Relative periodic solutions in conducting liquid films flowing down vertical fibres

Abstract

The dynamics of a conducting liquid film flowing down a cylindrical fibre, subjected to a radial electric field, is investigated using a long-wave model (Ding et al., J. Fluid Mech., vol. 752, 2014, p. 66). In this study, to account for the complicated interactions between droplets, we consider two large droplets in a periodic computational domain and find two distinct types of travelling wave solutions, which consist of either two identical droplets (type I) or two slightly different droplets (type II). Both are ‘relative’ equilibria, i.e. steady in a frame moving at their phase speed, and are stable in smaller domains when the electric field is weak. We also study relative periodic orbits, i.e. temporally recurrent dynamic solutions of the system. In the presence of the electric field, we show how these invariant solutions are linked to the dynamics, where the system can evolve into one of the steady travelling wave states, into an oscillatory state, or into a ‘singular structure’ (Wray et al., J. Fluid Mech., vol. 735, 2013, pp. 427–456; Ding et al., J. Fluid Mech., vol. 752, 2014, p. 66). We find that the oscillation between two similarly sized large droplets in the oscillatory state is well represented by relative periodic orbits. Varying the electric field strength, we demonstrate that relative periodic solutions arise as the dynamically important solution once the type-I or type-II travelling wave solutions lose stability. Oscillation can be either enhanced or impeded as the electric field’s strength increases. When the electric field is strong, no relative periodic solutions are found and a spike-like singular structure is observed. For the case where the electric field is not present, the oscillation is instead caused by the interaction between a large droplet and a nearby much smaller droplet. We show that this oscillation phenomenon originates from the instability of the type-I travelling wave solution in larger domains, and that the oscillatory state can again be represented by an exact relative periodic orbit. The relative periodic orbit solution is also compared with experimental study for this case. The present study demonstrates that the relative periodic solutions are better at capturing the wave speed and oscillatory dynamics than the travelling wave solutions in the unsteady flow regime.