Ecoulement de Fluides Complexes ‘en eaux peu profondes’: analyse mathématique et applications – SWECF
The mathematical and numerical analysis of models from fluids mechanics has recently known spectacular developments during the past ten years especially in the domain of complex fluids, compressible fluids mechanics and free surface flows. We propose to develop a project inside the Institut Camille Jordan (Univ. Lyon) that has strong connections with those research domains in order to model flows like rivers, snow avalanches, mud floods or pollutant derive,...The possible applications are likely to answer practical problems like the management of risks in relation with natural hazards like the design of defence structures, the determination of zones where there are risks of snow avalanches or mud floods. For a better understanding of those flows, we plan to develop numerical and mathematical tools that are complementary to the experimental and statistical approaches that are developed either in the Rhone Alpes region (laboratory of CEMAGREF, CEN, EPFL) but also at the international level within the european project SATSIE for avalanches. Let us deal with a pratical exemple, namely snow avalanches and dense snow flows. The snow is composed of water molecules that are in different physical states, thus snow can be considered as a complex fluid. Moreover this « fluid » is Non Newtonian, that is the deformation rate of the fluid is not proportional to the strain tensor applied to the fluid. The main problem here is to determine the rheology of the fluid that is precisely a relation between the strain applied to the fluid and its deformation under that constraint. Similarly to snow avalanches where two layers, one composed of powder snow and one composed of dense snow, coexist, the complexity of the fluid may come also from the superposition of several layer of fluids with different physical properties (such as density, rheology, physical state). One usual model the flows that we consider with the incompressible Navier-Stokes equations with a free surface: this problem is hardly analyse with standard mathematical and numerical tools due to the presence of that free surface and the complexity of the fluid that is considered (snow, muds,...). However, those flows have the common property that the characteristic fluid height is much small than the scale of the area where the flow takes place and are in some sense Shallow water type flows. With the help of that characteristic, one can obtain a simpler model: the Shallow Water Equations. It is a system of Partial Differential Equations that describes the evolution of the fluid height and the average speed along the fluid height in the main direction of propagation: as a consequence we can get rid of the free surface problem and decrease the dimension of the problem. The numerical simulations and mathematical analysis of the problem is then easier to handle with.
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