Numerical Prediction of Flow Recirculation Length Zone in an Artery with Double Stenosis at Low and High Reynolds Numbers
Keywords:
Numerical simulation, double stenosis, distal stenosis, proximal stenosis, recirculation zone length, wall shear stress
Abstract
This work presents simulation of arterial stenosis utilizing ANSYS (Fluent) Computational Fluid Dynamics (CFD) software package for better understanding of blood flow dynamics and to estimate the risk of complication. The aim of the paper is to investigate and study the effects of low and high Reynold’s number on recirculation zone length in arteries with varying levels of double stenosis. Blood flow was numerically simulated to predict the recirculation zone length and wall shear stress. An artery with double stenosis was studied and for the purpose of this investigation, stenosis levels of 15/15%, 75/75%, 15/75%, 75/15%, 15/60%, 20/60%, 45/60% 60/60%, and 75/60% in terms of proximal and distal stenosis are studied over the Reynolds number ranging from 150 to 3000. Blood was a Newtonian fluid flowing as a steady, three-dimensional, incompressible fluid. The velocity flow streamlines, and wall shear stress contours are presented. Results revealed that when varying both distal and proximal stenosis, as Reynolds number increases, recirculation zone length decreases in lower levels of stenosis between 15/60% and 45/60% while it remains relatively constant for higher levels of stenosis 60/60% and 75/60%. It was also revealed that when varying both distal and proximal stenosis, as Reynolds number increases, maximum wall shear stress increases gradually at lower levels of stenosis with almost equal values. At higher levels of stenosis, there was rapid increase in maximum wall shear stress as Reynolds number increases.
References
Figure 5: Velocity profile for 75-75% stenosis at (a) Re = 150, (b) Re = 500, (c) Re = 1000, (d) Re = 2000.
Figure 6: WSS contours at Re = 500 for various levels of (a) 15/60%, (b) 20/60%, (c) 45/60%, (d) 60/60%…, (e) 75/60% … stenosis
4.2 Effects of Reynolds Number on Axial Velocity
4.2.1 Effects of Reynolds Number on Axial Velocity For 20/60%, 45/60%, 60/60% and 75/60%
The effects of Reynolds number on axial velocity of varying levels of stenosis was studied. Figure 7 below shows the effects of Reynolds number on axial velocity of varying levels of distal stenosis. For a 20/60%, at Reynolds number of 150, a minimum axial velocity of 0.65203 m/s was observed and a maximum axial velocity of 11. 561 m/s was observed. This was very similar to the values obtained for 45/60% stenosis where the minimum and maximum axial velocities were 0.65116 m/s and 11.411 m/s respectively. However, for a 75/60% stenosis, a significant increase was observed with its maximum axial velocity reaching up to 25.471 m/s and a minimum axial velocity of 1.4514 m/s. This shows that at higher levels of distal stenosis, there is a significant increase in velocity due to a greater constriction. In the arteries, when axial velocity increases, it is associated with increased shear stress on the endothelial cells lining the coronary arteries. Increased velocity could also lead to a plaque rupture. Plaque rupture can result in the formation of a blood clot, which can partially or completely block the coronary artery and lead to a heart attack medically known as myocardial infarction. Furthermore, increased blood velocity can elevate the oxygen demand of the heart. If the coronary arteries cannot supply sufficient blood to meet this demand, it can result in ischemia and cause chest pain, known as angina.
Figure 7: Graph of axial velocity against Re for 20/60%, 45/60%, 60/60% and 75/60% stenosis
4.2.2 Effects of Reynolds Number On Axial Velocity For 15-75% And 75-15%
The effects of Reynolds number on a 15/75% and a 75/15% stenosis was also studied. As seen from Figure 8 below, a similar axial velocity variance was observed. For a 15/75% stenosis, a minimum axial velocity of 1.3818 m/s was observed with a maximum axial velocity of 25.528 m/s. For a 75/15% stenosis, minimum and maximum axial velocities of 1.4168 m/s and 26.003 m/s respectively were recorded. In addition to the effects of axial velocity discussed in section 4.2.1 above, research has shown that persistent high blood velocity can lead to hypertension. Over time, hypertension can lead to heart failure or death.
