Summary: Viscous flows in domains with boundaries forming two-dimensional corners are considered. We examine the case where on each side of the corner the boundary condition for the tangential velocity is formulated in terms of stress. It is shown that computing such flows numerically by straightforwardly applying well-tested algorithms (and numerical codes based on their use, such as COMSOL Multiphysics) can lead to spurious multivaluedness and mesh-dependence in the distribution of the fluid’s pressure. The origin of this difficulty is that, near a corner formed by smooth parts of the boundary, in addition to the solution of the formulated inhomogeneous problem, there also exists an eigensolution. For obtuse corner angles this eigensolution (a) becomes dominant and (b) has a singular radial derivative of velocity at the corner. Despite the bulk pressure in the eigensolution being constant, when the derivatives of the velocity are singular, numerical errors in the velocities calculation near the corner give rise to pressure spikes, whose magnitude increases as the mesh is refined. A method is developed that uses the knowledge about the eigensolution to remove the artifacts in the pressure distribution. The method is first explained in the simple case of a Stokes flow in a corner region and then generalized for the Navier – Stokes equations applied to describe steady and unsteady free-surface flows encountered in problems of dynamic wetting.