Grigory Panasenko (Université de Saint-Etienne, France).
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 the non-steady Navier-Stokes equations 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 in [1-3]. 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: [4-8]. 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 (see [6]).
The present work is supported by the grant number 14-11-00306 of Russian Scientific Foundation, by the Research Federative Structures MODMAD FED 4169 and FR CNRS 3490.
1. Panasenko G.P. 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
2. Panasenko G.P. Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure , C.R.Acad.Sci.Paris, t. 326, Série IIb, 1998, pp. 893-898
3. Panasenko G.P. Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht, 2005, 398 pp.
4. Panasenko G.,Pileckas K., Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe, Applicable Analysis, 2012, 91, 3, 559-574
5. Panasenko G., Pileckas K., Divergence equation in thin-tube structure, Applicable Analysis, 94,7, pp. 1450-1459, 2015, doi 10.1080/00036811.2014.933476.
6. Panasenko G., Pileckas K., Flows in a tube structure: equation on the graph, Journal of Mathematical Physics, 55, 081505 (2014); doi: 10.1063/1.4891249.
7. Panasenko G., Pileckas K., 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
8. Panasenko G., Pileckas K., Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. 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