Background

For the case of a viscous thread in an infinite viscous medium, Tomotika² extended Lord Rayleigh's¹ pioneering study to the breakup of Newtonian liquid cylinders in a Newtonian liquid matrix for the case in which no overall flow field is present. Initially, a liquid cylinder of radius R₀ = D₀/2 (D₀ is the thread diameter) is subject to thermally induced sinusoidal distortions of arbitrary wave lengths . For fluctuations (of amplitude ) and wavelength, , there is a decrease in the total interfacial area with increasing for the case > 2R₀. This decrease in area provides the driving force for growth of the instability. The dimensionless wavenumber X is defined by

There is a critical wavenumber, X_c = 1, separating fluctuations that decay (X > X_c) from those that grow in time (X < X_c). A linear stability analysis shows that in the early-stage growth fluctuations grow exponentially with time:

where ₀ is the initial amplitude and the growth rate of this distortion, q, is given by

where is the interfacial tension, _m is the viscosity of the matrix, p is the thread/matrix viscosity ratio, and R₀ is the initial thread diameter. The function, (X,p), can be obtained from Tomotika's original paper.² For a given viscosity ratio p, there is one dominant wavelength X_m at which the amplitude grows fastest; the distortion having this wavelength consequently leads the thread to break up into droplets.

At first it seems counterintuitive that a fluctuation, which increases the contour length of the thread, can cause a decrease in surface area. The total area of an axisymmetric thread per average unit length is approximately

in the limit of / 1. Applying conservation of volume to the distorted thread (which has a circular cross section everywhere) leads to a decrease in the average radius of the thread:

Thus, there are two competing effects for the interfacial area: the contour length increase vs the average decrease of the thread radius. It can be seen that for X < 1 there is an overall reduction in surface area. Thus, the decrease in surface area is due to the geometry of the thread, namely, the fact that the cross-sectional area scales as r². In the present problem we shall see that the confinement causes the fluctuation to grow asymmetrically. Because of the asymmetry, the decrease in interfacial area becomes a weaker effect compared to the increase in contour length; consequently, the wavelength of the fastest growing mode increases while the growth rate is reduced.