Abstract. Equations governing the flow of fluid containing visco-hyperelastic particles are developed in an Eulerian framework. The novel feature introduced here is to write an evolution equation for the strain. It is envisioned that this will simplify numerical codes which typically compute the strain on Lagrangian meshes moving through Eulerian meshes. Existence results for the flow of linear visco-hyperelastic particles in a Newtonian fluid are established using a Galerkin scheme.

The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen-Leslie equations is considered. Convergence of finite element approximations is established under appropriate regularity hypotheses, and numerical experiments exhibiting the interaction of singularities and the coupling of the director and momentum equations are presented.

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Abstract. The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen-Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve H 2 (Ω) norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.Mathematics Subject Classification. 65M60, 76A15.

a b s t r a c tWe describe a model based on continuum mechanics that reduces the study of a significant class of problems of discrete dislocation dynamics to questions of the modern theory of continuum plasticity. As applications, we explore the questions of the existence of a Peierls stress in a continuum theory, dislocation annihilation, dislocation dissociation, finite-speed-of-propagation effects of elastic waves vis-a-vis dynamic dislocation fields, supersonic dislocation motion, and short-slip duration in rupture dynamics.