The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

In this work, an extended discontinuous Galerkin (extended DG/XDG also called unfitted DG) solver for two-dimensional flow problems exhibiting moving contact lines is presented. The generalized Navier boundary condition is employed within the XDG discretization for the handling of the moving contact lines. The spatial discretization is based on a symmetric interior penalty method and the numerical treatment of the surface tension force is done via the Laplace-Beltrami formulation. The XDG method adapts the approximation space conformal to the position of the interface and allows a sub-cell accurate representation within the sharp interface formulation. The interface is described as the zero set of a signed-distance level-set function and discretized by a standard DG method. No adaption of the level-set evolution algorithm is needed for the extension to moving contact line problems. The developed solver is validated against typical two-dimensional contact line driven flow phenomena including droplet simulations on a wall and the two-phase Couette flow.