**PFEM for particle-laden flows**

Particle-laden, or particulate, flow refers to the dynamics of a **multi-component medium** where at least one component or phase is formed by a continuously connected fluid (the suspending fluid), and where at least one of the remaining phases is formed by a number of disconnected sub-domains (the **particles**). Generally, the particles are assumed to be immiscible, numerous, and small with respect to the characteristic size of the domain considered. Furthermore, other simplifying hypotheses, such as considering only rigid particles of spherical shapes, are generally assumed.

The PFEM, if combined with a discrete element solver, such as the **discrete element method (DEM)**, enables the solution of particle-laden flows also in presence of large motions and changes of shape of the fluid domain. This enables the application of coupled PFEM-DEM methods to a wide range of industrial and engineering problems.

*Simulation of a hypothetical tsunami scenario with PFEM-DEM [1]Rotator drum simulation modeled with a coupled PFEM-DEM strategy (3D view)Mixing process using a PFEM-DEM coupled method *

In the first PFEM–DEM methods presented in the literature [1, 2, 3], the PFEM mesh nodes were assumed to be coincident with the suspended spherical elements.

*Erosion and deposition processes modeled with a PFEM-DEM coupled method [1]*

Recently, in [4], a new formulation has been proposed. In this case, the solid particles are free to move across the fluid elements of the PFEM mesh, such as in the traditional particle-in cell scheme. This is extremely helpful in order to maintain a good quality of the mesh during the analysis at a reasonable cost.

*Water tank discharge with embedded particles. PFEM-DEM method presented in [4] *

**[1]** Onate E, Idelsohn SR, Celigueta MA, Rossi R (2008). *Advances in the particle finite element method for the analysis of fluid–multibody interaction and bed erosion in free surface flows*. Computer methods in applied mechanics and engineering, 197(19-20), 1777-1800.**[2]** Oñate E, Celigueta MA, Latorre S, Casas G, Rossi R, Rojek J (2014) *Lagrangian analysis of multiscale particulate flows with the particle finite element method*. Comput Particle Mech 1(1):85–102**[3]** Celigueta MA, Deshpande KM, Latorre S, Oñate E (2016) *A FEM-DEM technique for studying the motion of particles in non-Newtonian fluids. Application to the transport of drill cuttings in wellbores*. Comput Particle Mech 3(2):263–276**[4]** Franci A, De-Pouplana I, Casas G, Celigueta MA, Gonzalez J, Oñate E (2020) *PFEM-DEM for particle-laden flows with free surface*, Computational Particle Mechanics, 7 (1), 101-120,