Modélisation continue de la rhéologie des suspensions et de la migration

Abstract : Suspensions which are heterogeneous continuous media composed by a solid granular phaseand a liquid phase present many applications from natural and industrial sciences.Modelling the phenomenon involved in those applications suggests studyingvery complex cases whereas the simplest one present non trivial behaviour that are notperfectly understood. This is why we put forward some progresses in the understanding ofa reference material, mono-disperse suspension of hard spheres. Firstly we compute averagedversions of conservation laws at the scale of the grain. Hence we get continuous quantities atthe scale of an elementary reference volume and it allows us to revisit mathematical buildingof continuous biphasic models. Hence, we provide a system of continuous conservation laws.Nevertheless, we have to add several hypothesis to get a well posedmathematical problem from our system. With that mind, we choose to focus on non-colloidalsuspensions with a Newtonian suspending fluid. Thus, we propose a new rheological modelincluding a texture tensor which is an ancillary variable modelling the average deformationof the local cages formed by neighbouring particles. Hence, we can quantitatively reproducetwo effects that have been experimentally measured. The first one is about the anisotropy ofthe microstructure in a sheared suspension, measured by the averaged pair distribution function.The second one is the evolution of apparent viscosity that drops brutally before relaxing to asteady state during a shear-reversal experiment. This first model provides a link between twocontinuous quantities, the averaged pair distribution function and the apparent viscosity.However, it reproduces badly the evolution of some macroscopic quantities, the normal stressdifferences. This is why we extend our first model, improving those bad predictions. On largertime scales, it has been observed that the particles do not follow strictly the flow lines,for instance, they leave sheared zones. This phenomenon is called migration. Our extendedrheological model allows to enhance migration phenomenon predictions because normal stressdifferences play a crucial role in granular phase motion. In that mind, we integrate ournew rheology in the system of conservation laws that we stated in the beginning, then, weprocess it into a new mathematical problem.We put forward a system including both a bulk velocity and a granular phase one.We introduce an unilateral constraint in order to model the inelastic contact interactionbetween two rigid spheres at the macroscopic level. It constitutes the originality of ourproposal. Finally, we get a problem that interprets as two coupled sub-problems.The first one is like a visco-elastic fluid model and the second one is a compressiblecongested viscous system, like those used to modelcrowd motions.
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Olivier Ozenda. Modélisation continue de la rhéologie des suspensions et de la migration. Analyse numérique [math.NA]. Université Grenoble Alpes, 2019. Français. ⟨NNT : 2019GREAM011⟩. ⟨tel-02101366v2⟩

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