Break-up of a set of liquid threads under influence of surface tension

In this paper it is shown how the long-standing problem of the break-up of a cylindrical interface due to surface tension can be generalized to an arbitrary number of interacting interfaces in an arbitrary configuration. A system of immersed threads starting with two types of configurations is studied, i.e. , a system of threads on a row and a system of threads at triangular vertices. From these cases, which are worked out in detail, it becomes clear how the stability of an arbitrary configuration can be determined. The (in)stability of the configuration is discussed in terms of the so-called disturbance growth rate. It turns out that the threads break up in specific phase patterns in which neighbouring threads are either in-phase or out-of-phase. For L threads, in principle 2L phase patterns are possible. However, it is shown that the stability of the system directly follows from L so-called basic phase patterns. Special attention is paid to the special case of threads and fluid having equal viscosity. Then, the growth rate can be calculated analytically using Hankel transformations. An estimate for the growth rate in this case, which turns out to be quite sharp, is derived.