Geodesic
A mathematical model of an arterial bifurcation
Ural mathematical journal, Tome 5 (2019) no. 1, pp. 109-126 Cet article a éte moissonné depuis la source Math-Net.Ru

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An asymptotic model of an arterial bifurcation is presented. We propose a simple approximate method of calculation of the pressure drop matrix. The entries of this matrix are included in the modified transmission conditions, which were introduced earlier by Kozlov and Nazarov, and which give better approximation of 3D flow by 1D flow near a bifurcation of an artery as compared to the classical Kirchhoff conditions. The present modeling takes into account the heuristic Murrey cubic law.
Keywords: Stokes’ flow, bifurcation of a blood vessel, modified Kirchhoff conditions, pressure drop matrix, Murrey’s law. 
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     author = {German L. Zavorokhin},
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German L. Zavorokhin. A mathematical model of an arterial bifurcation. Ural mathematical journal, Tome 5 (2019) no. 1, pp. 109-126. http://geodesic.mathdoc.fr/item/UMJ_2019_5_1_a10/

[1] Amick C. J., “Steady solutions of the Navier–Stokes equations in unbounded channels and pipes”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 4:3 (1977), 473–513 http://www.numdam.org/item/?id=ASNSP_1977_4_4_3_473_0 | MR | Zbl

[2] Amick C. J., “Properties of steady Navier–Stokes solutions for certain unbounded channels and pipes”, Nonlinear Anal. Theory, Meth., Appl., 2:6 (1978), 689–720 | DOI | MR | Zbl

[3] Berntsson F., Karlsson M., Kozlov V. A., Nazarov S. A., “A Modification to the Kirchhoff Conditions at a Bifurcation and Loss Coefficients”, LiU electronic press, 2018, 1–10 https://www.diva-portal.org/smash/get/diva2:1204214/FULLTEXT01.pdf

[4] Bogovskii M. E., “On solution of certain problems of vector analysis associated with operators $\rm div$ and $\rm grad$”, Tr. Semin. im. S.L. Soboleva, 1 (1980), 5–40 | MR

[5] Fung Y. C., Biomechanics. Circulation, 2-nd ed., Springer-Verlag, New York, 2011, 572 pp. | DOI

[6] Il'in A. M., Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Am. Math. Soc., 1992, 281 pp. | MR | Zbl

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

[8] Kassab G. S., Rider C. A., Tang N. J., Fung Y. C., “Morphometry of pig coronary arterial trees”, Am. J. Physiol. Heart Circ. Physiol., 265:1 (1993), H350–H365 | DOI

[9] Kozlov V. A., Nazarov S. A., “Transmission conditions in a one-dimensional model of bifurcating arteries with elastic walls”, J. Math. Sci. (N.Y.), 224:1 (2017), 94–118 | DOI | MR | Zbl

[10] Kozlov V. A., Nazarov S. A., “A one-dimensional model of flow in a junction of thin channels, including arterial trees”, Sb. Math., 208:8 (2017), 1138–1186 | DOI | MR | Zbl

[11] Kozlov V. A., Nazarov S. A., Zavorokhin G. L., “A fractal graph model of capillary type systems”, Complex Var. Elliptic Equ., 63:7–8 (2018), 1044–1068 | DOI | MR | Zbl

[12] Kozlov V. A., Nazarov S. A., Zavorokhin G. L., “Pressure drop matrix of a bifurcation of an artery with defects”, Eurasian J. Math. Comput. Appl., 7:3 (2019) (Accepted) | Zbl

[13] Kufner A., Weighted Sobolev Spaces, Teubner, 1980, 152 pp. | MR | Zbl

[14] Ladyzhenskaya O. A., Solonnikov V. A., “Determination of the solutions of boundary value problems for stationary Stokes and Navier-Stokes equations having an unbounded Dirichlet integral”, J. Sov. Math., 21:5 (1983), 728–761 | DOI | Zbl

[15] Maz'ya V. G., Nazarov S. A., Plamenevskij B. A., Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. II, Ser. Oper. Theory Adv. Appl., Vol. 112, Birkhäuser, Verlag, Basel, 2000 | DOI | MR | Zbl

[16] Murray C. D., “The physiological principle of minimum work. I. The vascular system and the cost of blood volume”, Proc. Natl. Acad. Sci. USA, 12:3 (1926), 207–214 | DOI

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

[18] Nazarov S. A., Pileckas K., “Asymptotic conditions at infinity for the Stokes and Navier-Stokes problems in domains with cylindrical outlets to infinity”, Advances in Fluid Dynamics, Quaderni di Matematica, 4 (1999), 141–243 | MR | Zbl

[19] Pólya G., Szegö G., Isoperimetric Inequalities in Mathematical Physics, Ser. Ann. of Math. Stud., vol. 27, Princeton University Press, Princeton, USA, 1951, 279 pp. https://www.jstor.org/stable/j.ctt1b9rzzn | MR

[20] Simakov S. S., “Modern methods of mathematical modeling of blood flow using reduced order methods”, Comput. Res. Model., 10:5 (2018), 581—604 (in Russian) | DOI | MR

[21] Solonnikov V. A., “On the solvability of boundary and initial-boundary value problems for the Navier–Stokes system in domains with noncompact boundaries”, Pacific J. Math., 93:2 (1981), 443–458 https://projecteuclid.org/euclid.pjm/1102736272 | MR | Zbl

[22] Van Dyke M., Perturbation Methods in Fluid Mechanics, Academic Press, New York, London, 1964, 229 pp. | MR | Zbl