  $H^2$-spatial regularity of stationary and non-stationary problems for
Bingham fluids formulated with the pseudo-stress tensor is discussed. The
problem is mathematically described by an elliptic or parabolic variational
inequality of the second kind, to which weak solvability in the Sobolev space
$H^1$ is well known. However, higher regularity up to the boundary in a bounded
smooth domain seems to remain open. This paper indeed shows such
$H^2$-regularity if the problems are supplemented with the so-called perfect
slip boundary condition. For the stationary Bingham-Stokes problem, the key of
the proof lies in a priori estimates for a regularized problem avoiding
investigation of higher pressure regularity, which seems difficult to get in
the presence of a singular diffusion term. The $H^2$-regularity for the
stationary case is then directly applied to establish strong solvability of the
non-stationary Bingham-Navier-Stokes problem, based on discretization in time
and on the truncation of the nonlinear convection term.
