G.P. Panasenko | Non-Newtonian flows in thin tube structures
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2022-08-01

O.A. Ladyzhenskaya centennial conference on PDE’s, St. Petersburg, July 16 – July 22, 2022
Grigory P. Panasenko (University Jean Monnet) 
Non-Newtonian flows in thin tube structures

Thin tube structures are finite unions of thin cylinders depending on the
small parameter, ratio of the diameter of the cross section to the length of the cylinder. Flows in such domains model blood flow in a network of vessels. The asymptotic expansion of the solution of the steady Stokes and Navier-Stokes equations in these domains with no slip boundary condition was constructed in the papers [1], [2], and the book [3]. However, the blood exhibits a nonNewtonian rheology, when the viscosity depends on the strain rate. In the present talk we consider such rheology. Applying the Banach fixed point theorem we prove the existence and uniqueness of a solution and its regularity. An asymptotic approximation is constructed and justified by an error estimate. The results are published in the set of three papers [4], [5], [6]. It is a joint work with K.Pileckas and B.Vernescu.
References
[1] G. Panasenko, Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure, C.R. Acad. Sci. Paris, S´erie IIb 326 (1998), 867–872.
[2] F. Blanc, O. Gipouloux, G. Panasenko and A.M. Zine, Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes Equation in tube structure, Mathematical Models and Methods in Applied Sciences 9 (1999), no.9, 1351–1378.
[3] G. Panasenko, Multi-Scale Modelling for Structures and Composites,
Springer, Dordrecht, 2005.
[4] G. Panasenko, K. Pileckas and B. Vernescu, Steady state non-Newtonian flow in thin tube structure: equation on the graph, Algebra i Analiz 33 (2021), no.2, 197–214.
[5] G. Panasenko, K. Pileckas and B. Vernescu, Steady state non-Newtonian
flow with strain dependent viscosity in domains with cylindrical outlets at
infinity, Nonlinear Analysis: Modelling and control 26 (2021), no.6, 1166–
1199.
[6] G. Panasenko, K. Pileckas and B. Vernescu, Steady state non-Newtonian
flow with strain rate dependent viscosity in thin tube structure with no
slip boundary condition, Mathematical Modeling of Natural Phenomena,
to appear, 2022.