A one-dimensional model of flow in a junction of thin channels, including arterial trees

V. A. Kozlov   ; S. A. Nazarov

Sbornik. Mathematics, Tome 208 (2017) no. 8, pp. 1138-1186

Résumé

We study a Stokes flow in a junction of thin channels (of diameter $O(h)$) for fixed flows of the fluid at the inlet cross-sections and fixed peripheral pressure at the outlet cross-sections. On the basis of the idea of the pressure drop matrix, apart from Neumann conditions (fixed flow) and Dirichlet conditions (fixed pressure) at the outer vertices, the ordinary one-dimensional Reynolds equations on the edges of the graph are equipped with transmission conditions containing a small parameter $h$ at the inner vertices, which are transformed into the classical Kirchhoff conditions as $h\to+0$. We establish that the pre-limit transmission conditions ensure an exponentially small error $O(e^{-\rho/h})$, $\rho>0$, in the calculation of the three-dimensional solution, but the Kirchhoff conditions only give polynomially small error. For the arterial tree, under the assumption that the walls of the blood vessels are rigid, for every bifurcation node a ($2\times2$)-pressure drop matrix appears, and its influence on the transmission conditions is taken into account by means of small variations of the lengths of the graph and by introducing effective lengths of the one-dimensional description of blood vessels whilst keeping the Kirchhoff conditions and exponentially small approximation errors. We discuss concrete forms of arterial bifurcation and available generalizations of the results, in particular, the Navier-Stokes system of equations. Bibliography: 59 titles.

Détail

Keywords: junction of thin channels, bifurcation of a blood vessel, Reynolds equation, modified Kirchhoff conditions, pressure drop matrix, effective length of a one-dimensional image of a blood vessel.

Bibliographie

[1] G. Pólya, G. Szegö, Isoperimetric inequalities in mathematical physics, Ann. of Math. Stud., 27, Princeton Univ. Press, Princeton, NJ, 1951, xvi+279 pp. | DOI | MR | MR | Zbl | Zbl

[2] Y. C. Fung, Biomechanics. Circulation, 2nd ed., Springer, New York–Berlin, 2011, xviii+572 pp.

[3] Y. C. Fung, Biomechanics. Mechanical properties of living tissues, Springer, New York–Berlin, 1993, 568 pp.

[4] S. Čanić, A. Mikelić, “Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arterias”, SIAM J. Appl. Dyn. Syst., 2:3 (2003), 431–463 | DOI | MR | Zbl

[5] J. Tambača, S. Čanić, A. Mikelić, “Effective model of the fluid flow through elastic tube with variable radius”, XI. Mathematikertreffen Zagreb–Graz, Grazer Math. Ber., 348, Karl-Franzens-Univ. Graz, Graz, 2005, 91–112 | MR | Zbl

[6] S. Čanić, J. Tambača, G. Guidoboni, A. Mikelić, C. J. Hartley, D. Rosenstrauch, “Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow”, SIAM J. Appl. Math., 67:1 (2006), 164–193 | DOI | MR | Zbl

[7] V. A. Kozlov, S. A. Nazarov, “Surface enthalpy and elastic properties of blood vessels”, Dokl. Phys., 56:11 (2011), 560–566 | DOI | MR

[8] V. A. Kozlov, S. A. Nazarov, “Asymptotic models of the blood flow in arteries and veins”, J. Math. Sci. (N. Y.), 194:1 (2013), 44–57 | DOI | MR | Zbl

[9] B. Muha, S. Canić, “Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls”, Arch. Ration. Mech. Anal., 207:3 (2013), 919–968 | DOI | MR | Zbl

[10] F. Berntsson, M. Karlsson, V. Kozlov, S. A. Nazarov, “A one-dimensional model of viscous blood flow in an elastic vessel”, Appl. Math. Comput., 274 (2016), 125–132 | DOI | MR

[11] F. J. H. Gijsen, F. N. van de Vosse, J. D. Janssen, “The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model”, J. Biomech., 32:6 (1999), 601–608 | DOI

[12] V. A. Kozlov, S. A. Nazarov, “One-dimensional model of viscoelastic blood flow through a thin elastic vessel”, J. Math. Sci. (N. Y.), 207:2 (2015), 249–269 | DOI | MR | Zbl

[13] V. A. Kozlov, C. A. Nazarov, “Usloviya sopryazheniya v odnomernoi modeli razvetvlyayuscheisya arterii s uprugimi stenkami”, Matematicheskie voprosy teorii rasprostraneniya voln. 45, Zap. nauch. sem. POMI, 438, POMI, SPb., 2015, 138–177 | MR

[14] A. Bressan, S. Čanić, M. Garavello, M. Herty, B. Piccoli, “Flows on networks: recent results and perspectives”, EMS Surv. Math. Sci., 1:1 (2014), 47–111 | DOI | MR | Zbl

