The elastocapillary Landau–Levich problem

We study the classical Landau-Levich dip-coating problem for the case in which the interface possesses both elasticity and surface tension. The aim of the study is to develop a complete asymptotic theory of the elastocapillary Landau-Levich problem in the limit of small flow speeds. As such, the paper also extends our previous study on purely elastic Landau-Levich flow (Dixit & Homsy J. Fluid Mech., vol. 732, 2013, pp. 5-28) to include the effect of surface tension. The elasticity of the interface is described by the Helfrich model and surface tension is modelled in the usual way. We define an elastocapillary number, epsilon, which represents the relative strength of elasticity to surface tension. Based on the size of epsilon, we can define three different regimes of interest. In each of these regimes, we carry out asymptotic expansions in the small capillary (or elasticity) numbers, which represents the balance of viscous forces to surface tension (or elasticity).