---
title: "Liquid Filament Necking Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/filament-necking/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/filament-necking/"
last_modified: "2026-10-28"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

## The neck and the growing ripple

Cyan: numerical simulation records. Dashed amber: the independent small-ripple solution of this slender-jet model. The vertical marker follows the displayed frame.

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.

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

Physics tutorial

## How surface tension grows a neck

**Background** A thin water stream often develops beads before it separates into drops. A cylindrical surface curves around its axis even when its side profile looks straight.

**Why it matters** The two surface curvatures compete. Their balance explains why some disturbances fade and others carry liquid out of a neck.

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 misconception** Surface tension must always flatten a visible bump.

**Better mental model** Surface 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

1. 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.

2. 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.

3. 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.

Physical conclusion

The geometry determines whether capillarity restores a cylinder or drains its neck; viscosity sets how quickly that motion develops.

## Related experiments and articles

Dynamic interfaces · Reading path

### Curvature, films, thermal flow and liquid threads

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a thin stream grows into beads**](https://sandboxphysics.com/stories/dynamic-interfaces/liquid-thread/)
2. [Article **When should a shrinking-neck calculation stop?**](https://sandboxphysics.com/stories/dynamic-interfaces/resolution-and-singularities/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)