Mathematical Analysis of Macroscopic Models for Slow Dense Granular Flow
- In this dissertation we present analysis of macroscopic models for slow dense granular flow. Models are derived from plasticity theory with yield condition and flow rule. Corner stone equations are conservation of mass and conservation of momentum with special constitutive law. Such models are considered in the class of generalised Newtonian fluids, where viscosity depends on the pressure and modulo of the strain-rate tensor. We showed the hyperbolic nature for the evolutionary model in 1D and ill-posed behaviour for 2D and 3D. The steady state equations are always hyperbolic. In the 2D problem we derived a prototype nonlinear backward parabolic equation for the velocity and the similar equation for the shear-rate. Analysis of derived PDE showed the finite blow up time. Blow up time depends on the initial condition. Full 2D and antiplane 3D model were investigated numerically with finite element method. For 2D model we showed the presence of boundary layers. Antiplane 3D model was investigated with the Runge Kutta Discontinuous Galerkin method with mesh addoption. Numerical results confirmed that such a numerical method can be a good choice for the simulations of the slow dense granular flow.