Bingham-plastic model: The visco-plastic fluids have an “un-sheared” or “solid” zone under their yield stress point and their constitutive law represents a multi-valued function. In fact under their yield stress point they behave like a solid and above this point they behave like a Newtonian fluid with constant viscosity. The constitutive law for these fluids is given by:
In this paper, the solid zone is approximated via a highly viscous fluid whose viscosity is much (say, times) greater than the main fluid. This condition is used as long as stress is below the yield value, that is, . Above this limit, the following relations are used:
In fact, is calculated according to the gradient of velocity for each particle at each time-step and then it is used in the above criterion.
3.1.3 Herschel-Bulkley model. The constitutive law for the…show more content… A reasonably agreement is observed between the obtained result by the proposed ACISPH method, experimental  and previous SPH results  which show the accuracy of the numerical method implemented.It is observed in Bingham model, that the flow tends to the fixed-point’ with increasing time. It is known that in Bingham dam-breaking model with increasing time, the shear stress is below the yield stress in the flow domain. It was also reported in other researches [24,10, rrt]. For a general greater velocity of the front, the slight discrepancy with the experimental results can be seen; primarily, this can be due to the differences in the experimental setup. Unfortunately the geometry is not perfectly clear in reference.In the second , the treatment of the boundaries is not perfect, because no-slip boundary conditions in SPH represent a challenging problem yet. The final shape of the front and of the shear zone are finally well