Plateau-Rayleigh Instability of a Cylinder of Viscous Liquid (Rayleigh vs. Chandrasekhar)

TL;DR

This paper clarifies and simplifies the complex dispersion equations for the Plateau-Rayleigh instability of viscous liquid cylinders, enabling easier numerical analysis and practical estimation of jet breakup.

Contribution

It demonstrates the equivalence of Rayleigh's and Chandrasekhar's dispersion equations and simplifies them for arbitrary viscosities, providing a practical computational tool.

Findings

Derived simplified dispersion equation for arbitrary viscosity

Provided Mathematica code for numerical analysis

Estimated jet breakup using the simplified model

Abstract

In 1982, in his classical work, L. Rayleigh considered the instability of a cylinder of viscous liquid under capillary force, the so-called Plateau-Rayleigh instability. In this work, in linear approximation, he obtained a dispersion equation describing the increment of this instability as a function of wavelength, the radius of cylinder, the mass density, surface tension, and viscosity of the liquid. Hundreds of authors referred to this work, but none of them used his dispersion equation in its complete form; they used only the asymptotic solutions of his equation for zero and infinitely large viscousities. A reason for this is, probably, that Rayleigh's writing is very difficult and his dispersion equation is quite complex. Then, in 1961, S Chandrasekhar, in his monograph, also considered the stability of a viscous cylindrical jet and obtained his dispersion equation which is also quite complex and differs from the one obtained by Rayleigh. As in the case of Rayleigh's dispersion equation, other works use only the asymptotic solution of Chandrasekhar's equation that corresponds to the case where the viscosity is very large in comparison to inertia. In this paper, I demonstrate that Chandrasekhar's dispersion equation is equivalent to Rayleigh's and then simplify their dispersion equations to a form which can be easily solved numerically for arbitrary values of viscosity. I also present Mathematica code to calculate the maximum increment of the Plateau-Rayleigh instability for given parameters of the jet. To illustrate how the code works, I apply it to a cylindrical jet to estimate its breakup.