For the first time, a Newton method is proposed for the unregularized viscoplastic fluid flow problem. It leads to a superlinear convergence for Herschel-Bulkley fluids when 0 < n < 1, where n is the power law index. Performances are enhanced by using the inexact variant of the Newton method and, for solving the Jacobian system, by using an efficient preconditioner based on the regularized problem. A demonstration is provided by computing a viscoplastic flow in a pipe with a square cross section. Comparisons with the augmented Lagrangian algorithm show a dramatic reduction of the required computing time while this new algorithm provides an equivalent accuracy for the prediction of the yield surfaces. (C) 2016 Elsevier B.V. All rights reserved.