In this dissertation we consider mesoscale based models for flow driven fibre orientation dynamics in suspensions. Models for fibre orientation dynamics are derived for two classes of suspensions. For concentrated suspensions of rigid fibres the Folgar-Tucker model is generalized by incorporating the excluded volume effect. For dilute semi-flexible fibre suspensions a novel moments based description of fibre orientation state is introduced and a model for the flow-driven evolution of the corresponding variables is derived together with several closure approximations. The equation system describing fibre suspension flows, consisting of the incompressible Navier-Stokes equation with an orientation state dependent non-Newtonian constitutive relation and a linear first order hyperbolic system for the fibre orientation variables, has been analyzed, allowing rather general fibre orientation evolution models and constitutive relations. The existence and uniqueness of a solution has been demonstrated locally in time for sufficiently small data. The closure relations for the semiflexible fibre suspension model are studied numerically. A finite volume based discretization of the suspension flow is given and the numerical results for several two and three dimensional domains with different parameter values are presented and discussed.
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.