---
title: "Dynamic interfaces"
description: "Learn why liquid films, heated surfaces and thin threads smooth or amplify disturbances, with eight chapters and three interactive experiments."
canonical_url: "https://sandboxphysics.com/stories/dynamic-interfaces/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/dynamic-interfaces/"
last_modified: "2026-10-28"
---

[Dynamic interfaces](https://sandboxphysics.com/stories/dynamic-interfaces/)

FR18

# Dynamic interfaces

Sandbox Physics · 2026-10-28

A water drop rounds up, a thin coating opens into patches, and a thread develops a neck. The same surface tension takes part in all three. Follow the geometry and the motion of liquid to see when a smooth surface survives and when a disturbance grows.

## A route from drops to research papers

Begin with Chapter 1 for surface energy, curvature and wetting. Chapters 2–4 develop slow thin-film flow, instability and contact-line regularization. Chapter 5 adds a prescribed temperature field; Chapter 6 starts an axisymmetric thread with inertia. Chapters 7–8 turn the model into a reproducible calculation and open two papers beside the results.

Chapters 2–4 need derivatives and integrals; Chapter 3 explains Fourier amplitudes before using them. For the temperature branch, read Chapters 1–2 first. The thread branch requires Chapter 1 and mass conservation from Chapter 2. [Common textbook plan](https://github.com/yangzichao/sandboxphysics/issues/27)

1. L1 · 1

   ## [Why surface tension pulls water into beads](https://sandboxphysics.com/stories/dynamic-interfaces/surface-energy/)

   Start with the work needed to create an interface, then calculate curvature pressure, contact angles and the capillary length.

2. L2 · 2

   ## [Reduce a flowing layer to a height equation](https://sandboxphysics.com/stories/dynamic-interfaces/lubrication-flow/)

   Derive film flux from a velocity profile and see why thickness enters the flow rate as a cube.

3. L2 · 3

   ## [Which wavelength grows out of noise first?](https://sandboxphysics.com/stories/dynamic-interfaces/mode-selection/)

   Derive a film dispersion relation, estimate growth from stored heights and predict the first visible spacing.

4. L2 · 4

   ## [How a drop edge moves across a solid](https://sandboxphysics.com/stories/dynamic-interfaces/contact-lines/)

   Explain the moving-contact-line difficulty, then derive a precursor model, equilibrium angle and energy decay.

5. L3 · 5

   ## [How a temperature difference moves a liquid surface](https://sandboxphysics.com/stories/dynamic-interfaces/thermal-surface-flow/)

   Derive Marangoni velocity and film flux from surface stress, then calculate surface flow and a pressure-driven return current.

6. L3 · 6

   ## [Why a thin stream grows into beads](https://sandboxphysics.com/stories/dynamic-interfaces/liquid-thread/)

   Compare surface areas at fixed volume, derive slender-thread growth, and follow a conservative simulation to a resolved neck before breakup.

7. L3 · 7

   ## [When should a shrinking-neck calculation stop?](https://sandboxphysics.com/stories/dynamic-interfaces/resolution-and-singularities/)

   Use volume, energy and grid convergence to assess an interface simulation and separate numerical stopping from physical breakup.

8. L4 · 8

   ## [How to compare growth rates with nanofilm spectra](https://sandboxphysics.com/stories/dynamic-interfaces/paper-reading/)

   Recalculate Thiele’s Figure 1, check slender-thread dispersion and read a strong-slip thermal-fluctuation spectrum.

## Three moving interfaces

- [Watch a noisy film select growing wavelengths](https://sandboxphysics.com/labs/thin-film-instability/)
- [Move a warm patch and follow the surface flow](https://sandboxphysics.com/labs/marangoni-flow/)
- [Follow a liquid thread towards an unresolved neck](https://sandboxphysics.com/labs/filament-necking/)

The static capillary experiment supplies the equilibrium starting point. The three new experiments calculate model-generated records. The film models are small-slope and one-dimensional; the thread is axisymmetric. Each calculation stops at its stated validity limit. [Explore static wetting](https://sandboxphysics.com/labs/capillary-wetting/)

## Sources and calculation scope

Examples and diagrams are newly authored. The paper comparison recalculates analytic relations; the research reading distinguishes those curves from experimental photographs and molecular simulations.

1. [Huh & Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci. 35, 85–101 (1971)](https://doi.org/10.1016/0021-9797(71%2990188-3)

   The moving sharp contact line with no slip produces divergent stress and dissipation. The course explains the scaling argument; the full wedge solution is not recalculated.

2. [Yixin Zhang, Linear stability theory and molecular simulations of nanofilm dewetting with disjoining pressure, strong liquid–solid slip, and thermal fluctuations, J. Fluid Mech. 996, A19 (2024)](https://arxiv.org/html/2406.08179v1)

   Read alongside the published version, DOI 10.1017/jfm.2024.701. Figures 3–4 and sections 2, 4–5 define slip calibration and the ensemble spectrum. Reading only: molecular data and stochastic strong-slip curves are not reproduced.

3. [John W. M. Bush, Surface Tension Module, MIT lecture notes](https://web.mit.edu/1.63/www/Lec-notes/Surfacetension/Lecture1.pdf)

   Lecture 1, sections 1.1–1.3: interfacial energy, normal and tangential stress, and capillary scaling. Teaching source; the examples here are newly calculated.

4. [Oron, Davis & Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69, 931 (1997)](https://doi.org/10.1103/RevModPhys.69.931)

   Published review and framework for long-wave reductions. Specific implemented equations are cross-checked against the accessible author papers below.

5. [U. Thiele, Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth, J. Phys.: Condens. Matter 22, 084019 (2010)](https://www.uni-muenster.de/Physik.TP/~thiele/Paper/Thie2010jpm.pdf)

   Author-hosted published article, equations (1)–(3), section 2 potential and equation (5), Figure 1. Recalculation covers the analytic linear dispersion only; nonlinear continuation branches in Figures 2–3 are a reading topic.

6. [Bowen, Sur, Bertozzi & Behringer, Nonlinear dynamics of two-dimensional undercompressive shocks, Physica D 209, 36–48 (2005)](https://www.math.ucla.edu/~bertozzi/papers/BowenPhysicaD05.pdf)

   Section 2, equation (2.1), identifies the thermal-stress, gravity and capillary fluxes. The horizontal, prescribed-temperature browser model retains the thermal and capillary terms. The paper’s inclined fronts and experimental images are not reproduced.

7. [Eggers & Dupont, Drop formation in a one-dimensional approximation of the Navier–Stokes equation, J. Fluid Mech. 262, 205–221 (1994)](https://arxiv.org/pdf/physics/0110081v1)

   arXiv:physics/0110081v1 is a later repository copy of the 1994 paper. Section 2, equations (17)–(18), curvature (15) and dispersion (27)–(32); Figure 1 caption states the water-jet geometry. The course derives the periodic slender-thread benchmark, not the nozzle experiment or satellite drops.