The goal of the project is the development of multiscale models based on an asymptotic analysis of the partial differential equations in domains with complex geometry, such as thin tube structures, multistructures, eventually taking into consideration a fluid-wall interaction. The motivation of the project is related to the bio-medical applications (hemodynamics), as well as to some industrial applications (pipelines). Mostly the Navier-Stokes equations (NSE) will be used for the fluid motion description. In some cases, other more complex equations will be considered: Brinkmann flows; coupling the fluid motion equations with the diffusion-convection equations (for example, for the blood cells transfer); dependence of the viscosity on the particles (blood cells) concentrations and of the diffusivity on the shear strain rate particle models. The main assumption is a hypothesis of smallness of the ratio ε of the characteristic sizes of microscopic and macroscopic scales. Complex geometry of the vessels, presence of multiple scales and smallness of the microscopic characteristic size make a direct numerical solution of the NSE type boundary value problems set in all vessels at the microscopic level impossible because of a tremendous volume of computational resources which is needed for such implementation. The main idea is the multiscale analysis combining an asymptotic reduction of 3D models to 1D ones and the three-dimensional zooms in the neighborhood of bifurcations and other singularities of the domain. The expected results of the project are: new asymptotic decomposition method for NSE in thin tube structures; new time-periodic Reynolds’ type equation on the graph and numerical method of its solution; new multiscale models for NSE and non-Newtonian equations in thin structures combining an asymptotic reduction and three-dimensional zooms; numerical experiments comparing the exact solution and an approximate solution to the problem of hybrid dimension.

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