Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment

Kubugha Wilcox Bunonyo; Emeka Amos; Jason Biobaragha Goldie

doi:doi:10.11648/j.ijamtp.20200602.12

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Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment

Kubugha Wilcox Bunonyo, Emeka Amos, Jason Biobaragha Goldie

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 6, Issue 2)

Received: 31 May 2020 Accepted: 11 June 2020 Published: 20 June 2020

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Abstract

The aim of the present article is to model atherosclerotic blood flow through double stenosis in the arterial lumen with application of treatment subject to some boundary conditions in the channel. The research focused also on mathematical formulation of momentum equation governing the blood flow, driven by the oscillatory pressure gradient due to the contraction of muscles of the heart tissues and externally applied magnetic field. The nonlinear partial differential equation and geometries of the double stenosis are scaled using some quantities which made them dimensionless with some pertinent physical parameters such as the womersley number, the treatment parameter, Hartmann number, Darcy number at the region. The oscillatory perturbation method was adopted to further reduce the dimensionless equation to a Bessel differential equation of order zero. An exact solution of equation governing the flow is obtained with quantities coefficients. Mathematica codes were developed to investigate the influence of the pertinent parameters on the velocity profile, shear stress and volumetric flow rate of the first segment of stenosis δ₁, while we considered the second stenosis δ₂ segment fixed and present the results graphically. In conclusion, it is observed from the results and analysis that the pertinent parameters influenced the blood flow through a double stenosed region of an arterial lumen withholding the second stenosis fixed. These results could help in raising the awareness and the need for treatment of patients with double stenosis focusing on the most delicate.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 6, Issue 2)
DOI	10.11648/j.ijamtp.20200602.12
Page(s)	19-25
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment

References

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[3]	Adams, R. J., Albers, G., Alberts, M. J., Benavente, O., Furie, K., Goldstein, L. B.,... & Katzan, I. (2008). Update to the AHA/ASA recommendations for the prevention of stroke in patients with stroke and transient ischemic attack. Stroke, 39 (5), 1647-1652.
[4]	Brott, T. G., Hobson, R. W., Howard, G., Roubin, G. S., Clark, W. M., Brooks, W.,... & Howard, V. J. (2010). Stenting versus endarterectomy for treatment of carotid-artery stenosis. New England Journal of Medicine, 363 (1), 11-23.
[5]	Birchall, D., Zaman, A., Hacker, J., Davies, G., & Mendelow, D. (2006). Analysis of haemodynamic disturbance in the atherosclerotic carotid artery using computational fluid dynamics. European radiology, 16 (5), 1074-1083.
[6]	Sharafeev, A., Kutuzova, E., Tazyukov, F., Layek, G., & Garifullin, F. Blood flow peculiarities in vessels bifurcation.
[7]	Mandal, S. M., Mukhopadhyayy, S., & Layek, G. C. Z. (2012). Pulsatile flow of shear-dependent fluid in a stenosed artery. Theoretical and Applied Mechanics, 39 (3), 209-231.
[8]	Huh, H. K., Ha, H., & Lee, S. J. (2015). Effect of non-Newtonian viscosity on the fluid-dynamic characteristics in stenotic vessels. Experiments in Fluids, 56 (8), 167.
[9]	Cho, Y. I., & Kensey, K. R. (1991). Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: Steady flows. Biorheology, 28 (3-4), 241-262.
[10]	Moravec, S., & Liepsch, D. (1983). Flow investigations in a model of a three-dimensional human artery with Newtonian and non-Newtonian fluids. Part I. Biorheology, 20 (6), 745-759.
[11]	Perktold, K., Peter, R., & Resch, M. (1989). Pulsatile non-Newtonian blood flow simulation through a bifurcation with an aneurysm. Biorheology, 26 (6), 1011-1030.
[12]	Perktold, K. (1990). Non-Newtonian blood flow simulation and wall shear stress in an arterial bifurcation. In Biofluid Mechanics (pp. 471-477). Springer, Berlin, Heidelberg.
[13]	Steffan, H., Brandstätter, W., Bachler, G., & Pucher, R. (1990). Comparison of Newtonian and non-Newtonian blood flow in stenotic vessels using numerical simulation. In Biofluid Mechanics (pp. 479-485). Springer, Berlin, Heidelberg.
[14]	Tazyukova, A. F., Tazyukov, F. K., Salman, H. D., Kutuzova, E. R., Kutuzov, A. G., & Yushko, S. V. Blood flow through the circulatory system element affected by double stenotic lesions.
[15]	Sharma, M. K., Sharma, P. R., & Nasha, V. (2013). Pulsatile MHD Arterial Blood Flow in the Presence of Double Stenoses.
[16]	Bunonyo, K. W., Israel-Cookey, C., & Amos, E. (2018). Modeling of Blood Flow through Stenosed Artery with Heat in the Presence of Magnetic Field. Asian Research Journal of Mathematics, 1-14.

