FR18 · Capillary instability

Liquid Filament Necking

Why does surface tension turn a liquid thread into bulges and a thinning neck? A long bulge can lower the surface area of a liquid cylinder. Its narrow neck has higher capillary pressure, so liquid flows into the bulges and makes the neck thinner. Short ripples add too much axial curvature and flatten instead; viscosity slows the motion.

Watch a liquid thread carry fluid away from its narrow middle. Compare long and short disturbances, then slow the motion with viscosity.

Capillary time $2.08\,t_c$

Minimum radius $0.788\,R_0$

Linear growth exponent $0.225/t_c$

Retained volume $100\,\%$

Model assumptions

An axisymmetric one-dimensional Eggers–Dupont model evolves radius and axial velocity in a periodic cell, with constant density, viscosity and surface tension. Gravity, exterior-fluid stress, thermal noise and molecular forces are omitted. Length uses the initial mean-area radius; time uses the inertial capillary scale. The calculation stops before the neck becomes unresolved or the long-wave geometry becomes too steep. It does not continue through breakup. The amber markers follow the computed axial flow; their size is schematic. All records are simulations.

Minimum radius over time

r_{\min}/R_0

Measured cosine amplitude and linear reference

\lvert a_1\rvert/R_0

Units and the stopping point

The initial mean cross-sectional area defines the radius unit. The capillary time uses density, radius and surface tension. A computation stops when the neck radius reaches one and a half grid spacings or 16 percent of the initial radius, whichever is larger, or the axial slope exceeds 1.2.

t_c=\sqrt{\rho R_0^3/\gamma},\qquad \mathrm{Oh}=\mu/\sqrt{\rho\gamma R_0}

Start with the essentials

Focus question: Why does surface tension turn a liquid thread into bulges and a thinning neck?
One-sentence intuition: Long disturbances lower surface energy by moving liquid from narrow necks into broad bulges.

Core mathematical model

The volume moving along the thread

\partial_t(r^2)+\partial_z(r^2u)=0

The coordinate runs along the thread. Radius is measured in the initial mean-area radius, and velocity in that radius per capillary time. The area flux is radius squared times axial velocity; the factor of pi cancels.

Pressure from two curvatures

\kappa=\frac{1}{r\sqrt{1+r_z^2}}-\frac{r_{zz}}{(1+r_z^2)^{3/2}}

The first curvature wraps around the axis. The second follows the side profile. A narrow region increases the first term; axial curvature can oppose it for a short ripple. Dimensionless capillary pressure equals this curvature.

The small-ripple growth test

s^2+3\mathrm{Oh}\,k^2s=\tfrac12 k^2(1-k^2),\qquad k=\frac{2\pi R_0}{\lambda}

A small ripple grows exponentially when its dimensionless wavenumber is below one. The viscous term reduces the growth rate. This relation belongs to the one-dimensional model.

Common difficulties

Surface tension and smoothness

Typical misconceptionSurface tension must always flatten a visible bump.

Better mental modelSurface tension reduces surface area at fixed volume. A long cylindrical thread can lower its area by forming bulges, so smoothing every bump is not the same operation.

Run the experiment

01
Compare a long ripple with a short one
Watch A growing neck, then choose Short waves flatten.
What to observe: The long-wave neck empties. The short disturbance loses amplitude because axial curvature reverses the pressure difference.
02
Slow the same disturbance
Choose Viscosity slows the neck and compare the radius history.
What to observe: The instability threshold stays at the same wavelength, but the neck takes longer to narrow.
03
Find where the linear prediction leaves the curve
Choose Watch a small ripple grow. Drag the neck vertically to change the starting ripple.
What to observe: The simulated cosine amplitude initially follows the amber linear solution. As the shape develops higher harmonics, the two histories separate.

Liquid Filament Necking

The neck and the growing ripple

How surface tension grows a neck

Core mathematical model

The volume moving along the thread

Pressure from two curvatures

The small-ripple growth test

Common difficulties

Surface tension and smoothness

Run the experiment

Compare a long ripple with a short one

Slow the same disturbance

Find where the linear prediction leaves the curve

Related experiments and articles

Curvature, films, thermal flow and liquid threads