An investigation of grouping of two falling dissimilar droplets using the homotopy analysis method

The motion of two dissimilar spherical non-evaporating liquid droplets moving vertically along their line of centers is analysed mathematically. The stimulus for the study is evidence from the literature that, under appropriate operating conditions, droplet grouping can occur. Carefully controlled experiments, aimed at providing a paradigm for understanding this phenomenon, considered the configuration of the current work. The governing non-linear ordinary differential equation describing the behaviour of the distance between the droplets is solved analytically for the first time, using the homotopy analysis method. In order to successfully implement the latter, the fact that the two consecutive droplets considered are mildly different in size was exploited. This enabled a considerable reduction of the complicated expressions for the drag forces acting on the two droplets. Whilst retaining generality, further simplification was achieved using curve fitting for some of the key expressions appearing in the reduced form of the drag forces. The resulting nonlinear equation was then tractable for straightforward application of the homotopy analysis method. For almost identical droplets, validation of the solution is afforded by experimental data from the literature. Comparison between the predictions of the new analytical solution and a numerical solution of the relevant ODE yielded excellent agreement thereby ratifying the proposed combined approach. The results indicate that the larger droplet of the pair approaches the leading droplet more slowly than if the two droplets are equal in size.