In the present work, we describe the dynamics of a moving contact line for thin films flowing down an inclined plane. Our focus is the problem of triple point located at the interface between the solid wall, the moving fluid and air, for example the spreading of a drop on a plane dry wall (horizontal or inclined) due to gravity and capillarity. In the first part, we explain how we can reduce to the Stokes equations and why the resulting problem is ill-posed. This singularity is removed by permitting the fluid to slip along the wall close to the contact line. Thus we manage to find a solution in H1 constructed by asymptotic expansions. Then we focus on the upstream dynamic of the flow, which is set to a thin film flow. We develop the classical system of Shallow-water equations (Saint-Venant equations) from the full Navier-Stokes system using the classical long-wavelength expansion. We obtain a set coupled equations for the flow depth and the flow-rate. In the neightboorhood of the contact line, we develop an asymptotic expansion of the steady Stokes system with slip at bottom in function of the capillary number. The solution in the vicinity of the contact line is developped in the inner region and the outer region. Then, a direct matching can be done (assuming dynamic and static angles are equals) or using an intermediate region (with different angles). This leads to two different families of models. Bringing together the upstream Shallow-water equations and the contact line models, we write a new Shallow-water model taking into account the dynamic of the moving contact line. Then, we deduce a simplier one-equation model for the film thickness. This model extends existing models with no slip at bottom to models with slip. Direct numerical simulations of the last models are performed, combining a moving contact line and a thin liquid film.