Figure 8: Graph of Axial Velocity Against Reynolds number for 15/75% and 75/15% stenosis
4.3 Effects of Varying Level of Stenosis on Wall Shear Stress
4.3.1 Effects of Varying Level of Distal Stenosis On Wall Shear Stress
Figure 9 shows the comparison of maximum wall shear stress with various levels of distal stenosis while the proximal stenosis is kept constant. From the graph, it can be observed that for all cases of Reynolds number, between 15-60% and 60-60% level of distal stenosis, wall shear stress remains constant. As the level of distal stenosis increases between 60-60% and 75-60%, wall shear stress increases exponentially with 75% level of distal stenosis having the highest Wall shear stress value of 3500 Pa. The lowest Wall shear stress was recorded at 15% level of distal stenosis with a value of 14 Pa. It can also be observed that very little change is recorded between Reynolds numbers of 150 and 500. This is since flow is purely laminar within this region. As Reynolds number increases beyond Re=500, there is a significant change in Wall Shear Stress. Hence, it can be inferred that Wall Shear Stress increases with an increasing level of distal stenosis. It is noteworthy that high wall shear stress is associated with formation of vulnerable plaque phenotype and promotes plaque development. High WSS is also associated with damage to the endothelial cells lining the arteries. This damage can lead to endothelial dysfunction, reducing the artery's ability to regulate blood flow and pressure effectively. An excessive WSS also leads to high blood pressure which can in turn leads to heart attack or even death of the individual.
Figure 9: Graph of Maximum wall shear stress against percentage level of stenosis
4.3.2 Effects of Varying Level of Proximal Stenosis on Wall Shear Stress
Figure 10 also shows the comparison of maximum wall shear stress with varying level of proximal stenosis i.e., keeping the distal stenosis constant at 60% and varying the proximal stenosis between 15 and 75%. It is evident from the graph that for all cases of Reynolds numbers, maximum wall shear stress remains constant between 15 and 45% levels of proximal stenosis. At 75% stenosis and Re=3000, the wall shear stress reaches the peak of 3004 Pa. As previously stated, an excessive WSS also leads to high blood pressure which can in turn leads to heart attack or even death of the individual.
Figure 10: Graph of Maximum wall shear stress against percentage level of stenosis
4.4 Effects of Varying Reynolds Number on Wall Shear Stress
Furthermore, the comparison between wall shear stress and Reynolds number was analyzed. This was conducted on four geometries: 60-15%, 60-20%,60-45%,70-75%. For stenosis levels of 60-15 to 60-45%, the WSS was almost equal for 60-15% to 60-45% and increased gradually with the minimum values at an average of 15 Pa and maximum values at an average of 731 Pa. For a stenosis level of 60-75%, significant changes were recorded . minimum and maximum values of 52 Pa and 3004 Pa respectively. This shows that higher levels of stenosis, there is a drastic increase in WSS as Reynolds number increases. This is line with the research carried out by Kabir et al., 2021. This is shown in Figures 11 (a), (b), (c), (d) and (e).
These results have shown that as Reynolds number increases, indicating higher blood velocity, WSS increases. Extremely high WSS can damage the endothelial cells lining the arteries. This can lead to endothelial dysfunction and further cause more plaque formation.
(a) (b)
(c) (d)
(e)
Figure 11: Graph of Maximum wall shear stress against Reynolds number. for (a) 60-15%, (b) 60-20%, (c) 60-45%, (d)60-60%, (e) 60-75% level of proximal stenosis
4.5 Effects of Reynolds Number on Both Proximal and Distal Recirculation Zone Length
The effects of Reynolds no. on recirculation zone length were studied for 15-60% and 20-60% stenosis and the results are presented in Figure 12(a) and (b) respectively. The graphs show that as Reynolds number increases, dimensionless recirculation zone length decreases but this occurs for only the proximal stenotic site. It can be seen in Figure 12(a) that at 02000, the proximal recirculation zone length remains relatively constant at a value of 0.8D.
