--- title: "Why a thin stream grows into beads" description: "Compare surface areas at fixed volume, derive slender-thread growth, and follow a conservative simulation to a resolved neck before breakup." canonical_url: "https://sandboxphysics.com/stories/dynamic-interfaces/liquid-thread/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/dynamic-interfaces/liquid-thread/" last_modified: "2026-10-28" --- [Dynamic interfaces](https://sandboxphysics.com/stories/dynamic-interfaces/) L3 · Chapter 6 # Why a thin stream grows into beads Sandbox Physics · 2026-10-28 A long ripple can reduce the area of a liquid cylinder. Capillary pressure then pushes liquid from the neck into the bulges, which deepens the neck. A short ripple costs area and flattens. Geometry sets the instability threshold; viscosity changes how quickly it develops. ## Before you begin Surface energy and curvature pressure; derivatives of a sine or cosine. Exponential growth is introduced here. A free liquid thread has a different flux law from a film resting on a solid. - Derive the wavelength threshold by comparing areas at fixed volume. - Obtain growth rates from coupled radius and velocity equations. - Recognize linear growth, nonlinear necking and the numerical stopping point in saved radius records. ## Hold the volume fixed before comparing areas A long cylinder can enclose its volume with less area by becoming a series of rounded bulges. To decide whether an initially tiny ripple starts that process, compare an entire wavelength of surface. Looking only at the narrowest point misses the axial curvature. Let the original radius be $R_0$, with coordinate $z$ along the axis and dimensional wavenumber $q=2\pi/\lambda$. Adding a cosine alone increases the average cross-sectional area. A second-order shift in the mean radius removes that extra volume. $$ r(z)=R_0\left[1+\epsilon\cos(qz)-\frac{\epsilon^2}{4}\right]+O(\epsilon^3) $$ The cosine averages to zero and its square averages to one half. The mean-radius correction preserves volume through second order. $$ \frac{A-A_0}{A_0}=\frac{\epsilon^2}{4}\left[(qR_0)^2-1\right]+O(\epsilon^3) $$ Expand the axisymmetric area element for a small slope and integrate over one wavelength. Sources: John W. M. Bush · Eggers & Dupont ## Two curvatures decide which way liquid moves Write the dimensionless wavenumber as $k=qR_0$. The area falls when $k<1$, or when the wavelength exceeds the circumference of the cylinder. This is the Rayleigh–Plateau threshold. Short ripples add too much axial curvature to lower the area. The same argument appears locally as a pressure difference. A narrow neck curves more tightly around the axis, increasing one contribution to capillary pressure. The side profile supplies the other signed curvature. For a short ripple, that axial contribution can reverse the pressure difference. Take a cylinder convex towards the gas as having positive curvature. In the dimensional expression below, subscripts denote derivatives, $\kappa$ has units of $\mathrm{m^{-1}}$, and the capillary pressure contribution is $\gamma\kappa$. $$ \kappa=\frac{1}{r\sqrt{1+r_z^2}}-\frac{r_{zz}}{(1+r_z^2)^{3/2}} $$ A straight cylinder has zero axial curvature and total curvature equal to inverse radius. Sources: Eggers & Dupont ## Evolve the radius and velocity together Liquid leaving a neck must enter its neighbours. In the leading slender, axisymmetric approximation, axial velocity is nearly uniform across a section. Multiplying that velocity by cross-sectional area gives the volume flux. Radius squared is therefore a useful conserved variable. Measure length in $R_0$ and time in the inertial capillary scale $t_c=\sqrt{\rho R_0^3/\gamma}$. Coordinates, radius and velocity in the following equations are dimensionless. Positive $u$ points along increasing axial coordinate. The Ohnesorge number $\mathrm{Oh}=\mu/\sqrt{\rho\gamma R_0}$ compares viscosity with inertial capillary effects. Mass conservation relates radius change to incoming and outgoing flux. Axial momentum balances acceleration, capillary pressure gradients and viscous extensional stress. The factor of three comes from axisymmetric extension. We use the Eggers–Dupont equations with full curvature, zero gravity and periodic boundaries. $$ a=r^2,\qquad \partial_ta+\partial_z(au)=0 $$ The factor of pi cancels from the area balance. Integrating radius squared over the periodic cell gives a constant. $$ \partial_tu+u\partial_zu=-\partial_z\kappa+\frac{3\mathrm{Oh}}{a}\partial_z(a\partial_zu) $$ Liquid accelerates down a capillary pressure gradient. Viscous extension resists velocity gradients. Exterior-fluid stress and molecular forces are omitted. Sources: Eggers & Dupont ## Extract the growth rate from the coupled equations Set $r=1+\delta$ near a uniform cylinder and retain