Modelling moving contact lines on inextensible elastic sheets in two dimensions

Elastocapillarity has attracted increasing interest in recent years due to its important roles in many industrial applications. In this work, we derive a thermodynamically consistent continuum model for the dynamics of two immiscible fluids on a thin and inextensible elastic sheet in two dimensions. With the sheet being modelled by a deformable curve with the Wilmore energy and local inextensibility constraint, we derive a two-phase hydrodynamics model with the interfacial and boundary conditions consistent with the second law of thermodynamics. In particular, the boundary conditions on the sheet and at the moving contact line take the form of force balances involving the fluid stress, surface tensions, the sheet bending force and sheet tension, as well as friction forces arising from the slip of fluids on the sheet. The resulting model obeys an energy dissipation law. To demonstrate its capability of modelling complex elastocapillary interactions, we consider two applications: (1) the relaxation dynamics of a droplet on an elastic sheet and (2) the transport of a droplet driven by bendotaxis in a channel bounded by elastic sheets. Numerical solutions for the coupled fluid-sheet dynamics are obtained using the finite element method. The detailed information provided by the full hydrodynamics model allows us to better understand the dynamical processes as compared to other simplified models that were used in previous work.