Numerical study of the equation on the graph for the steady state non-Newtonian flow in thin tube structure
Abstract
The dimension reduction for the viscous flows in thin tube structures leads to equations on the graph for the macroscopic pressure with Kirchhoff type junction conditions in the vertices. Non-Newtonian rheology of the flow generates nonlinear equations on the graph. A new numerical method for second order nonlinear differential equations on the graph is introduced and numerically tested.
Keyword : non-Newtonian flow, strain rate dependent viscosity, asymptotic dimension reduction, quasi-Poiseuille flows, equation on the graph
How to Cite
Kozulinas, N., Panasenko, G., Pileckas, K., & Šumskas, V. (2023). Numerical study of the equation on the graph for the steady state non-Newtonian flow in thin tube structure. Mathematical Modelling and Analysis, 28(4), 581–595. https://doi.org/10.3846/mma.2023.18311
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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G. Panasenko, K. Pileckas and B. Vernescu. Steady state non-Newtonian flow with strain rate dependent viscosity in domains with cylindrical outlets to infinity. Nonlinear Analysis: Modeling and Controle, 26(6):1166–1199, 2021. https://doi.org/10.15388/namc.2021.26.24600
G. Panasenko, K. Pileckas and B. Vernescu. Steady state non-Newtonian flow in thin tube structure: equation on the graph. St. Petersburg Mathematical Journal, 33:327–340, 2022. https://doi.org/10.1090/spmj/1702
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, 17(18):1–36, 2022. https://doi.org/10.1051/mmnp/2022005
C. Poelma. Exploring the potential of blood flow network data. Meccanica, 52:489–502, 2017. https://doi.org/10.1007/s11012-015-0255-4
A.R. Pries, T.W. Secomb, T. Gessner, M.B. Sperandio, J.F. Gross and P. Gaehtgens. Resistance to blood flow in microvessels in vivo. Circ. Res., 75:904–915, 1994. https://doi.org/10.1161/01.RES.75.5.904
A. Quarteroni and F. Saleri. Calcul Scientifique. Springer-Verlag Italia, Milano, 2006.
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(09):1351–1378, 1999. https://doi.org/10.1142/S0218202599000609
R. Bunoiu and A. Gaudiello. On the Bingham flow in a thin Y-like shaped structure. J. Math. Fluid Mech., 24:20, 2022. https://doi.org/10.1007/s00021-021-00657-0
R. Bunoiu, A. Gaudiello and A. Leopardi. Asymptotic analysis of a Bingham fluid in a thin T-like shaped structure. J. Math. Pures Appl., 123:148–166, 2019. https://doi.org/10.1016/j.matpur.2018.01.001
C. D’Angelo and A. Quarteroni. On the coupling of 1D and 3D diffusion-reaction equations: application to tissue perfusion problems. Arch. Rat. Mech. Anal., 18(08):1481–1504, 2008. https://doi.org/10.1142/S0218202508003108
T. Köppl, E. Vidotto and B. Wohlmuth. A 3D-1D coupled blood flow and oxygen transport model to generate microvascular networks. Int. J. for Num. Meth. in Biomed. Eng., 36(10):e3386, 2020. https://doi.org/10.1002/cnm.3386
V. Kozlov, S. Nazarov and G. Zavorokhin. A fractal graph model of capillary type systems. Complex Variables and Elliptic Equations, 63(7-8):1044–1068, 2018. https://doi.org/10.1080/17476933.2017.1349117
B. Liu and D. Tang. Influence of non-Newtonian properties of blood on the wall shear stress in human atherosclerotic right coronary arteries. Mol. Cell Biotech, 8(1):73–90, 2011.
E. Marušić-Paloka and I. Pažanin. A note on Kirchhoff junction rule for power-law fluids. Zeitschrift fu¨r Naturforschung A, 70(9):695–702, 2015. https://doi.org/10.1515/zna-2015-0148
C. Murray. The physiological principle of minimum work applied to the angle of branching of arteries. J. Gen. Physiol., 9(6):835–841, 1926. https://doi.org/10.1085/jgp.9.6.835
C. Murray. A relationship between circumference and weight in trees and its bearing on branching angles. The Journal of general physiology, 10(5):725, 1927. https://doi.org/10.1085/jgp.10.5.725
D. Notaro, L. Cattaneo, L. Formaggia, A. Scotti and P. Zunino. A mixed finite element method for modeling the fluid exchange between microcirculation and tissue interstitium, pp. 3–25. Springer, 2016. https://doi.org/10.1007/978-3-319-41246-7_1
G. Panasenko. Multi-scale Modeling for Structures and Composites. Springer, Dordrecht, 2005.
G. Panasenko. Initiation à l’Analyse Numérique. Editions Universitaires Européennes, Chisinau, Springer, Dordrecht, 2022.
G. Panasenko and K. Pileckas. Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe. C.R. Acad. Sci. Paris, 326(12):867–872, 1998.
G. Panasenko and K. Pileckas. Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe. Applicable Analysis, 91(3):559–574, 2012. https://doi.org/10.1080/00036811.2010.549483
G. Panasenko and K. Pileckas. Flows in a tube structure: equation on the graph. Journal of Mathematical Physics, 55(8):081505, 2014. https://doi.org/10.1063/1.4891249
G. Panasenko and K. Pileckas. Asymptotic analysis of the non-steady NavierStokes equations in a tube structure. I. The case without boundary layer-in-time. Nonlinear Analysis, Series A, Theory, Methods and Applications, 122:125–168, 2015. https://doi.org/10.1016/j.na.2015.03.008
G. Panasenko and K. Pileckas. Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. II. General case. Nonlinear Analysis, Series A, Theory, Methods and Applications, 125:582–607, 2015. https://doi.org/10.1016/j.na.2015.05.018
G. Panasenko and K. Pileckas. Periodic in time flow in thin structures: Equations on the graph. Journal of Mathematical Analysis and Applications, 490(2):1–8, 2020. https://doi.org/10.1016/j.jmaa.2020.124335
G. Panasenko, K. Pileckas and B. Vernescu. Steady state non-Newtonian flow with strain rate dependent viscosity in domains with cylindrical outlets to infinity. Nonlinear Analysis: Modeling and Controle, 26(6):1166–1199, 2021. https://doi.org/10.15388/namc.2021.26.24600
G. Panasenko, K. Pileckas and B. Vernescu. Steady state non-Newtonian flow in thin tube structure: equation on the graph. St. Petersburg Mathematical Journal, 33:327–340, 2022. https://doi.org/10.1090/spmj/1702
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, 17(18):1–36, 2022. https://doi.org/10.1051/mmnp/2022005
C. Poelma. Exploring the potential of blood flow network data. Meccanica, 52:489–502, 2017. https://doi.org/10.1007/s11012-015-0255-4
A.R. Pries, T.W. Secomb, T. Gessner, M.B. Sperandio, J.F. Gross and P. Gaehtgens. Resistance to blood flow in microvessels in vivo. Circ. Res., 75:904–915, 1994. https://doi.org/10.1161/01.RES.75.5.904
A. Quarteroni and F. Saleri. Calcul Scientifique. Springer-Verlag Italia, Milano, 2006.