_i − P_j)·Δt, moving water from the higher-pressure (smaller) bubble to the lower-pressure (larger) one each frame, on top of a small constant gravity term that drains liquid toward the bottom of the canvas.

What triggers a bubble merge?

The model estimates a film-thickness proxy between each pair of neighbours from their combined water content divided by the distance between their centres. When that value drops below the Critical film hcrit slider, the shared film is treated as ruptured and the two bubbles are merged into one larger cell with combined area and water — visually this is foam coarsening, the same process by which real bubbles disappear and neighbours grow.

What do the Surface tension, Drainage rate and Critical film sliders control?

Surface tension γ scales every bubble's Laplace pressure, so higher γ produces stronger pressure gradients and faster flow. Drainage rate κ is a direct multiplier on the flow between neighbours, so it speeds up or slows down the whole simulation. Critical film hcrit sets the rupture threshold — raise it and films rupture (and bubbles merge) far more readily, producing a drier, more chaotic foam.

Is this a physically exact foam simulation?

No — it is a simplified, qualitatively faithful model built for intuition rather than a full solution of the foam drainage equation or real 3D Plateau border geometry. The Voronoi cells approximate bubble shapes, radius and water content are proxies for true curvature and film thickness, and the pairwise flow/rupture rules capture the right trends (drainage from small to large bubbles, coarsening via rupture) without solving the underlying continuum equations exactly.