Grigory Panassenko: Partial dimension reduction: Navier-Stokes problem in networks of thin tubes (blood vessels)

Abstract

Partial differential equations in thin domains combining thin plates and thin rods or pipes (so-called multi-structures) are extensively studied in mathematical solid and fluid mechanics (see e.g. [1-3]). The talk briefly presents the results on asymptotic analysis for viscous flows in thin tube structures. These domains are idealized geometrical models for networks of thin blood vessels, tubes in catalytic converters, pipelines etc. The asymptotic analysis of the flows in these structures allowed to introduce the hybrid dimension models. They combine one-dimensional and multidimensional description of the flow with asymptotically exact coupling conditions between 3D and 1D parts of the model (see [4,5] and a recent monograph [6]). Another approach for junction of models of different dimensions for blood flow in arteries was proposed in [7, 8]. Thus, hybrid dimension models provide the one-dimensional description in the main part of the domain and make small full-dimensional zooms. These zooms give detailed description of the flow in the zones of interest such as the bifurcations of vessels, zones of blood clot formation, stents and so on. The hybrid dimension models allow substantially accelerate computations without loss of accuracy. They are justified via asymptotic analysis of the full-dimensional problem in the whole domain of the flow and the proof of estimates for the difference between the exact solution of the full-dimensional problem and the solution to the hybrid dimension model. The classical method of asymptotic decomposition of the domain [5] deals with coupled problems in the zoomed parts of the domain. The new method of partial asymptotic dimension reduction [9] allows the parallelization of the computations in the zoomed parts solving some special problem on the graph of the network.

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9. G. Panasenko, K. Pileckas, Partial asymptotic dimension reduction for steady state non-Newtonian flow with strain rate dependent viscosity in thin tube structure, J.Math. Fluid. Mech., 25:11, 2023.