" Dynamics Days Latin America and the Caribbean " Puebla , México , October 24 - November 01 , 2016 PROGRAM SUNDAY 23

We study the dynamics of liquids drops on a solid substrate [1] employing a long-wave evolution equation for partially wetting liquids of small equilibrium contact angles. After a brief introduction of the modelling of capillarity and wettability in a mesoscopic hydrodynamic theory we focus on the case where lateral driving forces, e.g., gravity for an inclined substrate results in sliding ridges (two-dimensional drops) [2] and three-dimensional drops [3]. First, we analyse periodic trains of stationary sliding droplets and the transformations of their individual droplets in dependence of the driving force employing continuation techniques [4]. We show that a number of shape transitions of stationary sliding droplets occur at saddle-node bifurcations. Further there is a global bifurcation that results in dynamic states where a main sliding droplet emits small satellite droplets at its rear (pearling instability) that subsequently coalesce with the next main droplet. These pearling states show the period-doubling route to chaos. Second, we conduct direct numerical simulations on a large spatial domain and examine the interaction of many sliding drops. As the sliding velocity of the drops depends on their volume, larger drops overtake smaller drops and merge with them. The ongoing merging and pearling behavior results in a stationary distribution of drop sizes, whose shape depends on the inclination angle of the substrate and the overall volume of liquid in the system. We illustrate that aspects of the steady long-time drop size distribution may be deduced from the bifurcation diagrams for individual drops. Finally, we return to the pearling-coalescence cycles and pose the question whether the illustrated droplet interactions indicate that sliding droplets may actually be considered as another type of dissipative solitons. [1] T. Podgorski, J.-M. Flesselles and L. Limat, Phys. Rev. Lett. 87, 036102 (2001). [2] U. Thiele et al., Phys. Rev. E 64, 061601 (2001). [3] S. Engelnkemper, M. Wilczek, S. Gurevich and U. Thiele, preprint at http://arxiv.org/abs/1607.05482 (2016). [4] Using PDE2PATH, see H. Uecker, D. Wetzel and J.D.M Rademacher, Numer. Math.-Theory Methods Appl. 7, 58-106 (2014). Dynamics Days Latin America and the Caribbean 2016 Puebla, México, October 24 November 01, 2016 40 Dynamics of Large Eddy Simulation of box turbulence