Why Water Streams Break into Droplets — Plateau–Rayleigh Instability and Breakup Regimes

Open your kitchen faucet halfway. Initially, you see a smooth liquid column. However, starting about 10 cm down, it breaks into regular droplets. It is a common sight, yet answering why a continuous medium spontaneously fragments is not trivial. This post follows the classic explanations provided by Joseph Plateau in 1849 and Lord Rayleigh in 1879, and explores how the Weber number (inertia/surface tension ratio) and Ohnesorge number (viscosity/inertia-surface tension ratio)—still used today in engine nozzle and inkjet design—extend those results. By the end, you will be able to answer "Why do some jets break into large droplets while others shatter like mist?" using a single regime map.

Why a Continuum Chooses to Fragment#

Surface tension is a force that seeks to minimize surface area. For a fixed volume of liquid, a spherical shape has the smallest surface area. This is why a mass of water forms a sphere in zero gravity.

Now, consider a long liquid cylinder versus a chain of droplets created by cutting that cylinder at regular intervals. Which has a smaller surface area? Counter-intuitively, the chain of droplets has a smaller area—but only when the cutting interval is longer than the cylinder's circumference. This is why a water column fragments; it is a natural result of falling into a lower energy state.

Plateau's Geometric Intuition#

Plateau calculated the change in surface area for a cylinder of radius $R$ when subjected to a weak axial perturbation of wavelength $\lambda$.

Here, $2\pi R/\lambda$ is the dimensionless wavenumber $kR$ (the product of wavenumber $k = 2\pi/\lambda$ and the radius). If this value is less than 1, $\Delta A < 0$, meaning the surface area decreases as the perturbation grows, which is energetically favorable. Conversely, if it is greater than 1, $\Delta A > 0$, and the perturbation subsides.

Rewriting the condition is simple:

Only wavelengths longer than the cylinder's circumference grow. This criterion is known as the Plateau limit.

Rayleigh's Linear Stability#

Plateau only answered "which wavelengths can grow." Thirty years later, Rayleigh identified "which wavelength grows the fastest" by inserting a small perturbation $\eta = \epsilon e^{\sigma t} \cos(kx)$ into the inviscid Navier–Stokes equations to find the dispersion relation.

Where $\sigma$ is the growth rate (unit $s^{-1}$), $\sigma_{s}$ is the surface tension (N/m), $\rho$ is the liquid density, and $I_{0}, I_{1}$ are modified Bessel functions. The term $1-(kR)^{2}$ in parentheses is exactly the Plateau limit.

The maximum value occurs at $kR \approx 0.697$, corresponding to a wavelength of $\lambda_{\max} \approx 9.02 R$. A water column "prefers" to be cut at a period of about 9 times its radius. This period explains the spacing of actual faucet droplets almost perfectly.

Including viscosity, as organized by Chandrasekhar, reduces $\sigma$, but the instability boundary $kR < 1$ remains unchanged. Viscosity only slows down the growth; it does not affect the necessary conditions for fragmentation.

Weber and Ohnesorge: The Map of Breakup Regimes#

While the above explanation suffices for slow flows like a faucet, high-speed jets in rocket combustors or diesel injectors involve the inertia of the surrounding gas. Two dimensionless numbers emerge here:

$We$ is the ratio of gas inertia to surface tension, and $Oh$ is the ratio of liquid viscosity to (inertia and surface tension). Summarizing experimental data, a jet passes through four regimes as its velocity increases.

If $Oh$ is small ($< 0.1$), the regime is determined by $We$ alone. As $Oh$ increases, viscosity corrections are needed, and in practice, boundaries are drawn using composite numbers like $Oh(We)^{0.5}$. Faucets and inkjets usually fall within the Rayleigh breakup region. Diesel injectors target Atomization, raising $We$ above $10^{5}$.

Visualizing the Growth Rate Curve#

Try interacting with the simulation below. Use the slider to change the wavenumber $kR$, which seeds the jet with a perturbation of that wavelength, and see the point on the growth rate curve move accordingly.

Increasing $kR$ from 0.2 to 0.7 causes the growth rate to spike, leading to faster fragmentation into droplets. Passing $kR = 1.1$ brings the curve to zero, causing perturbations to subside and keeping the column smooth. Increasing Ohnesorge from 0 to 0.4 slows the growth itself, but the unstable $kR$ range remains the same—you can see for yourself that viscosity "cannot erase the boundary."

Finding the Fastest Wavelength with Python#

We can numerically search for the optimal wavenumber by translating Rayleigh's dispersion relation using Bessel functions. We use i0 and i1 from scipy.

import numpy as np
from scipy.special import i0, i1
from scipy.optimize import minimize_scalar
 
def rayleigh_sigma(kR, sigma_s=0.072, rho=1000.0, R=1e-3):
    """Inviscid Plateau-Rayleigh growth rate (s^-1)."""
    if kR <= 0 or kR >= 1:
        return 0.0
    coeff = sigma_s / (rho * R**3)
    return np.sqrt(coeff * kR * (1 - kR**2) * i1(kR) / i0(kR))
 
def fastest_mode(R, sigma_s=0.072, rho=1000.0):
    """Wavenumber and corresponding wavelength that grow the fastest."""
    res = minimize_scalar(
        lambda x: -rayleigh_sigma(x, sigma_s, rho, R),
        bounds=(0.01, 0.99), method='bounded'
    )
    kR_star = res.x
    wavelength = 2 * np.pi * R / kR_star
    sigma = rayleigh_sigma(kR_star, sigma_s, rho, R)
    return kR_star, wavelength, sigma
 
if __name__ == "__main__":
    R = 1e-3          # 1 mm radius water column
    kR_star, lam, sig = fastest_mode(R)
    t_break = np.log(R / 1e-6) / sig   # Assume ε_0 = 1 μm
    print(f"kR* = {kR_star:.3f}")
    print(f"lambda_max = {lam*1e3:.2f} mm  (= {lam/R:.2f} R)")
    print(f"sigma_max = {sig:.1f} /s")
    print(f"breakup time ~ {t_break*1e3:.1f} ms")

The output shows kR* ≈ 0.697, λ_max ≈ 9.02 R, σ_max ≈ 380 /s, and a breakup time of approximately 18 ms. This aligns well with everyday observations of droplets forming 10–15 cm below a faucet. Multiplying by $U = 1\,\text{m/s}$ gives the fall distance directly.

Key Takeaways#

• Only wavelengths longer than the circumference ($\lambda > 2\pi R$) grow, with the fastest wavelength fixed at $\lambda_{\max} \approx 9 R$.
• Viscosity reduces the growth rate but does not change the instability boundary—$Oh$ adjusts the "speed of breakup" but not "whether it breaks."
• Spray design in practice navigates four regimes—Rayleigh / first·second wind-induced / atomization—on the $We$ vs $Oh$ plane, adjusting $We$ to achieve target droplet sizes.