# Fluid Thread Breakup - Linear Stability of Inviscid Liquids

## Description

Fluid thread breakup is the process by which a single mass of fluid breaks into several smaller fluid masses. The process is characterized by the elongation of the fluid mass forming thin, thread-like regions between larger nodules of fluid. The thread-like regions continue to thin until they break, forming individual droplets of fluid.

Thread breakup occurs where two fluids or a fluid in a vacuum form a free surface with surface energy. If more surface area is present than the minimum required to contain the volume of fluid, the system has an excess of surface energy. A system not at the minimum energy state will attempt to rearrange so as to move toward the lower energy state, leading to the breakup of the fluid into smaller masses to minimize the system surface energy by reducing the surface area. The exact outcome of the thread breakup process is dependent on the surface tension, viscosity, density, and diameter of the thread undergoing breakup.

Linear stability of inviscid liquids

The linear stability of low viscosity liquids was first derived by Plateau in 1873. However, his solution has become known as the Rayleigh-Plateau instability due to the extension of the theory by Lord Rayleigh to include fluids with viscosity. Rayleigh-Plateau instability is often used as an introductory case to hydrodynamic stability as well as perturbation analysis.

Plateau considered the stability of a thread of fluid when only inertial and surface tension effects were present. By decomposing an arbitrary disturbance on the free surface into its constitutive harmonics/wavelengths, he was able to derive the a condition for the stability of the jet in terms of the perturbation that is shown here.

Related formulas## Variables

ω | growth rate of the perturbation (dimensionless) |

σ | surface tension of the fluids(N/m^2) (dimensionless) |

k | wavenumber of perturbation(1/m) (dimensionless) |

ρ | fluid density(m) (dimensionless) |

a | initial radius of the unperturbed fluid(m) (dimensionless) |

I_{1} | modified Bessel function of the first kind (dimensionless) |

I_{0} | modified Bessel function of the first kind (dimensionless) |