We discretize the Lagrange multiplier formulation of the obstacle problem by mixed and stabilized finite element methods. A priori and a posteriori error estimates are derived and numerically verified.

This article is concerned with the existence of rigid freely floating structures capable of supporting trapped modes (time-harmonic water waves of finite energy in an unbounded domain). Under the usual assumptions of linear water-wave theory, a condition guaranteeing the existence of trapped modes is derived, and structures satisfying this geometric condition are shown to exist in a three-dimensional water channel. The sufficient condition arises from the application of variational principles to a conveniently formulated linear spectral problem, the main effort being the construction of a reduction scheme that turns the quadnic operator pencil associated with the original coupled system into a linear self-adjoint spectral problem. An example of floating bodies supporting at least four trapped modes is given.

The interaction of linear water waves with totally or partially submerged obstacles is considered in a two-layer fluid consisting of two immiscible liquid layers of different densities. A sufficient condition for the existence of trapped modes is established by introducing a trace operator that restricts the solutions to the free surface and the interface. The modes correspond to localized solutions of a spectral problem, decaying at large distances from the obstacles and belonging to the discrete spectrum below a positive cut-off value of the continuous spectrum. The sufficient condition is a simple relation between the cut-off value and some geometrical constants, namely the surface integrals taken over the cross sections of the submerged parts of the obstacles and the line integrals along the parts of the free surface and the interface pierced by the obstacles.

Summary In this article, we study the existence of solutions for the problem of interaction of linear water waves with an array of three-dimensional fixed structures in a density-stratified multi-layer fluid, where in each layer the density is assumed to be constant. Considering time-harmonic small-amplitude motion, we present recursive formulae for the coefficients of the eigenfunctions of the spectral problem associated with the water-wave problem in the absence of obstacles and for the corresponding dispersion relation. We derive a variational and operator formulation for the problem with obstacles and introduce a sufficient condition for the existence of propagating waves trapped in the vicinity of the array of obstacles. We present several (arrays of) structures supporting trapped waves and discuss the possibility of approximating the continuously stratified fluid by a multi-layer model.

We derive a novel and rigorous correction to the classical Reynolds lubrication approximation for fluids with viscosity depending upon the pressure. Our analysis shows that the pressure dependence of viscosity leads to additional nonlinear terms related to the shear-rate and arising from a non negligible cross-film pressure. We present a numerical comparison between the classical Reynolds equation and our modified equation and conclude that the modified equation leads to the prediction of higher pressures and viscosities in the flow domain.