[15] S. A. Nazarov, K. Pileckas, “Asymptotic conditions at infinity for the Stokes and Navier–Stokes problems in domains with cylindrical outlets to infinity”, Advances in fluid dynamics, Quad. Mat., 4, Dept. Math., Seconda Univ. Napoli, Caserta, 1999, 141–243 | MR | Zbl

[16] S. A. Nazarov, B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Exp. Math., 13, Walter de Gruyter Co., Berlin–New York, 1994, viii+525 pp. | DOI | MR | Zbl

[17] M. Specovius-Neugebauer, “Approximation of the Stokes Dirichlet problem in domains with cylindrical outlets”, SIAM J. Math. Anal., 30:3 (1999), 645–677 | DOI | MR | Zbl

[18] V. A. Kondrat'ev, “Boundary problems for elliptic equations in domains with conical or angular points”, Trans. Moscow Math. Soc., 16 (1967), 227–313 | MR | Zbl

[19] C. J. Amick, “Steady solutions of the Navier–Stokes equations in unbounded channels and pipes”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4:3 (1977), 473–513 | MR | Zbl

[20] S. A. Nazarov, B. A. Plamenevskii, “Radiation conditions for selfadjoint elliptic problems”, Soviet Math. Dokl., 41:2 (1991), 274–277 | MR | Zbl

[21] S. A. Nazarov, B. A. Plamenevskii, “A generalized Green's formula for elliptic problems in domains with edges”, J. Math. Sci. (N. Y.), 73:6 (1995), 674–700 | DOI | MR | Zbl

[22] S. A. Nazarov, K. I. Pileckas, “Reynolds flow of a fluid in a thin three-dimensional channel”, Lithuanian Math. J., 30:4 (1990), 366–375 | DOI | MR | Zbl

[23] G. Panasenko, K. Pileckas, “Flows in a tube structure: equation on the graph”, J. Math. Phys., 55:8 (2014), 081505, 11 pp. | DOI | MR

[24] M. van Dyke, Perturbation methods in fluid mechanics, Appl. Math. Mech., 8, Academic Press, New York–London, 1964, x+229 pp. | MR | Zbl | Zbl

[25] A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems, Transl. Math. Monogr., 102, Amer. Math. Soc., Providence, RI, 1992, x+281 pp. | MR | MR | Zbl | Zbl

[26] V. Maz'ya, S. Nazarov, B. Plamenevskii, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, v. 1, Oper. Theory Adv. Appl., 111, Birkhäuser Verlag, Basel, 2000, xxiv+435 pp. | MR | MR | Zbl

[27] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, v. 1, Travaux et Recherches Mathématiques, 17, Dunod, Paris, 1968, xx+372 pp. | MR | MR | Zbl | Zbl

[28] I. S. Zorin, S. A. Nazarov, “Edge effect in the bending of a thin three-dimensional plate”, J. Appl. Math. Mech., 53:4 (1989), 500–507 | DOI | MR | Zbl

[29] S. A. Nazarov, “On three-dimensional effects near the vertex of a crack in a thin plate”, J. Appl. Math. Mech., 55:3 (1991), 407–415 | DOI | MR | Zbl

[30] S. A. Nazarov, “Boundary layers and boundary hinge-support conditions for thin plates”, J. Math. Sci. (N. Y.), 108:5 (2002), 806–850 | DOI | MR | Zbl

[31] C. A. Nazarov, Yu. A. Romashev, “Izmenenie koeffitsienta intensivnosti pri razrushenii peremychki mezhdu dvumya kollinearnymi treschinami”, Izv. AN ArmSSR. Mekhanika, 35:4 (1982), 30–40 | Zbl

[32] S. A. Nazarov, “Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions”, Proceedings of the St. Petersburg Mathematical Society, v. V, Amer. Math. Soc. Transl. Ser. 2, 193, Amer. Math. Soc., Providence, RI, 1999, 77–125 | DOI | MR | Zbl

[33] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Math. Appl., 2, 2nd eng. ed., rev. and enlarged, Gordon and Breach Science Publishers, New York–London–Paris, 1969, xviii+224 pp. | MR | MR | Zbl | Zbl

[34] G. Panasenko, K. Pileckas, “Divergence equation in thin-tube structures”, Appl. Anal., 94:7 (2015), 1450–1459 | DOI | MR | Zbl

[35] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II”, Comm. Pure Appl. Math., 17:1 (1964), 35–92 | DOI | MR | Zbl

[36] V. G. Mazya, B. A. Plamenevskii, “Otsenki v $L_p$ i v klassakh Geldera i printsip maksimuma Miranda–Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Math. Nachr., 81:1 (1978), 25–82 | DOI | MR | Zbl

[37] V. A. Kozlov, S. A. Nazarov, “Asymptotic models of anisotropic heterogeneous elastic walls of blood vessels”, J. Math. Sci. (N. Y.), 213:4 (2016), 561–581 | DOI | MR | Zbl

