On Plateau–Rayleigh Instability of a Cylinder of Viscous Liquid

Abstract

In 1892, 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 or infinitely large viscosities. A reason for this is, probably, that Rayleigh’s writing is 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 article, the author demonstrates that Chandrasekhar’s dispersion equation is equivalent to Rayleigh’s and then simplifies their dispersion equations to a form which can be easily solved numerically for arbitrary values of viscosity. He also presents a Mathematica code to calculate the maximum increment of the Plateau‐Rayleigh instability for given parameters of the jet. To illustrate how the code works, he applies it to a cylindrical jet to estimate its breakup.