When should a shrinking-neck calculation stop?

Check the quantity that should stay fixed

A periodic film has no inflow, outflow or evaporation, so its integrated height stays fixed. A periodic thread conserves integrated cross-sectional area: square radius before integrating. Adding radii would test the wrong quantity.

Divide the volume change by the initial volume to obtain relative error. A small error validates this conservation property, but a method can move liquid in the wrong direction and still preserve its total volume.

V_{\rm film}=\int h\,\mathrm dx,\qquad V_{\rm thread}=\pi\int r^2\,\mathrm dz,\qquad e_V=\frac{V(t)-V(0)}{V(0)}

Film volume is per unit transverse width. Thread volume uses axisymmetric sections.

Energy provides a second independent constraint

Write the energy derivative as $\mu_{\rm chem}=-\gamma h_{xx}+f\prime(h)$ , distinguishing it from the viscosity symbol in the thermal-flow chapter. Multiply the unforced height equation by $\mu_{\rm chem}$ and integrate by parts. Periodic boundaries remove the end term, leaving a negative integral of a square. A bump can grow as total energy falls: reduced wetting energy can pay for additional curvature energy.

A maintained temperature gradient can supply mechanical power through surface traction. Capillary energy need not decrease in the Marangoni experiment; compare surface work and viscous dissipation. The thread also carries kinetic energy, so surface area alone is insufficient.

\frac{\mathrm dF}{\mathrm dt}=\int\mu_{\rm chem} h_t\,\mathrm dx=-\int M(h)[(\mu_{\rm chem})_x]^2\,\mathrm dx\le0

Positive mobility, periodic or sealed boundaries and absence of external drive are essential.

Sources: U. Thiele · Eggers & Dupont · Bowen

Worked example: quantify a refinement

Use the dimensionless example of Chapter 3, whose maximum continuum growth rate is one quarter. Suppose the same continuous initial state at the same time gives growth rates $0.240,0.2475,0.249375$ on $N=64,128,256$ points. Successive differences are $0.0075$ and $0.001875$ . Their factor of four is consistent with second-order spatial error. These are constructed teaching values.

If time error is smaller, Richardson extrapolation gives $0.249375+(0.249375-0.2475)/3=0.250$ . This can be compared with an independent continuum relation. Two similar-looking pictures would not reveal that quantitative agreement.

Refine the same continuous initial condition. Regenerating grid noise adds new short wavelengths and mixes initial-state changes with numerical error. Compare at the same physical time, not each run’s final frame before its own cutoff.

p_{\rm obs}=\log_2\left|\frac{Q_{\Delta x}-Q_{\Delta x/2}}{Q_{\Delta x/2}-Q_{\Delta x/4}}\right|

An observed order is meaningful only in the asymptotic convergence range, above rounding noise.

Record numerical and physical limits separately

Necks, precursor transition regions and temperature patches need enough grid points across them. A neck cutoff tied to spacing is a resolution rule. Steep slopes, appreciable neglected inertia or molecular thickness can instead invalidate the reduced physics. Refining a grid does not add missing physics.

Save the last accepted finite, positive-thickness profile and the reason for stopping. Clipping negative thickness to a small positive value changes volume and curvature. A single-valued thread radius describes one connected interface; continuation through breakup requires a new treatment of topology.

Sources: Eggers & Dupont · U. Thiele

A regularization parameter changes the problem

Changing precursor thickness can change both short-range forces and apparent contact angle. State which macroscopic quantities remain fixed before comparing motion. Treating precursor thickness only as a numerical stabilizer hides its physical effect.

A reproducible record contains model version, units, boundaries, grid, time-step rule, seed, parameters and unmodified height or radius histories. Store analytic references and fit intervals separately so another reader can repeat the measurement.

Predict, change and explain

Follow the neck up to the stopping condition

Choose A growing neck, record its decreasing radius, then read the stopping reason. Identify the geometric or resolution condition involved.
Raise viscosity to recalculate the trajectory. Compare radii and linear rates at the same capillary time, and identify the term changed by viscosity.

Exercises and explanations

Does volume error near machine precision establish correct dynamics?

Read the explanation

No. It constrains total volume. Check equilibrium, energy, independent dispersion and refinement of the actual motion as well.

If a finer grid stops later, has the breakup time converged?

Read the explanation

No. Compare profiles during a common resolved interval and track the stated cutoff. The cutoff records the resolving ability of that grid; predicting topological breakup needs additional physics and numerical treatment.

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.

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.