Contact lines that form when fluid-fluid interfaces intersect with solid boundaries are ubiquitous in industrial and natural applications, such as micro-nanofluidics and immiscible flows in porous media. Significant progress has been made recently on the modeling of dynamic contact line problems, yet they remain computationally difficult to tackle in that they involve length scales ranging from molecular to macroscopic scales. Here we present a review of the moving contact line paradox and present two approaches to eliminate it. The first approach is based on the inclusion of fluid-solid intermolecular interactions into the Navier-Stokes equations, and the second approach is based on a Volume-of-Fluid model where the boundary conditions are modified by introducing a slip model. In both models, we suggest a manner in which simulations can be carried out, despite the singularity at the contact line.
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