FR18
Dynamic interfaces
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
L1 · 1
Why surface tension pulls water into beads
Start with the work needed to create an interface, then calculate curvature pressure, contact angles and the capillary length.
L2 · 2
Reduce a flowing layer to a height equation
Derive film flux from a velocity profile and see why thickness enters the flow rate as a cube.
L2 · 3
Which wavelength grows out of noise first?
Derive a film dispersion relation, estimate growth from stored heights and predict the first visible spacing.
L2 · 4
How a drop edge moves across a solid
Explain the moving-contact-line difficulty, then derive a precursor model, equilibrium angle and energy decay.
L3 · 5
How a temperature difference moves a liquid surface
Derive Marangoni velocity and film flux from surface stress, then calculate surface flow and a pressure-driven return current.
L3 · 6
Why a thin stream grows into beads
Compare surface areas at fixed volume, derive slender-thread growth, and follow a conservative simulation to a resolved neck before breakup.
L3 · 7
When should a shrinking-neck calculation stop?
Use volume, energy and grid convergence to assess an interface simulation and separate numerical stopping from physical breakup.
L4 · 8
How to compare growth rates with nanofilm spectra
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
- Move a warm patch and follow the surface flow
- Follow a liquid thread towards an unresolved neck
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
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.
- Huh & Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci. 35, 85–101 (1971)
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.
- 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)
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.
- John W. M. Bush, Surface Tension Module, MIT lecture notes
Lecture 1, sections 1.1–1.3: interfacial energy, normal and tangential stress, and capillary scaling. Teaching source; the examples here are newly calculated.
- Oron, Davis & Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69, 931 (1997)
Published review and framework for long-wave reductions. Specific implemented equations are cross-checked against the accessible author papers below.
- U. Thiele, Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth, J. Phys.: Condens. Matter 22, 084019 (2010)
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.
- Bowen, Sur, Bertozzi & Behringer, Nonlinear dynamics of two-dimensional undercompressive shocks, Physica D 209, 36–48 (2005)
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.
- Eggers & Dupont, Drop formation in a one-dimensional approximation of the Navier–Stokes equation, J. Fluid Mech. 262, 205–221 (1994)
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.