in thin tube structures.

Grigory Panassenko est professeur et directeur de la Structure Fédérative de Recherche Modélisation Mathématique et Aide à la Décision, de l'université de Saint-Etienne. Il est invité à parler au séminaire du laboratoire MICS coencadré par la Fédération de Mathématiques de CentraleSupélec.

Titre : Asymptotic reduction and numerical strategy for Newtonian flows in thin tube structures.

Résumé :

Thin structures are some finite unions of thin rectangles (in 2D settings) or cylinders (in 3D settings) depending on small parameter $$\epsilon << 1$$ that is, the ratio of the thickness of the rectangle (cylinder) to its length. We consider a steady and then a non-steady Navier-Stokes equation in thin structures with the no-slip boundary condition at the lateral boundary and with the inflow and outflow conditions with the given velocity of order one. The steady state Navier-Stokes equations in thin structures were considered first in the paper by G.Panasenko Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure , C.R.Acad.Sci.Paris, t. 326, Série IIb, 1998, pp. 867-872 and then developed in the book by the same author “Multi-Scale Modelling for Structures and Composites”, Springer, 2005.

The asymptotic expansion of the solution is constructed. For the steady state case it consists of the Poiseuille flows within the tubes and the exponentially decaying boundary layer (in-space) correctors. The gradient drops in each tube are defined by a steady elliptic problem on a graph of the structure. The error estimates for high order asymptotic approximations are proved. Asymptotic analysis is applied for an asymptotically exact condition of junction of 1D and 2D (or 3D) models. These results are generalized (in co-authorship with K.Pileckas) to the case of a non-steady Navier-Stokes equations in tube structures, see G.Panasenko, K.Pileckas, Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure.I. The case without boundary layer-in-time. Nonlinear Analysis, Series A, Theory, Methods and Applications, 122, 2015, 125-168, http://dx.doi.org/10.1016/j.na.2015.03.008; II. General case. Nonlinear Analysis, Series A, Theory, Methods and Applications, 125, 2015, 582-607, http://dx.doi.org/10.1016/j.na.2015.05.018. The structure of the asymptotic expansion is more complex: the Poiseuille type flow now depends on time and the boundary layer-in-space is now completed by two fast boundary layers: in-time only and in-time-and-in space. The fast-in-time pressure drops are now described by a new non-local in time problem on the graph. This new problem for pressure on the graph has multiple applications for periodic in time flows.

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