  We study, both theoretically and numerically, the equilibrium of a hinged
rigid leaflet with an attached rotational spring, immersed in a stationary
incompressible fluid within a rigid channel. Through a careful investigation of
the properties of the functional describing the angular momentum exerted by the
fluid on the leaflet (which depends on both the leaflet angular position and
its thickness), we identify sufficient conditions on the spring stiffness
function for the existence (and uniqueness) of equilibrium positions. We
propose a numerical technique that exploits the mesh flexibility of the Virtual
Element Method (VEM). A (polygonal) computational mesh is generated by cutting
a fixed background grid with the leaflet geometry, and the problem is then
solved with stable VEM Stokes elements of degrees $1$ and $2$ combined with a
bisection algorithm. We present a large array of numerical experiments to
document the accuracy and robustness with respect to degenerate geometry of the
proposed methodology.
