Scaling maximum spreading of droplet impacting on flexible substrates

Abstract

We numerically study the impact of a droplet on superhydrophobic flexible plates, aiming to understand how the flexible substrate influences the maximum spreading of the droplet. Compared with the rigid case, the vertical movement of the flexible substrate due to droplet impact reduces the maximum spreading. Besides, the average acceleration $a$ during droplet spreading changes significantly. Arising from energy conservation, we rescale the acceleration $a$ for cases with different bending stiffness $K_B$ and mass ratio $M_r$ . Moreover, through theoretical analysis, we propose a scaling for the droplet's maximum spreading diameter ratio $\beta _{max}$ . In the scaling, based on the derived $a$ , an effective Weber number $We_m$ is well defined, which accounts for the substrate properties without any adjustable parameters. In the ( $\beta _{max}, We_m$ ) plane, the two-dimensional numerical results of different $K_B$ , $M_r$ and rigid cases all collapse into a single curve, as do the experimental and three-dimensional (3-D) results. In particular, the collapsed 3-D data can be well represented by the universal rescaling of $\beta _{max}$ proposed by Lee et al. (J. Fluid Mech., vol. 786, 2016, R4). Furthermore, an a posteriori energy analysis confirms the validation of our a priori scaling law.