[38] C. D. Murray, “The physiological principle of minimum work. I. The vascular system and the cost of blood volume”, Proc. Natl. Acad. Sci. U.S.A., 12 (1926), 207–214 | DOI

[39] G. S. Kassab, C. A. Rider, N. J. Tang, Y. C. Fung, “Morphometry of pig coronary aterial trees”, Amer. J. Physiol., 265 (1993), H350–H365

[40] G. S. Kassab, Yuan-Cheng B. Fung, “The pattern of coronary arteriolar bifurcation and the uniform shear hypothesis”, Ann. Biomed. Eng., 23:1 (1995), 13–20 | DOI

[41] V. A. Kozlov, S. A. Nazarov, “A simple one-dimensional model of a false aneurysm in the femoral artery”, J. Math. Sci. (N. Y.), 214:3 (2016), 287–301 | DOI | MR | Zbl

[42] J. Alastruey, K. H. Parker, S. J. Sherwin, “Arterial pulse wave haemodynamics”, 11th international conference on pressure surges, Virtual PiE Led t/a BHR Group, Lisbon, 2012, 401–442

[43] J. P. Mynard, J. J. Smolich, “One-dimensional haemodynamic modeling and wave dynamics in the entire adult circulation”, Ann. Biomed. Eng., 43:6 (2015), 1443–1460 | DOI

[44] G. Panasenko, R. Stavre, “Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel”, Netw. Heterog. Media, 5:4 (2010), 783–812 | DOI | MR | Zbl

[45] L. Formaggia, A. Quarteroni, A. Veneziani, “Multiscale models of the vascular system”, Cardiovascular mathematics, MS Model. Simul. Appl., 1, eds. L. Formaggia, A. Quarteroni, A. Veneziani, Springer Italia, Milan, 2009, 395–446 | DOI | MR | Zbl

[46] J. Peiró, A. Veneziani, “Reduced models of the cardiovascular system”, Cardiovascular mathematics, MS Model. Simul. Appl., 1, eds. L. Formaggia, A. Quarteroni, A. Veneziani, Springer Italia, Milan, 2009, 347–394 | DOI | MR | Zbl

[47] G. P. Panasenko, “Asymptotic expansion of the solution of Navier–Stokes equation in tube structure and partial asymptotic decomposition of the domain”, Appl. Anal., 76:3-4 (2000), 363–381 | DOI | MR | Zbl

[48] F. Blanc, O. Gipouloux, G. Panasenko, A. M. Zine, “Asymptotic analysis and partial asymptotic decomposition of domain for Stokes equation in tube structure”, Math. Models Methods Appl. Sci., 9:9 (1999), 1351–1378 | DOI | MR | Zbl

[49] G. Panasenko, Multi-scale modelling for structures and composites, Springer, Dordrecht, 2005, xiv+398 pp. | MR | Zbl

[50] G. Panasenko, K. Pileckas, “Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe”, Appl. Anal., 91:3 (2012), 559–574 | DOI | MR | Zbl

[51] G. Panasenko, K. Pileckas, “Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structures I. The case without boundary-layer-in-time”, Nonlinear Anal., 122 (2015), 125–168 | DOI | MR | Zbl

[52] G. Panasenko, K. Pileckas, “Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structures II. General case”, Nonlinear Anal., 125 (2015), 582–607 | DOI | MR | Zbl

[53] V. I. Lebedev, V. I. Agoshkov, Operatory Puankare–Steklova i ikh prilozheniya v analize, Otd. vychisl. matem. AN SSSR, M., 1983, 184 pp. | MR | Zbl

[54] S. V. Tsynkov, “Numerical solution of problems on unbounded domains. A review”, Appl. Numer. Math., 27:4 (1998), 465–532 | DOI | MR | Zbl

[55] O. Reynolds, “On the dynamical theory of incompressible viscous fluids and the determination of the criterion”, Philos. Trans. Roy. Soc. London Ser. A, 186 (1895), 123–164 | DOI | MR | Zbl

[56] S. A. Nazarov, “Asymptotic solution of the Navier–Stokes problem on the flow of a thin layer of fluid”, Sib. Math. J., 31:2 (1990), 296–307 | DOI | MR | Zbl

[57] S. A. Nazarov, J. H. Videman, “A modified nonlinear Reynolds equation for thin viscous flows in lubrication”, Asymptot. Anal., 52:1-2 (2007), 1–36 | MR | Zbl

[58] S. Nazarov, J. H. Videman, “Reynolds type equation for a thin flow under intensive transverse percolation”, Math. Nachr., 269/270:1 (2004), 189–209 | DOI | MR | Zbl

[59] V. Kozlov, S. Nazarov, G. Zavarokhin, “A fractal graph model of cappilary type systems”, Complex Var. Elliptic Equ. (to appear) , published online 2017 | DOI