Compared with the distal recirculation zone length, Figure 12(a) below shows that recirculation zone length remains constant for 0
(a) (b)
Figure 12: Graph of dimensionless recirculation zone length against Reynolds number. for (a) 15/60% and (b) 20/60% level of stenosis
4.6 Comparison of Effects of Recirculation Zone Length on Varying Level of Distal and Proximal Stenosis
Figure 13(a) below illustrates the relationship between recirculation and Re for varying levels of distal stenosis. From the graph, it can be observed that for all levels of stenosis, as Reynolds number increases, recirculation zone length decreases, this is noticeable in lower levels of stenosis between 15/60% and 45/60%. However, it can be observed that for higher levels of stenosis i.e, 60/60% and 75/60%, the recirculation zone length remains relatively constant. This may indicate persistent flow disturbances that do not significantly change as Reynolds number increases. Furthermore, recirculation zone length remains constant at higher levels of stenosis because the severe reduction in cross sectional area of the artery creates a significant flow disturbance, thereby leading to immediate separation and stable recirculation patterns. The flow disturbance is so pronounced as such that it overrides the influence of other factors that can change the recirculation zone length i.e, a further increase in the Reynolds number will not cause any significant change in the recirculation zone length. As a result, even as Reynolds number increases, the recirculation zone length, the RZL remains relatively constant. This shows a saturation effect where the stenosis itself dictates the flow behaviour. The same applies for varying levels of proximal stenosis shown in Figure 13(b), only that the recirculation zones form earlier and remain relatively constant.
(a) (b)
Figure 13:Comparison of effect of recirculation zone length on varying level of (a) distal and (b) proximal stenosis
5.0 CONCLUSION
A computational fluid dynamics study was conducted to investigate the effects of double stenosis on heamodynamic indicators such as recirculation zone length and wall shear stress in double stenotic arteries of various levels. The simulation was carried out on 11 geometries – 15/60%, 20/60%, 45/60%, 75/60%, 60/15%, 60/20%, 60/45%, 60/60%, 60/75%, 15/75%, 75/15%, using ANSYS FLUENT. Five Reynolds numbers were considered – Re = 150, 500, 1000, 2000, 3000 and the major findings in this research are:
1. At low Reynolds number, recirculation zone length is high while it is lower at higher Reynolds number, i.e., as Reynolds number increases, recirculation zone length decreases.
2. When varying both distal and proximal stenosis, as Reynolds number increases, recirculation zone length decreases in lower levels of stenosis between 15/60% and 45/60% while it remains relatively constant for higher levels of stenosis i.e., 60/60% and 75/60%.
3. As Reynolds number increases, maximum wall shear stress increases.
4. When varying both distal and proximal stenosis, as Reynolds number increases, maximum wall shear stress increases gradually at lower levels of stenosis with almost equal values. At higher levels of stenosis, there was rapid increase in maximum wall shear stress as Reynolds number increases.
5. The velocity flow streamlines and the WSS contours were also presented.
This study assumed that blood was a Newtonian fluid for simplicity of computation. However, blood is a non-Newtonian fluid. Using a non-Newtonian model such as Casson model could influence wall shear stress distributions and flow streamlines. Future studies should incorporate non-Newtonian behaviours to get a more accurate representation.
REFERENCES
Alshare, A., Bourhan, T., Hossam H. E. (2013). Computational Modeling of Non-Newtonian Blood Flow Through Stenosed Arteries in the Presence of Magnetic Field. [DOI: 10.1115/1.4025107].
Alshare, A., Tashtoush, B. (2015). Simulations of Magnetohemodynamics in Stenosed Arteries in Diabetic or Anemic Models. Computational and Mathematical Methods in Medicine Volume 2016, Article ID 8123930, 13 pages http://dx.doi.org/10.1155/2016/8123930.
Changdar, S.,and De, S. (2015). Numerical Simulation of Nonlinear Pulsatile Newtonian Blood Flow through a Multiple Stenosed Artery. Hindawi Publishing Corporation International Scholarly Research Notices Volume 2015, Article ID 628605, 10 pages http://dx.doi.org/10.1155/2015/62860
Daniel, J. T., Jeroen, F., Krzysztof, C., Ian, H., D.R., Hose, Rebecca, G., Louise, A. , Marcel, v. V., Danielle, C. J. K., Pim, T., Michel, R., Julian P. G., and Paul D. M. (2023). Validation of a novel numerical model to predict regionalized blood flow in the coronary arteries. European Heart Journal - Digital Health (2023) 4, 81–89 https://doi.org/10.1093/ehjdh/ztac077
Dattani, S., Samborska, V., Ritchie, H. Roser, M. (2021). Cardiovascular Diseases. https://ourworldindata.org/cardiovasculardiseases.
