  We developed a computational framework for simulating thin fluid flow in
narrow interfaces between contacting solids, which is relevant for a range of
engineering, biological and geophysical applications. The treatment of this
problem requires coupling between fluid and solid mechanics equations, further
complicated by contact constraints and potentially complex geometrical features
of contacting surfaces. We developed a monolithic finite-element framework for
handling mechanical contact, thin incompressible viscous flow and fluid-induced
tractions on the surface of the solid, suitable for both one- and two-way
coupling approaches. Additionally, we consider the possibility of fluid
entrapment in "pools" delimited by contact patches and its pressurisation
following a non-linear compressibility constitutive law. Furthermore, image
analysis algorithms were adapted to identify the local status of each interface
element within the Newton-Raphson loop. First, an application of the proposed
framework for a problem with a model geometry is given, and the robustness is
demonstrated by the residual-wise and status-wise convergence. The full
capability of the developed two-way coupling framework is demonstrated on a
problem of a fluid flow in contact interface between a solid with
representative rough surface and a rigid flat. The evolution of the contact
pressure, fluid flow pattern and the morphology of trapped fluid zones until
the complete sealing of the interface is displayed. Additionally, we
demonstrated an almost mesh-independent result of a refined post-processing
approach to the real contact-area computation. The developed framework permits
not only to study the evolution of effective properties of contact interfaces,
but also to highlight the difference between one- and two-way coupling
approaches and to quantify the effect of multiple trapped fluid "pools" on the
coupled problem.