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APA Style

Kubugha Wilcox Bunonyo, Emeka Amos, Jason Biobaragha Goldie. (2020). Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment. International Journal of Applied Mathematics and Theoretical Physics, 6(2), 19-25. https://doi.org/10.11648/j.ijamtp.20200602.12

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ACS Style

Kubugha Wilcox Bunonyo; Emeka Amos; Jason Biobaragha Goldie. Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment. Int. J. Appl. Math. Theor. Phys. 2020, 6(2), 19-25. doi: 10.11648/j.ijamtp.20200602.12

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AMA Style

Kubugha Wilcox Bunonyo, Emeka Amos, Jason Biobaragha Goldie. Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment. Int J Appl Math Theor Phys. 2020;6(2):19-25. doi: 10.11648/j.ijamtp.20200602.12

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@article{10.11648/j.ijamtp.20200602.12,
  author = {Kubugha Wilcox Bunonyo and Emeka Amos and Jason Biobaragha Goldie},
  title = {Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {6},
  number = {2},
  pages = {19-25},
  doi = {10.11648/j.ijamtp.20200602.12},
  url = {https://doi.org/10.11648/j.ijamtp.20200602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20200602.12},
  abstract = {The aim of the present article is to model atherosclerotic blood flow through double stenosis in the arterial lumen with application of treatment subject to some boundary conditions in the channel. The research focused also on mathematical formulation of momentum equation governing the blood flow, driven by the oscillatory pressure gradient due to the contraction of muscles of the heart tissues and externally applied magnetic field. The nonlinear partial differential equation and geometries of the double stenosis are scaled using some quantities which made them dimensionless with some pertinent physical parameters such as the womersley number, the treatment parameter, Hartmann number, Darcy number at the region. The oscillatory perturbation method was adopted to further reduce the dimensionless equation to a Bessel differential equation of order zero. An exact solution of equation governing the flow is obtained with quantities coefficients. Mathematica codes were developed to investigate the influence of the pertinent parameters on the velocity profile, shear stress and volumetric flow rate of the first segment of stenosis δ1, while we considered the second stenosis δ2 segment fixed and present the results graphically. In conclusion, it is observed from the results and analysis that the pertinent parameters influenced the blood flow through a double stenosed region of an arterial lumen withholding the second stenosis fixed. These results could help in raising the awareness and the need for treatment of patients with double stenosis focusing on the most delicate.},
 year = {2020}
}

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TY  - JOUR
T1  - Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment
AU  - Kubugha Wilcox Bunonyo
AU  - Emeka Amos
AU  - Jason Biobaragha Goldie
Y1  - 2020/06/20
PY  - 2020
N1  - https://doi.org/10.11648/j.ijamtp.20200602.12
DO  - 10.11648/j.ijamtp.20200602.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 19
EP  - 25
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20200602.12
AB  - The aim of the present article is to model atherosclerotic blood flow through double stenosis in the arterial lumen with application of treatment subject to some boundary conditions in the channel. The research focused also on mathematical formulation of momentum equation governing the blood flow, driven by the oscillatory pressure gradient due to the contraction of muscles of the heart tissues and externally applied magnetic field. The nonlinear partial differential equation and geometries of the double stenosis are scaled using some quantities which made them dimensionless with some pertinent physical parameters such as the womersley number, the treatment parameter, Hartmann number, Darcy number at the region. The oscillatory perturbation method was adopted to further reduce the dimensionless equation to a Bessel differential equation of order zero. An exact solution of equation governing the flow is obtained with quantities coefficients. Mathematica codes were developed to investigate the influence of the pertinent parameters on the velocity profile, shear stress and volumetric flow rate of the first segment of stenosis δ1, while we considered the second stenosis δ2 segment fixed and present the results graphically. In conclusion, it is observed from the results and analysis that the pertinent parameters influenced the blood flow through a double stenosed region of an arterial lumen withholding the second stenosis fixed. These results could help in raising the awareness and the need for treatment of patients with double stenosis focusing on the most delicate.
VL  - 6
IS  - 2
ER  -

