In order to study suspensions of deformable particles, a hybrid numerical technique was developed that combined a lattice-Boltzmann (LB) fluid solver with a finite element (FE) solid-phase solver. The LB method accurately recovered Navier-Stokes hydrodynamics, while the linear FE method accurately modeled deformation of fluid-filled elastic capsules for moderate levels of deformation. The LB/FE technique was extended using the Message Passing Interface (MPI) to allow scalable simulations on leading-class distributed memory supercomputers. An extensive series of validations were conducted using model problems, and the LB/FE method was found to accurately capture proper capsule dynamics and fluid hydrodynamics. The dilute-limit rheology was studied, and the individual normal stresses were accurately measured. An extension to the analytical theory for viscoelastic spheres [R. Roscoe. J. Fluid Mech., 28(02):273-93, 1967] was proposed that included the isotropic pressure disturbance. Single-body deformation was found to have a small negative (tensile) effect on the particle pressure. Next, the rheology and microstructure of dense suspensions of elastic capsules were probed in detail. As elastic deformation was introduced to the capsules, the rheology exhibited rapid changes. Moderate amounts of shear thinning were observed, and the first normal stress difference showed a rapid increase from a negative value for the rigid case, to a positive value for moderate levels of deformation. The particle pressure also demonstrated a decrease in compressive stresses as deformation increased. The corresponding changes in microstructure were quantified. Changes in particle self-diffusivity were also noted.

The Eulerian description of dispersed two-phase flows results in a system of partial differential equations describing characteristics of the flow, namely volume fraction, density and velocity of the two phases, around any point in space over time. When pressure forces are neglected or a same pressure is considered for both phases, the resulting system is weakly hyperbolic and solutions may exhibit vacuum states (regions void of the dispersed phase) or localized unbounded singularities (delta shocks) that are not physically desirable. Therefore, it is crucial to find a physical way for preventing the formation of such undesirable solutions in weakly hyperbolic Eulerian two-phase flow models. This thesis focuses on the mathematical analysis of an Eulerian model for air- droplet flows, here called the Eulerian droplet model. This model can be seen as the sticky particle system with a source term and is successfully used for the prediction of droplet impingement and more recently for the prediction of particle flows in air- ways. However, this model includes only one-way momentum exchange coupling, and develops delta shocks and vacuum states. The main goal of this thesis is to improve this model, especially for the prevention of delta shocks and vacuum states, and the adjunction of two-way momentum exchange coupling. Using a characteristic analysis, the condition for loss of regularity of smooth solutions of the inviscid Burgers equation with a source term is established. The same condition applies to the droplet model. The Riemann problems associated, respectively, to the Burgers equation with a source term and the droplet model are solved. The characteristics are curves that tend asymptotically to straight lines. The existence of an entropic solution to the generalized Rankine-Hugoniot conditions is proven. Next, a way for preventing the formation of delta shocks and vacuum states in the model is identified and a new Eulerian droplet model is proposed. A new hierarchy of two-way coupling Eulerian models is derived. Each model is analyzed and numerical comparisons of the models are carried out. Finally, 2D computations of air-particle flows comparing the new Eulerian droplet model with the standard Eulerian droplet model are presented.