Geodesic
Some exact solutions of the problem of liquid flow in the contracting or expanding vessel
Matematičeskoe modelirovanie, Tome 31 (2019) no. 3, pp. 124-140.

Voir la notice de l'article provenant de la source Math-Net.Ru

Some exact analytical solutions of the quasi-onedimensional hemodynamic equations in the contracting or expanding vessel are considered in the article. Such equations appear in the modeling of lymph flow in the lymphatic system. Solutions of the linearised problem in the case of periodical changing of the vessel lumen with small amplitude are presented. Also the solution of the not linear problem is obtained in the case when crosssection area depends only on time. Analytical solutions are reproduced numerically.
Keywords: hemodynamic equations, quasi-onedimensional approach, analytical solutions.
Mots-clés : contractions 
@article{MM_2019_31_3_a8,
     author = {A. S. Mozokhina and S. I. Mukhin},
     title = {Some exact solutions of the problem of liquid flow in the contracting or expanding vessel},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {124--140},
     publisher = {mathdoc},
     volume = {31},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2019_31_3_a8/}
}
TY  - JOUR
AU  - A. S. Mozokhina
AU  - S. I. Mukhin
TI  - Some exact solutions of the problem of liquid flow in the contracting or expanding vessel
JO  - Matematičeskoe modelirovanie
PY  - 2019
SP  - 124
EP  - 140
VL  - 31
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2019_31_3_a8/
LA  - ru
ID  - MM_2019_31_3_a8
ER  - 
%0 Journal Article
%A A. S. Mozokhina
%A S. I. Mukhin
%T Some exact solutions of the problem of liquid flow in the contracting or expanding vessel
%J Matematičeskoe modelirovanie
%D 2019
%P 124-140
%V 31
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2019_31_3_a8/
%G ru
%F MM_2019_31_3_a8
A. S. Mozokhina; S. I. Mukhin. Some exact solutions of the problem of liquid flow in the contracting or expanding vessel. Matematičeskoe modelirovanie, Tome 31 (2019) no. 3, pp. 124-140. http://geodesic.mathdoc.fr/item/MM_2019_31_3_a8/

[1] E.I. Borziak, V. Ia. Bocharov, M.R. Sapin, Anatomiia cheloveka, Meditsina, M., 1993, 560 pp.

[2] V.M. Petrenko, Limfaticheskaia sistema. Anatomiia i razvitie, DEAN, Spb., 2010, 112 pp.

[3] N.P. Reddy, T.A. Krouskop, P.H. Jr. Newell, “Biomechanics of a lymphatic vessel”, Blood vessels, 12:5 (1975), 261–278

[4] C.M. Quick, A.M. Venugopal, A.A. Gashev, D.C. Zawieja, R.H. Stewart, “Intrinsic pumpconduit behavior of lymphangions”, Am. J. Physiol. Regul. Integr. Comp. Physiol., 292:4 (2007), R1510-R1518 | DOI | MR

[5] C.D. Bertram, C. Macaskill, J.E. Jr. Moore, “Simulation of a chain of collapsible contracting lymphangions with progressive valve closure”, J. Biomech. Eng., 133:1 (2011), 011008 | DOI

[6] A.J. Macdonald, K.P. Arkill, G.R. Tabor, N.G. McHale, C.P. Winlove, “Modeling flow in collecting lymphatic vessels: one-dimensional flow through a series of contractile elements”, Am. J. Physiol. Heart. Circ. Physiol., 295:1 (2008), 305–313 | DOI

[7] E. Rahbar, J.E. Jr. Moore, “A model of a radially expanding and contracting lymphangion”, J. Biomech., 44:6 (2011), 1001–1007 | DOI

[8] S. Uchida, H. Aoki, “Unsteady flows in a semi-infinite contracting or expanding pipe”, J. Fluid Mech., 82 (1977), 371–387 | DOI | MR | Zbl

