Rayleigh–Plateau Jet Dispersion

Worked Example

Consider an ideal liquid cylinder with a normalized surface tension γ = 1 and density ρ = 1. We analyze a perturbation with a reduced wavenumber q = 0.5 and a cylinder radius R = 1.

1. The ratio of modified Bessel functions of the first kind for q = 0.5 converges to approximately 0.2424996125808019.
2. The squared exponent is s² = (1/R³) * q * (1 - q²) * I₁(q) / I₀(q).
3. Substitute the values: s² = (1 / 1³) * 0.5 * (1 - 0.5²) * 0.2424996125808019 = 0.5 * 0.75 * 0.2424996125808019.
4. s² = 0.375 * 0.2424996125808019 = 0.09093735471780072.
5. Because s² is positive, the mode is growing. The real growth exponent is the square root of 0.09093735471780072, which is s ≈ 0.3015582111596378, with an imaginary part of 0.
6. The corresponding wavelength is 2πR / q = 2π(1) / 0.5 = 4π ≈ 12.566370614359172.

Experiment

You can interact with the dispersion properties of the capillary jet in the module. Follow these steps:

1. Open the Rayleigh–Plateau Jet Dispersion module.
2. Set the Reduced wavenumber control to 0.75.
3. Set the Radius control to 1.
4. Observe that the Bessel ratio increases, changing the growth exponent and lowering the wavelength.

Limitations

This model assumes an inviscid incompressible liquid cylinder with negligible inertia from the surrounding fluid. This implies that the outside fluid is treated effectively as a vacuum regarding its dynamical impact, making it suitable for a liquid jet but not for an air cavity where modified Bessel K functions must be used instead. The model predicts the linear growth rate of a small-amplitude axial surface perturbation but does not claim to describe the fully nonlinear pinch-off evolution of droplets. The long-wave limit q = 0 is a neutral boundary where wavelength becomes Unavailable.

References

For a detailed analytical and numerical study of fluid interfaces and the Rayleigh-Plateau instability, including derivations of the dispersion relation:

• Rayleigh-Plateau instability of anisotropic interfaces: Part 1