We analyze the equations of motion for incompressible fluids with the “viscous stress tensor” \(\mathbb {S}\) in a family which includes the Bingham model for viscoplastic fluids (more generally, the Herschel-Bulkley model). \(\mathbb {S}\) is the subgradient of a convex potential, which can depend on the space-time variables. The potential has its one-sided directional derivatives uniformly bounded from below and above by a p-power function. For \(p\geqslant 2.2\) we solve an initial boundary value problem for those fluid systems, in a bounded region in \(\mathbb {R}^3\) . We take a nonlinear boundary condition, which encompasses the Navier friction/slip boundary condition.