Pressure boundary conditions for viscous flows in thin tube structures: Stokes equations with locally distributed Brinkman term

Grigory Panasenko¹ ; Konstantinas Pileckas²

¹ Institute Camille Jordan UMR CNRS 5208, University Jean Monnet, 23 rue P. Michelon, 42023, Saint-Etienne, France and Institute of Applied Mathematics, Vilnius University, Naugarduko Str. 24, Vilnius 03225, Lithuania.
² Institute of Applied Mathematics, Vilnius University, Naugarduko Str. 24, Vilnius 03225, Lithuania.

Mathematical modelling of natural phenomena, Tome 18 (2023), article no. 17.

Résumé

The steady state Stokes-Brinkman equations in a thin tube structure is considered. The Brinkman term differs from zero only in small balls near the ends of the tubes. The boundary conditions are: given pressure at the inflow and outflow of the tube structure and the no slip boundary condition on the lateral boundary. The complete asymptotic expansion of the problem is constructed. The error estimates are proved. The method of partial asymptotic dimension reduction is introduced for the Stokes-Brinkman equations and justified by an error estimate. This method approximates the main problem by a hybrid dimension problem for the Stokes-Brinkman equations in a reduced domain.

Détail

DOI : 10.1051/mmnp/2023016

Bibliographie

[1] S. Agmon, A. Douglis, L. Nirenberg Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions Commun. Pure Appl. Math. 1959 623 727

[2] G. Allaire Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, volume distribution of holes Arch. Ration. Mech. Anal. 1991 209 259

[3] H. Amar, D. Givoli Mixed-dimensional modeling of time-dependent wave problems using the Panasenko construction J. Theor. Comput. Acoust. 2018 1850034

[4] V.L. Berdichevsky On averaging of periodic structures Pr. Mat. Mech. 1977 993 1006

[5] R. Bunoiu, A. Gaudiello, A. Leopardi Asymptotic analysis of a Bingham fluid in a thin T-like shaped structure J. Math Pures Appl. 2019 148 166

[6] R. Bunoiu, A. Gaudiello On the Bingham flow in a thin Y-like shaped structure J. Math. Fluid Mech. 2022

[7] G. Cardone, G. Panasenko, I. Sirakov Asymptotic analysis and numerical modelling of mass transport in tubular structure Math. Models Methods Appl. Sci. 2010 397 421

[8] C. Conca, F. Murat, O. Pironneau The Stokes and Navier-Stokes equations with boundary conditions involving pressure Jap. J. Math. 1994

[9] A.-C. Egloffe, Study of some inverse problems for the Stokes system. Application to the lungs, These de doctorat, l’Universite Pierre et Marie Curie-Paris VI, 2012.

[10] H.I. Ene, E. Sanchez-Palencia Equations de phenomenes de surface pour l’ecoulement dans un modele de milieu poreux J. Mécan. 1975 73 108

[11] L. Formaggia, D. Lamponi, A. Quarteroni One dimensional models for blood flow in arteries J. Eng. Math. 2003 251 276

[12] P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems. Springer (2011).

[13] P. Galdi, R. Rannacher, A.M. Robertson and S. Turek, Hemodynamical Flows, Modeling, Analysis and Simulation. Oberwolfach Seminars, Vol. 37. Birkhäuser, Basel, Boston, Berlin (2008).

[14] F. Golfier, D. Lasseur and M. Quintard, Etude du probléme de Darcy-Brinkman dans une structure fluide/poreux, 14 Congres Francais de Mécanique, 1999.

[15] F. Harary, Graph Theory. Addison-Wesley, Reading, Massachusetts (1971).

[16] J.G. Heywood, R. Rannacher, S. Turek Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations Int. J. Numer. Methods Fluids. 1996 325 352

[17] M.V. Korobkov, K. Pileckas, R. Russo Solvability in a finite pipe of steady-state Navier-Stokes equations with boundary conditions involving Bernoulli pressure Calc. Var. 2020

[18] V. Kozlov, V. Maz’ya and A. Movchan, Asymptotic Analysis of Fields in Multi-structures. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1999).

[19] V.A. Kozlov, S.A. Nazarov One-dimensional model for flows in the junction of thin channels, in particular, arterial trees Math. Sb 2017 56 105

[20] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluid. Gordon and Breach (1969).

[21] O.A. Ladyzhenskaya, V.A. Solonnikov On some problems of vector analysis and generalized formulations of boundary value problems for the Navier-Stokes equations Zapiski Nauchn. Sem. LOMI 1976 81 116

[22] Th. Levy, E. Sanchez-Palencia On boundary conditions of flow in porous media Int. J. Eng. Sci. 1975 923 940

[23] Th. Levy, E. Sanchez-Palencia Equations and interface conditions for acoustic phenomena in porous media J. Math. Anal. Appl. 1977 813 834

[24] E. Marušić-Paloka Steady flow of a non-Newtonian fluid in unbounded channels and pipes Math. Models Methods Appl. Sci. 2000 1425 1445

[25] E. Marušić-Paloka, I. Pažanin A note on Kirchhoff junction rule for power-law fluids Z. Naturf. A 2015 695 702

[26] V.G. Maz’Ya, A. Slutskii Asymptotic analysis of the Navier-Stokes system in a plane domain with thin channels Asymptotic Anal. 2000 59 89

[27] G. Panasenko Method of asymptotic partial decomposition of Domain Math. Models Methods Appl. Sci. 1998 139 156

[28] G. Panasenko Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure C.R. Acad. Sci. Paris Serie Ilb 1998 867 872

[29] G. Panasenko Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure C.R. Acad. Sci. Paris Seerie IIb 1998 893 898

[30] G. Panasenko, Multi-scale Modeling for Structures and Composites. Springer, Dordrecht (2005).

[31] G. Panasenko, K. Pileckas Divergence equation in thin-tube Structure Appl. Anal. 2015 1450 1459

[32] G. Panasenko, K. Pileckas Flows in a tube structure: equation on the graph J. Math. Phys. 2014 081505

[33] 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 Anal. Ser. A Theory Methods Applic. 2015 125 168

[34] G. Panasenko, K. Pileckas, B. Vernescu Steady state non-Newtonian flow with strain rate dependent viscosity in thin tube structure with no-slop boundary condition Math. Model. Natural Phenomena 2022

[35] O. Pironneau Conditions aux limites sur la pression pour les equations de Stokes et de Navier-Stokes C.R. Acad. Sci. Paris Ser. I Math. 1986 403 406

[36] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in Modelling of Living Systems, Handbook of Numerical Analysis Series, edited by N. Ayache (2002).

[37] A. Quarteroni, A. Veneziani Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations Multiscale Model. Simul. 2003 173 195

[38] E. Sanchez-Palencia, Equations aux derivees partielles. Solutions periodiques par rapport aux variables d’espace et applications. Compt. Rend. Acad. Sci. Paris, seer. A (1970) 1129–1132.

[39] E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory. Springer-Verlag, New York (1980).

[40] L. Tartar, Homogenization. Cours Peccot. College de France, Paris (1977).

[41] L. Tartar, Topics in Non Linear Analysis. Report of the University of Orsay, France (1978).

[42] R. Temam, Navier-Stokes Equations. North-Holland Pub. Company (1977).

[43] D.B. Volkov On averaging of some boundary value problems in domains with periodic structure USSR Comp. Math. and Math. Phys. 1982 117 128