<p>In this paper, we study numerical methods for the Navier-Stokes equations governing incompressible Bingham fluids, subject to nonsmooth and nonconvex frictional boundary conditions. The weak formulation of this problem leads to a system of mixed variational-hemivariational inequalities. We introduce a Lagrange multiplier to handle the convex term and derive an equivalent mixed hemivariational inequality. Furthermore, this paper discretizes the mixed hemivariational inequality by mixed finite element method and derives the error convergence order with the help of inf-sup conditions. Finally, the numerical test results are presented, which are consistent with the theoretical error estimates.</p>