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Author Information

Kubugha Wilcox Bunonyo

Department of Mathematics and Statistics, Federal University Otuoke, Otuoke, Nigeria
Emeka Amos

Department of Mathematics, Rivers State University, Port Harcourt, Nigeria
Jason Biobaragha Goldie

Department of Biochemistry, Bayelsa State Polytechnic, Yenagoa, Nigeria

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APA Style

Kubugha Wilcox Bunonyo, Emeka Amos, Jason Biobaragha Goldie. (2020). Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment. International Journal of Applied Mathematics and Theoretical Physics, 6(2), 19-25. https://doi.org/10.11648/j.ijamtp.20200602.12

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ACS Style

Kubugha Wilcox Bunonyo; Emeka Amos; Jason Biobaragha Goldie. Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment. Int. J. Appl. Math. Theor. Phys. 2020, 6(2), 19-25. doi: 10.11648/j.ijamtp.20200602.12

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AMA Style

Kubugha Wilcox Bunonyo, Emeka Amos, Jason Biobaragha Goldie. Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment. Int J Appl Math Theor Phys. 2020;6(2):19-25. doi: 10.11648/j.ijamtp.20200602.12

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@article{10.11648/j.ijamtp.20200602.12,
  author = {Kubugha Wilcox Bunonyo and Emeka Amos and Jason Biobaragha Goldie},
  title = {Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {6},
  number = {2},
  pages = {19-25},
  doi = {10.11648/j.ijamtp.20200602.12},
  url = {https://doi.org/10.11648/j.ijamtp.20200602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20200602.12},
  abstract = {The aim of the present article is to model atherosclerotic blood flow through double stenosis in the arterial lumen with application of treatment subject to some boundary conditions in the channel. The research focused also on mathematical formulation of momentum equation governing the blood flow, driven by the oscillatory pressure gradient due to the contraction of muscles of the heart tissues and externally applied magnetic field. The nonlinear partial differential equation and geometries of the double stenosis are scaled using some quantities which made them dimensionless with some pertinent physical parameters such as the womersley number, the treatment parameter, Hartmann number, Darcy number at the region. The oscillatory perturbation method was adopted to further reduce the dimensionless equation to a Bessel differential equation of order zero. An exact solution of equation governing the flow is obtained with quantities coefficients. Mathematica codes were developed to investigate the influence of the pertinent parameters on the velocity profile, shear stress and volumetric flow rate of the first segment of stenosis δ1, while we considered the second stenosis δ2 segment fixed and present the results graphically. In conclusion, it is observed from the results and analysis that the pertinent parameters influenced the blood flow through a double stenosed region of an arterial lumen withholding the second stenosis fixed. These results could help in raising the awareness and the need for treatment of patients with double stenosis focusing on the most delicate.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Modelling of an Atherosclerotic Blood Flow Through Double Stenosed Region with Application of Treatment
AU  - Kubugha Wilcox Bunonyo
AU  - Emeka Amos
AU  - Jason Biobaragha Goldie
Y1  - 2020/06/20
PY  - 2020
N1  - https://doi.org/10.11648/j.ijamtp.20200602.12
DO  - 10.11648/j.ijamtp.20200602.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 19
EP  - 25
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20200602.12
AB  - The aim of the present article is to model atherosclerotic blood flow through double stenosis in the arterial lumen with application of treatment subject to some boundary conditions in the channel. The research focused also on mathematical formulation of momentum equation governing the blood flow, driven by the oscillatory pressure gradient due to the contraction of muscles of the heart tissues and externally applied magnetic field. The nonlinear partial differential equation and geometries of the double stenosis are scaled using some quantities which made them dimensionless with some pertinent physical parameters such as the womersley number, the treatment parameter, Hartmann number, Darcy number at the region. The oscillatory perturbation method was adopted to further reduce the dimensionless equation to a Bessel differential equation of order zero. An exact solution of equation governing the flow is obtained with quantities coefficients. Mathematica codes were developed to investigate the influence of the pertinent parameters on the velocity profile, shear stress and volumetric flow rate of the first segment of stenosis δ1, while we considered the second stenosis δ2 segment fixed and present the results graphically. In conclusion, it is observed from the results and analysis that the pertinent parameters influenced the blood flow through a double stenosed region of an arterial lumen withholding the second stenosis fixed. These results could help in raising the awareness and the need for treatment of patients with double stenosis focusing on the most delicate.
VL  - 6
IS  - 2
ER  -

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