Doutel E., Carneiro J., Campos J.B.L.M., Miranda J. M. (2018). Artificial stenoses for computational hemodynamics
Dwidmuthe, P. D., Dastane, G. G., Mathpati, C. S., Joshi, J. B., Changdar, S., De, S. (2015). Study of Blood Flow in Stenosed Artery Model Using Computational Fluid Dynamics and Response Surface Methodology. doi: 10.1002/cjce.23991
Editors at Mount Sinai. (2024). Hardening of the arteries. https://www.mountsinai.org/health-library/disease-conditions/hardening-of-the-arteries
Editors at Vascular Interventional Radiology Clinic. (2011) https://virclinic.com/stroke/carotid-arterystenosis/
Fetuga, I. A., Olakoyejo, O.T., Oluwatusin, O., Adelaja, A. O., Gbegudu, J. K., Aderemi, K. S., Adeyemi, E. A., (2023). Computational Model of Nano-Pharmacological Particles for the Clinical Management of Stenotic and Aneurysmatic Coronary Artery in the Human Body. Nigerian Journal of Technological Development, Vol. 20, No. 1, March 2023. Online ISSN: 2437-2110.
Fetuga, I.A., Olakoyejo, O.T., Ewim, D. R. E., Oluwatusin, O., Adelaja A. O., Aderemi K.S. (2022). Numerical prediction of flow recirculation length zone in an artery with multiple stenoses at low and high Reynolds number. Series on Biomechanics, Vol.36 No.4 (2022), 10-24. DOI:10.7546/SB.03.04.2022
Hoffmann, K. A., and Chiang, S. T., Computational Fluid Dynamics Volume III, 4th Ed., EES, Wichita, KS, USA, 2000.
Hussain, A., Dar, M. N. R., Cheema, W. K., Tag-eldin, E. M., Kanwal, R. (2023). Numerical Simulation of Unsteady Generic Newtonian Blood Flow and Heat Transfer Through Discrepant Shaped Dilatable
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Figure 6: WSS contours at Re = 500 for various levels of (a) 15/60%, (b) 20/60%, (c) 45/60%, (d) 60/60%…, (e) 75/60% … stenosis
4.2 Effects of Reynolds Number on Axial Velocity
4.2.1 Effects of Reynolds Number on Axial Velocity For 20/60%, 45/60%, 60/60% and 75/60%
The effects of Reynolds number on axial velocity of varying levels of stenosis was studied. Figure 7 below shows the effects of Reynolds number on axial velocity of varying levels of distal stenosis. For a 20/60%, at Reynolds number of 150, a minimum axial velocity of 0.65203 m/s was observed and a maximum axial velocity of 11. 561 m/s was observed. This was very similar to the values obtained for 45/60% stenosis where the minimum and maximum axial velocities were 0.65116 m/s and 11.411 m/s respectively. However, for a 75/60% stenosis, a significant increase was observed with its maximum axial velocity reaching up to 25.471 m/s and a minimum axial velocity of 1.4514 m/s. This shows that at higher levels of distal stenosis, there is a significant increase in velocity due to a greater constriction. In the arteries, when axial velocity increases, it is associated with increased shear stress on the endothelial cells lining the coronary arteries. Increased velocity could also lead to a plaque rupture. Plaque rupture can result in the formation of a blood clot, which can partially or completely block the coronary artery and lead to a heart attack medically known as myocardial infarction. Furthermore, increased blood velocity can elevate the oxygen demand of the heart. If the coronary arteries cannot supply sufficient blood to meet this demand, it can result in ischemia and cause chest pain, known as angina.
Figure 7: Graph of axial velocity against Re for 20/60%, 45/60%, 60/60% and 75/60% stenosis
4.2.2 Effects of Reynolds Number On Axial Velocity For 15-75% And 75-15%
The effects of Reynolds number on a 15/75% and a 75/15% stenosis was also studied. As seen from Figure 8 below, a similar axial velocity variance was observed. For a 15/75% stenosis, a minimum axial velocity of 1.3818 m/s was observed with a maximum axial velocity of 25.528 m/s. For a 75/15% stenosis, minimum and maximum axial velocities of 1.4168 m/s and 26.003 m/s respectively were recorded. In addition to the effects of axial velocity discussed in section 4.2.1 above, research has shown that persistent high blood velocity can lead to hypertension. Over time, hypertension can lead to heart failure or death.
