Numerical simulation of powder flow by Finite Element methods

In contrast to most fluids, flowing powders do not exhibit viscosity such that a Newtonian rheology cannot accurately describe granular flow. Assuming that the material is incompressible, dry, cohesionless, and perfectly rigid-plastic, generalized Navier-Stokes equations (’Schaeffer Model’) have been derived where the velocity gradient has been replaced by the shear rate, and the viscosity depends on pressure and shear rate which leads to mathematically complex problems. In this report we present numerical algorithms to approximate these highly nonlinear equations based on finite element methods. First of all, a Newton linearization technique is applied directly to the continuous variational formulation. The approximation of the incompressible velocity field is treated by using stabilized nonconforming Stokes elements and we use a Pressure Schur Complement smoother as defect correction inside of a direct multigrid approach to solve the linear saddle-point problems with high numerical efficiency. The results of computational experiments for two prototypical flow configurations are provided. 1 PHYSICAL BACKGROUND 1.1 Mohr-Coulomb criterion for friction The Mohr theory suggests that the shear stress on a failure reaches some unique function of normal stress, τ = f(σ), where τ is the shear stress and σ is the normal stress. Coulomb found that for frictional motion the yield shear stress can be expressed as a combination of a normal stress dependent component and a stress independent component. While the normal stress dependent component is connected with the internal angle of friction φ, the former seems to be related to the intrinsic cohesion and is denoted by the symbol c. Then, the Coulomb equation reads τ = σ tanφ+ c, (1) where φ and c are the material constants defined as the angle of internal friction and the cohesive strength, respectively; a material is called non-cohesive if c = 0. Eq. (1) represents the simple law of friction of two solids sliding on each other with the shear force proportional to the normal force, η = tanφ being the friction coefficient. A similar condition also exists at the interface between the granular material and the walls of the container: only the angle of internal friction is replaced by the angle of wall friction, φW . The angle satisfies φW < φ since the wall is usually less rough than a powder layer; this is mainly due to the void fraction near the wall. 1.2 Regimes of powder flow Similar to fluid flow, where several characteristic numbers, like Froude number, Reynolds number, etc., can be used to characterize the qualitative flow behavior, the various powder regimes can be represented as a function of a dimensionless shear rate γ = γ[dp/g] 1/2 which contains a gravitational term g and a particle size dp (see Tardos et al. [13]). Based on such a characterization, one has the following 3 different regimes. 1.2.1 Quasi-static regime This regime is valid when the flow is slow enough that any movement between two static states can be neglected; then the static equilibrium equation can be applied. With this approach only stress and condition of the onset of flow can be computed, while no flow field can be predicted which circumscribes the range of applications of this approach. There is a large number of analytical and numerical solutions to this case