On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid

TLDR

Rayleigh’s classic analysis of cylindrical liquid columns showed that capillary‑driven instability occurs when the wavelength exceeds the circumference, with maximum growth at λ≈9.02a, and that for highly viscous columns the instability shifts to very long wavelengths; recent experiments by Taylor on viscous drag between two fluids clarified the mechanism of thread formation and emulsion creation. This study investigates the equilibrium of a long cylindrical thread of a viscous liquid surrounded by another viscous fluid, considering both interfacial surface tension and the viscous forces exerted by the surrounding fluid. Experiments revealed that when the viscosity ratio of the thread to the surrounding fluid.

Abstract

1—The dynamical theory of the instability of a long cylindrical column of an incompressible perfect liquid under the action of capillary force has been given by Rayleigh, neglecting the effect of the surrounding fluid. According to his results, if the column becomes varicose with wave-length λ , the equilibrium of the column is unstable, provided λ exceed the circumference 2π a of the cylinder, in accordance with the result of Plateau’s statical theory; and the degree of instability, as indicated by the value of q in the exponential e qt to which the motion is assumed to be proportional, depends upon the value of λ reaching a maximum when λ = 4.51 × 2 a . The case of a long cylindrical column of an incompressible viscous liquid has also been discussed by Rayleigh, again leaving out of consideration the effect of the surrounding fluid. Assuming the viscosity to be very great compared with the inertia and neglecting the effect of the latter, he has shown that for a very viscous liquid column the maximum instability occurs when the wave-length of the varicosity is very large in comparison with the radius of the cylinder, i. I . e . when λ = ∞ theoretically. Quite recently G. I. Taylor has made interesting experimental researches, together with some theoretical investigations, upon the mode of formation of the cylindrical thread by the disruptive effect of the viscous drag of one fluid on the other, by putting a small drop of a viscous liquid in definable shearing fields of flow of another viscous liquid. He has thus thrown much light upon the mechanism of the formation of emulsions. In the course of his experiments he observed an interesting phenomenon, in one case when the ratio of the viscosity of the liquid forming the thread to that of the surrounding liquid is 0.91, that after the apparatus which was used to produce the field of flow was stopped the final thread gradually broke up into a number of small drops spaced at nearly regular intervals, although it had seemed quite stable while the apparatus was in motion. In connection with this interesting phenomenon, Professor G. I. Taylor kindly suggested to the writer a problem of investigating the character of the equilibrium of a long cylindrical thread of a viscous liquid surrounded by an­other viscous fluid under the action of interfacial surface tension as well as under the effect of viscous forces acting on the liquid inside the column by the surrounding viscous fluid. The effect of the latter is expected to play some important role in the phenomenon under con­sideration, although, as mentioned already, its effect had been neglected by Rayleigh in his investigation.