ORIGINAL CONTRIBUTION Numerical simulations of complex yield-stress fluid flows Evan Mitsoulis 1 & John Tsamopoulos 2 Received: 2 August 2016 /Revised: 30 October 2016 /Accepted: 1 November 2016 # Springer-Verlag Berlin Heidelberg 2016 Abstract Viscoplasticity is characterized by a yield stress, be- low which the materials will not deform and above which they will deform and flow according to different constitutive rela- tions. Viscoplastic models include the Bingham plastic, the Herschel-Bulkley model and the Casson model. All of these ideal models are discontinuous. Analytical solutions exist for such models in simple flows. For general flow fields, it is nec- essary to develop numerical techniques to track down yielded/ unyielded regions. This can be avoided by introducing into the models a regularization parameter, which facilitates the solution process and produces virtually the same results as the ideal models by the right choice of its value. This work reviews several benchmark problems of viscoplastic flows, such as en- try and exit flows from dies, flows around a sphere and a bubble and squeeze flows. Examples are also given for typical process- ing flows of viscoplastic materials, where the extent and shape of the yielded/unyielded regions are clearly shown. The above- mentioned viscoplastic models leave undetermined the stress and elastic deformation in the solid region. Moreover, devia- tions have been reported between predictions with these models and experiments for flows around particles using Carbopol, one of the very often used and heretofore widely accepted as a simple “viscoplastic” fluid. These have been partially remedied in very recent studies using the elastoviscoplastic models pro- posed by Saramito. Keywords Viscoplastic fluids . Bingham plastics . Herschel-Bulkley fluids . Viscoplastic models . Simulations . Yield stress . Yielded/unyielded regions . Elastoviscoplastic fluids Introduction During the past several decades, the emphasis in rheology and continuum mechanics had been on single-phase materials, with particular attention to polymer solutions and polymer melts (Bird et al. 1983). Slurries, pastes and suspensions, frequently encountered in industrial problems, had received less attention than they deserved. Many of these materials have a yield stress, a critical value of stress below which they do not flow; they are sometimes called viscoplastic materials or Bingham plastics, after Bingham (1922), who was the first to describe several types of paint in this way in 1919. They constitute an important class of non-Newtonian materials. The situation appears to have gradually changed after the appearance in 1983 of a seminal review paper by Bird et al. (1983). In that work, the authors provide a list of several ma- terials exhibiting yield, including examples from the food in- dustry, such as margarine, mayonnaise and ketchup, examples of suspensions in Newtonian fluids, etc. They have also made a comprehensive effort to collect most of the works related to viscoplastic materials up to 1980, amounting to 214 references. They have included theoretical developments based on viscoplastic models. The models presented for such viscoplastic materials included the Bingham ( 1922 ), Special Issue to celebrate the centennial anniversary of the seminal Bingham paper * John Tsamopoulos tsamo@chemeng.upatras.gr Evan Mitsoulis mitsouli@metal.ntua.gr 1 School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, 157 80 Athens, Greece 2 Laboratory of Fluid Mechanics and Rheology, Department of Chemical Engineering, University of Patras, Rio, 265 04 Patras, Greece Rheol Acta DOI 10.1007/s00397-016-0981-0