Figure 8: Graph of Axial Velocity Against Reynolds number for 15/75% and 75/15% stenosis
4.3 Effects of Varying Level of Stenosis on Wall Shear Stress
4.3.1 Effects of Varying Level of Distal Stenosis On Wall Shear Stress
Figure 9 shows the comparison of maximum wall shear stress with various levels of distal stenosis while the proximal stenosis is kept constant. From the graph, it can be observed that for all cases of Reynolds number, between 15-60% and 60-60% level of distal stenosis, wall shear stress remains constant. As the level of distal stenosis increases between 60-60% and 75-60%, wall shear stress increases exponentially with 75% level of distal stenosis having the highest Wall shear stress value of 3500 Pa. The lowest Wall shear stress was recorded at 15% level of distal stenosis with a value of 14 Pa. It can also be observed that very little change is recorded between Reynolds numbers of 150 and 500. This is since flow is purely laminar within this region. As Reynolds number increases beyond Re=500, there is a significant change in Wall Shear Stress. Hence, it can be inferred that Wall Shear Stress increases with an increasing level of distal stenosis. It is noteworthy that high wall shear stress is associated with formation of vulnerable plaque phenotype and promotes plaque development. High WSS is also associated with damage to the endothelial cells lining the arteries. This damage can lead to endothelial dysfunction, reducing the artery's ability to regulate blood flow and pressure effectively. An excessive WSS also leads to high blood pressure which can in turn leads to heart attack or even death of the individual.
Figure 9: Graph of Maximum wall shear stress against percentage level of stenosis
4.3.2 Effects of Varying Level of Proximal Stenosis on Wall Shear Stress
Figure 10 also shows the comparison of maximum wall shear stress with varying level of proximal stenosis i.e., keeping the distal stenosis constant at 60% and varying the proximal stenosis between 15 and 75%. It is evident from the graph that for all cases of Reynolds numbers, maximum wall shear stress remains constant between 15 and 45% levels of proximal stenosis. At 75% stenosis and Re=3000, the wall shear stress reaches the peak of 3004 Pa. As previously stated, an excessive WSS also leads to high blood pressure which can in turn leads to heart attack or even death of the individual.
Figure 10: Graph of Maximum wall shear stress against percentage level of stenosis
4.4 Effects of Varying Reynolds Number on Wall Shear Stress
Furthermore, the comparison between wall shear stress and Reynolds number was analyzed. This was conducted on four geometries: 60-15%, 60-20%,60-45%,70-75%. For stenosis levels of 60-15 to 60-45%, the WSS was almost equal for 60-15% to 60-45% and increased gradually with the minimum values at an average of 15 Pa and maximum values at an average of 731 Pa. For a stenosis level of 60-75%, significant changes were recorded . minimum and maximum values of 52 Pa and 3004 Pa respectively. This shows that higher levels of stenosis, there is a drastic increase in WSS as Reynolds number increases. This is line with the research carried out by Kabir et al., 2021. This is shown in Figures 11 (a), (b), (c), (d) and (e).
These results have shown that as Reynolds number increases, indicating higher blood velocity, WSS increases. Extremely high WSS can damage the endothelial cells lining the arteries. This can lead to endothelial dysfunction and further cause more plaque formation.
(a) (b)
(c) (d)
(e)
Figure 11: Graph of Maximum wall shear stress against Reynolds number. for (a) 60-15%, (b) 60-20%, (c) 60-45%, (d)60-60%, (e) 60-75% level of proximal stenosis
4.5 Effects of Reynolds Number on Both Proximal and Distal Recirculation Zone Length
The effects of Reynolds no. on recirculation zone length were studied for 15-60% and 20-60% stenosis and the results are presented in Figure 12(a) and (b) respectively. The graphs show that as Reynolds number increases, dimensionless recirculation zone length decreases but this occurs for only the proximal stenotic site. It can be seen in Figure 12(a) that at 0
Compared with the distal recirculation zone length, Figure 12(a) below shows that recirculation zone length remains constant for 0
(a) (b)
Figure 12: Graph of dimensionless recirculation zone length against Reynolds number. for (a) 15/60% and (b) 20/60% level of stenosis
4.6 Comparison of Effects of Recirculation Zone Length on Varying Level of Distal and Proximal Stenosis
Figure 13(a) below illustrates the relationship between recirculation and Re for varying levels of distal stenosis. From the graph, it can be observed that for all levels of stenosis, as Reynolds number increases, recirculation zone length decreases, this is noticeable in lower levels of stenosis between 15/60% and 45/60%. However, it can be observed that for higher levels of stenosis i.e, 60/60% and 75/60%, the recirculation zone length remains relatively constant. This may indicate persistent flow disturbances that do not significantly change as Reynolds number increases. Furthermore, recirculation zone length remains constant at higher levels of stenosis because the severe reduction in cross sectional area of the artery creates a significant flow disturbance, thereby leading to immediate separation and stable recirculation patterns. The flow disturbance is so pronounced as such that it overrides the influence of other factors that can change the recirculation zone length i.e, a further increase in the Reynolds number will not cause any significant change in the recirculation zone length. As a result, even as Reynolds number increases, the recirculation zone length, the RZL remains relatively constant. This shows a saturation effect where the stenosis itself dictates the flow behaviour. The same applies for varying levels of proximal stenosis shown in Figure 13(b), only that the recirculation zones form earlier and remain relatively constant.
