Static interface profiles for contact lines on an elastic membrane with the Willmore energy

We consider a fluid interface in contact with an elastic membrane and study the static profiles of the interface and the membrane. Equilibrium conditions are derived by minimizing the total energy of the system with volume constraints. The total energy consists of surface energies and the Willmore energy; the latter penalizes the bending of the membrane. It is found that, while the membrane is locally flat at the contact line with the contact angle satisfying the Young-Dupré equation, the gradient of the mean curvature of the membrane exhibits a jump across the contact line. This jump balances the surface tension of the fluid interface in the normal direction of the membrane. Asymptotic solutions are obtained for two-dimensional systems in the limits as the reduced bending modulus ν tends to +∞ and 0, respectively. In the stiff limit as ν→+∞, the leading-order solution is given by that of a droplet sitting on a rigid substrate with the contact angle satisfying the Young-Dupré equation; in contrast, in the soft limit as ν→0, a transition layer appears near the contact line and the interfaces have constant curvatures in the outer region with apparent contact angles obeying Neumann's law. These solutions are validated by numerical experiments.