The steady state of gravity-capillary problem with inclined walls

The gravity-capillary problem with inclined walls is a problem that describes an open fluid flowing over an angled wall. It has broad applications in science and engineering. In this paper, we study the steady states of the two-dimensional inclined-wall problem. The steady-state configurations are characterized as solutions of the Euler-Lagrange equation associated with a prescribed energy functional, subject to a fixed contact-angle boundary condition. By parameterizing the free surface using an appropriately chosen maximal point, we construct solutions to this Euler-Lagrange equation via a shooting method, with the fluid volume serving as the shooting parameter. The construction is valid for arbitrary contact angles and arbitrary inclined angles of the walls.