(a) (b)
Figure 13:Comparison of effect of recirculation zone length on varying level of (a) distal and (b) proximal stenosis
5.0 CONCLUSION
A computational fluid dynamics study was conducted to investigate the effects of double stenosis on heamodynamic indicators such as recirculation zone length and wall shear stress in double stenotic arteries of various levels. The simulation was carried out on 11 geometries – 15/60%, 20/60%, 45/60%, 75/60%, 60/15%, 60/20%, 60/45%, 60/60%, 60/75%, 15/75%, 75/15%, using ANSYS FLUENT. Five Reynolds numbers were considered – Re = 150, 500, 1000, 2000, 3000 and the major findings in this research are:
1. At low Reynolds number, recirculation zone length is high while it is lower at higher Reynolds number, i.e., as Reynolds number increases, recirculation zone length decreases.
2. When varying both distal and proximal stenosis, as Reynolds number increases, recirculation zone length decreases in lower levels of stenosis between 15/60% and 45/60% while it remains relatively constant for higher levels of stenosis i.e., 60/60% and 75/60%.
3. As Reynolds number increases, maximum wall shear stress increases.
4. When varying both distal and proximal stenosis, as Reynolds number increases, maximum wall shear stress increases gradually at lower levels of stenosis with almost equal values. At higher levels of stenosis, there was rapid increase in maximum wall shear stress as Reynolds number increases.
5. The velocity flow streamlines and the WSS contours were also presented.
This study assumed that blood was a Newtonian fluid for simplicity of computation. However, blood is a non-Newtonian fluid. Using a non-Newtonian model such as Casson model could influence wall shear stress distributions and flow streamlines. Future studies should incorporate non-Newtonian behaviours to get a more accurate representation.
REFERENCES
Alshare, A., Bourhan, T., Hossam H. E. (2013). Computational Modeling of Non-Newtonian Blood Flow Through Stenosed Arteries in the Presence of Magnetic Field. [DOI: 10.1115/1.4025107].
Alshare, A., Tashtoush, B. (2015). Simulations of Magnetohemodynamics in Stenosed Arteries in Diabetic or Anemic Models. Computational and Mathematical Methods in Medicine Volume 2016, Article ID 8123930, 13 pages http://dx.doi.org/10.1155/2016/8123930.
Changdar, S.,and De, S. (2015). Numerical Simulation of Nonlinear Pulsatile Newtonian Blood Flow through a Multiple Stenosed Artery. Hindawi Publishing Corporation International Scholarly Research Notices Volume 2015, Article ID 628605, 10 pages http://dx.doi.org/10.1155/2015/62860
Daniel, J. T., Jeroen, F., Krzysztof, C., Ian, H., D.R., Hose, Rebecca, G., Louise, A. , Marcel, v. V., Danielle, C. J. K., Pim, T., Michel, R., Julian P. G., and Paul D. M. (2023). Validation of a novel numerical model to predict regionalized blood flow in the coronary arteries. European Heart Journal - Digital Health (2023) 4, 81–89 https://doi.org/10.1093/ehjdh/ztac077
Dattani, S., Samborska, V., Ritchie, H. Roser, M. (2021). Cardiovascular Diseases. https://ourworldindata.org/cardiovasculardiseases.
Doutel E., Carneiro J., Campos J.B.L.M., Miranda J. M. (2018). Artificial stenoses for computational hemodynamics
Dwidmuthe, P. D., Dastane, G. G., Mathpati, C. S., Joshi, J. B., Changdar, S., De, S. (2015). Study of Blood Flow in Stenosed Artery Model Using Computational Fluid Dynamics and Response Surface Methodology. doi: 10.1002/cjce.23991
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Published
2025-04-11
How to Cite
engineering, J., & Olakoyejo, O. (2025). Numerical Prediction of Flow Recirculation Length Zone in an Artery with Double Stenosis at Low and High Reynolds Numbers. Journal of Engineering Research, 30(1), 1-21. Retrieved from http://jer.unilag.edu.ng/article/view/2444
Section
Articles
