In-phase and out-of-phase break-up of two immersed liquid threads under influence of surface tension

The dynamical behaviour of two infinitely long adjacent parallel liquid threads immersed in a fluid is considered under influence of small initial perturbations. Assuming all fluids to behave Newtonian, we used the creeping flow approximation, which resulted in Stokes equations. Applying cylindrical coordinates and separation of variables, and writing the dependence on the azimuthal direction in the form of a Fourier expansion, we obtained general representations of the equations for both the threads and the surrounding fluid. Substitution of these expressions into the boundary conditions leads to an infinite set of linear equations for the unknown coefficients. Its solutions for the lowest two orders of the Fourier expansion, the so-called zero- and first-order solutions, are presented. Much attention is paid to the (in)stability of the configuration, in terms of the so-called growth rate of the disturbance amplitudes. The growth rate of these amplitudes determines the behaviour of the break-up process of the threads. It turns out that this breaking up occurs either in-phase or out-of-phase. This depends on the viscosity ratio of the fluids and on the distance between the threads. These findings agree with experimental observations. The results of the present work also show that the zero-order solution yields the qualitatively correct insight in the break-up process. The extension to a one order higher expansion only leads to relatively small quantitative corrections.