Unlike ordinary liquids and elastic solids, complex fluids exhibit several puzzling behaviors that critically depend on the underlying structures that compose the fluids. Indeed, many complex fluids are made of microscopic entities (such as rigid or soft particles, biological cells, macromolecules etc...) which are suspended in a liquid, and whose individual and collective behavior strongly impacts on the overall rheological properties of the fluid at the global scale and gives to complex fluids nontrivial behaviors. It is this feedback from the microscale to the macroscale that continues to pose a formidable challenge to theoretical and numerical modeling. The proposed project is oriented towards the direct numerical simulation of complex systems composed of an incompressible newtonian fluid and entities (rigid or deformable), in connection with modeling at the microscopic level (e.g. close range interaction forces will be taken into account) and at the macroscopic level (comparison with continuous, non-newtonian models). The efforts of the mathematics/numerics part of the team will be dedicated to the numerical modeling of actual experiments on vesicle suspensions carried out by the experimental part of the team, and a special attention will be paid by experimentalists to measure characteristic quantities to be compared to their computed counterparts. The numerical challenge consists in solving the incompressible Stokes and Navier Stokes equations coupled to suspended, possibly deformable, possibly many entities, in order to investigate both the microscopic phenomena and the collective behavior of such mixtures, including the case of highly concentrated suspensions. Whereas the use of existing Arbitrary Lagrangian Approach is not excluded (Stokes and Navier-Stokes solvers have been developed independently by two members of the team) to validate some test cases and investigate the behavior of inclusions close to contact, the strategies we plan to develop are mainly based on the Eulerian approach (a fixed mesh covers the domain occupied by the mixture). Beside the difficulty introduced by the deformability of the membranes which delimit the inclusions (those membranes tend to minimize a so-called bending energy whose superficial density is the square of the mean curvature), a special attention will be given to close-range interactions (lubrication forces), which are likely to play a major role in the concentrated case. Lubrication forces, which depend singularly upon inter-object distances, will be integrated in the numerical models. The expected results are (i) physical, mathematical and numerical validations of the proposed methods, (ii) their use to extract the space-time organization of the assemblies, (iii) a systematic analysis of the micro-macro coupling regarding the rheological properties, (iv) comparison of the outcome from full numerical simulations with the phenomenological equations, like Oldroyd B model; (v) a systematic confrontation of the results with the experimental findings, especially for vesicles which are known, unlike rigid particles, to have various microscopic motions. The richness of the microscopic dynamics is expected to deliver various nontrivial behaviour regarding the global rheological properties; (vi) beyond 'passive' motion of the entities, the proposed project should open new lines of inquiries towards the study of active motions (like cellular motility).