  In this paper, we extend a dual-mixed formulation for a nonlinear generalized
Stokes problem to a Huber regularization of the Herschel-Bulkey flow problem.
The present approach is based on a two-fold saddle point nonlinear operator
equation for the corresponding weak formulation. We provide the uniqueness of
solutions for the continuous formulation and propose a discrete scheme based on
Arnold-Falk-Winther finite elements. The discretization scheme yields a system
of Newton differentiable nonlinear equations, for which a semismooth Newton
algorithm is proposed and implemented. Local superlinear convergence of the
method is also proved. Finally, we perform several numerical experiments to
investigate the behavior and efficiency of the method.