first-order terms. Conservation gives $\delta_t=-u_z/2$, and the curvature disturbance is $\delta\kappa=-\delta-\delta_{zz}$. Differentiate the first relation in time and insert the momentum equation to eliminate velocity. For a disturbance proportional to $e^{st}\cos(kz)$, two spatial derivatives produce $-k^2$ and a time derivative produces $s$. The growth rate follows from a quadratic. One root is positive for $k<1$. More viscosity reduces that root without shifting the threshold. This dispersion relation belongs to the one-dimensional slender model. The full inviscid three-dimensional Rayleigh theory contains modified Bessel functions and gives different rates at finite wavenumber. They agree in the long-wave limit. A numerical check must use the reference associated with the equations being solved. $$ s^2+3\mathrm{Oh}\,k^2s=\frac12k^2(1-k^2) $$ A positive real part means growth; a negative real part means decay. Complex roots also describe oscillation. Sources: Eggers & Dupont · John W. M. Bush ## Worked example: long and short initial ripples Take $\mathrm{Oh}=0.25$ and $k=0.65$, corresponding to $\lambda/R_0\simeq9.67$. The growing root is approximately $s=0.225$; the other is $-0.542$. The growing branch gains a factor of the natural exponential base in about $4.44t_c$. An initially resting thread needs both roots: its initial radius derivative must vanish. Starting with cosine amplitude $0.0100R_0$, the combined linear solution gives $0.0145R_0$ at $t=3t_c$. Raising viscosity to $\mathrm{Oh}=1$ reduces the growing root to $0.0899$. Now choose $k=1.3$. The right-hand side is negative. At $\mathrm{Oh}=0.25$, the roots are complex with real part about $-0.634$. The same initially resting ripple has amplitude about $0.00256R_0$ after $3t_c$. It has largely flattened. Sources: Eggers & Dupont ## Follow the neck beyond the linear shape A growing disturbance eventually stops resembling a scaled cosine. Bulges round out, the neck narrows, and higher spatial harmonics appear. The first cosine amplitude extracted from radius records then leaves the linear reference. Total volume remains a useful check. Without external work, viscosity should reduce the sum of kinetic and surface energy. The browser solver stores radius squared in cells and velocity between cells. Each cell gains its incoming flux and loses its outgoing flux. Liquid removed from a neck is therefore transferred to adjacent regions. Refining the grid checks whether the trajectory converges before the neck becomes unresolved. Breakup creates new free surfaces and needs a continuation rule. This calculation stops when the neck approaches its resolution cutoff or the axial slope exceeds the specified range. Its last frame shows the final neck this calculation resolves. Compare different grids over their common resolved interval. Sources: Eggers & Dupont ## Predict, change and explain ### [Read the radius histories](https://sandboxphysics.com/labs/filament-necking/) 1. Predict which ripple grows. Compare A growing neck with Short waves flatten, and connect the result to the fixed-volume area calculation. 2. Choose Watch a small ripple grow. Find when the simulated amplitude leaves the dashed amber reference. Drag the neck to change the starting ripple and compare that departure time. 3. Export the radius records. Sum radius squared across the grid at one time, then another. Check retained volume, grid spacing and the recorded stopping reason. ## Exercises and explanations 1. At fixed radius, change the wavelength from $10R_0$ to $5R_0$. What changes? Read the explanation The dimensionless wavenumber rises from about $0.628$ to $1.257$, crossing the threshold. The first ripple grows and the shorter one decays. Area and linear dynamics agree. 2. Why is one growing exponential insufficient for an initially resting thread? Read the explanation Its initial time derivative is nonzero. Initial rest requires the radius derivative to vanish too, so both growing and decaying branches are needed. Stable short waves use the corresponding decaying oscillatory solution. 3. Why does viscosity change the growth rate without moving the instability threshold? Read the explanation Viscous extensional stress slows the flow. The change in surface area at fixed volume sets the threshold; uniform viscosity changes the speed of the resulting motion. [Previous: How a temperature difference moves a liquid surface](https://sandboxphysics.com/stories/dynamic-interfaces/thermal-surface-flow/) [Next: When should a shrinking-neck calculation stop?](https://sandboxphysics.com/stories/dynamic-interfaces/resolution-and-singularities/) ## 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. [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. 2. [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.