[9] M.G. Blyth, P. Hall, D.T. Papageorgiou, “Chaotic flows in pulsating cylindrical tubes: a class of exact Navier-Stokes solutions”, J. Fluid Mech., 481 (2003), 187–213 | DOI | MR | Zbl

[10] F.M. Skalak, C.Y. Wang, “On the unsteady squeezing of a viscous fluid from a tube”, J. Austral. Math. Soc., 21 (1979), 65–74 | DOI | Zbl

[11] O.D. Makinde, “Collapsible tube flow: a mathematical model”, Rom. Journ. Phys., 2005, 493–506

[12] D.V. Kniazev, I.Iu. Kolpakov, “Tochnye resheniia zadachi o techenii viazkoi zhidkosti v tsilindricheskoi oblasti s meniaiushchimsia radiusom”, Nelineinaia dinamika, 11:1 (2015), 89–97 | Zbl

[13] S.A. Regirer, “Kvaziodnomernaia teoriia peristalticheskikh techenii”, Izv. AN SSSR, MZHG, 1984, no. 5, 89–97

[14] O.A. Dudchenko, Peristalticheskii transport v biologicheskikh sistemakh: bazovye modeli i iavnye asimptoticheskie resheniia, avtoreferat dissert. ... kand. fiz.-mat. nauk, MFTI, M., 2012

[15] M.V. Abakumov, N.B. Esikova, S.I. Mukhin, N.V. Sosnin, V.F. Tishkin, A.P. Favorskii, Raznostnaia schema resheniia zadach gemodinamiki na grafe, preprint, No 16, Dialog-MGU, M., 1998

[16] M.V. Abakumov, I.V. Ashmetkov, N.B. Esikova, V.B. Koshelev, S.I. Mukhin, N.V. Sosnin, V.F. Tishkin, A.P. Favorskii, A.B. Khrulenko, “Metodika matematicheskogo modelirovaniia serdechno-sosudistoi sistemy”, Matemat. modelir., 12:2 (2000), 106–117 | Zbl

[17] I.V. Ashmetkov, S.I. Mukhin, N.V. Sosnin, A.P. Favorskii, A.B. Khrulenko, Chislennoe issledovanie svoistv raznostnoi skhemy dlia uravnenii gemodinamiki, preprint, No 14, Dialog MGU, M., 1999

[18] I.V. Ashmetkov, S.I. Mukhin, N.V. Sosnin, A.P. Favorskii, A.B. Khrulenko, “Analysis and comparison of some analytic and numerical solutions of hemodynamic problems”, Differential Equations, 36:7 (2000), 1021–1026 | DOI | MR | Zbl

[19] M.V. Abakumov, K.V. Gavrilyuk, N.B. Esikova, A.V. Lukshin, S.I. Mukhin, N.V. Sosnin, V.F. Tishkin, A.P. Favorskii, “Mathematical model of the hemodynamics of the cardio-vascular system”, Differential Equations, 33:7 (1997), 895–901 | MR

[20] A.A. Samarskii, Iu.P. Popov, Raznostnye metody resheniia zagach gazovoi dinamiki, Nauka, M., 1992, 424 pp.

[21] E.C. Dauenhauer, J. Majdalani, “Unsteady Flows in Semi-Infinite Expanding Channels with Wall Injection”, 30th Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences, 1999

[22] I.V. Ashmetkov, S.I. Mukhin, N.V. Sosnin, A.P. Favorskii, A.B. Khrulenko, Chastnye resheniia uravnenii gemodinamiki, preprint, No 43, Dialog-MGU, M., 1999

[23] A.N. Tikhonov, A.A. Samarskii, Uravneniia matematicheskoi fiziki, Izd-vo MGU, M., 1999, 798 pp.

[24] A.S. Mozokhina, S.I. Mukhin, “Quasi-one-dimensional flow of a fluid with anisotropic viscosity in a pulsating vessel”, Differential Equations, 54:7 (2018), 938–944 | DOI | DOI | MR | Zbl