  We present a new approach to the problem of stationary viscoplastic duct flow
as modelled by the Herschel-Bulkley model, with Bingham fluids included as a
special case.
  While the mathematical formulation of this problem is conventionally based on
a variational inequality, or equivalently, on a nonsmooth minimisation problem
for the flow velocity, we suggest an alternative approach. Considering the
Lagrangian dual in terms of the stress, rather than the velocity, turns out to
be advantageous in numerous ways. The objective functional possesses higher
regularity, which ensures applicability of second order methods. Our numerical
experiments with a trust-region SQP algorithm also demonstrate clearly superior
performance compared to the widely used augmented Lagrangian method, although
no artificial regularisation is introduced into the problem.
  Hence, besides providing a new theoretical angle to a classical problem, our
results also pave the way for an entirely new class of numerical approaches to
simulating flows of viscoplastic fluids.
