# Mathematical Analysis of Unsteady Solute Dispersion with Chemical Reaction Through a Stenosed Artery

## Authors

• Nurul Aini Jaafar Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor, Johor Bahru, Malaysia
• Siti NurulAifa Mohd ZainulAbidin Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor, Johor Bahru, Malaysia
• Zuhaila Ismail Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor, Johor Bahru, Malaysia
• Ahmad Qushairi Mohamad Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor, Johor Bahru, Malaysia

## Keywords:

Blood flow, Dispersion function, Generalized dispersion model, Herschel-Bulkley, Non-Newtonian fluids, Stenosed artery

## Abstract

One major kind of arterial disease in blood flow that attracted many researchers is arterial stenosis. Arterial stenosis occurs when a lumen of artery is narrowed by the accumulation of fats, cholesterols and lipids plaques at the inner layer of the wall of an artery. To treat this arterial disease, the drug (solute) is injected into the blood vessels. Injection of the drug into the blood vessel cause the occurrence of chemical reaction between the drug and blood proteins and it affects the effectiveness of the solute transportation in blood flow. Hence, this study examines the unsteady dispersion of solute with the influence of chemical reaction and stenosis height through a very narrow artery with a cosine-curved stenosis. The blood is treating as Herschel-Bulkley (H-B) fluid. The momentum and constitutive equations are solved analytically to gain velocity of H-B blood flow. The convective-diffusion equation is solved by applying the generalized dispersion model to gain the dispersion function of solute. The influence of chemical reaction, power-law index, plug flow radius and stenosis height on the solute dispersion process is investigated. The results are validated with the previous solution without the effect of chemical reaction and stenosis. The results showed a good conformity between the two solutions. An increase in the chemical reaction coefficient, stenosis height, power-law index and plug flow radius reduces the dispersion function. It is observed that the solute dispersion in blood flow is affected by chemical reaction and stenosis height. H-B fluid is an appropriate fluid to investigate the blood velocity and transportation of the drug in blood flow to the targeted stenosed region through a very narrow artery for the treatment of arterial diseases. The results of the present study can potentially be used to predict the changes of blood flow behavior and dispersion process in blood flow.

## Author Biographies

### Nurul Aini Jaafar, Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor, Johor Bahru, Malaysia

nurulaini.jaafar@utm.my

### Siti NurulAifa Mohd ZainulAbidin, Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor, Johor Bahru, Malaysia

aifazainul94@gmail.com

zuhaila@utm.my

## References

Ali, N., A. Zaman, M. Sajid, J. J. Nieto, and A. Torres. "Unsteady non-Newtonian blood flow through a tapered overlapping stenosed catheterized vessel." Mathematical Biosciences 269 (2015): 94-103. https://doi.org/10.1016/j.mbs.2015.08.018

Jaafar, Nurul Aini, Yazariah Mohd Yatim, and D. S. Sankar. "Influence of chemical reaction on the steady dispersion of solute in blood flow-a mathematical model." Far East Journal of Mathematical Sciences 100, no. 4 (2016): 617. https://doi.org/10.17654/MS100040617

Taylor, Geoffrey Ingram. "Dispersion of soluble matter in solvent flowing slowly through a tube." Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 219, no. 1137 (1953): 186-203. https://doi.org/10.1098/rspa.1953.0139

Aris, Rutherford. "On the dispersion of a solute in a fluid flowing through a tube." Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 235, no. 1200 (1956): 67-77. https://doi.org/10.1098/rspa.1956.0065

Ananthakrishnan, Vs, Wr N. Gill, and Allen J. Barduhn. "Laminar dispersion in capillaries: Part I. Mathematical analysis." AIChE Journal 11, no. 6 (1965): 1063-1072. https://doi.org/10.1002/aic.690110620

Gill, W. N., and R. Sankarasubramanian. "Exact analysis of unsteady convective diffusion." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 316, no. 1526 (1970): 341-350. https://doi.org/10.1098/rspa.1970.0083

Dash, R. K., G. Jayaraman, and K. N. Mehta. "Shear augmented dispersion of a solute in a Casson fluid flowing in a conduit." Annals of Biomedical Engineering 28, no. 4 (2000): 373-385. https://doi.org/10.1114/1.287

Rana, Jyotirmoy, and P. V. S. N. Murthy. "Unsteady solute dispersion in Herschel-Bulkley fluid in a tube with wall absorption." Physics of Fluids 28, no. 11 (2016): 111903. https://doi.org/10.1063/1.4967210

Rana, Jyotirmoy, and P. V. S. N. Murthy. "Unsteady solute dispersion in small blood vessels using a two-phase Casson model." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2204 (2017): 20170427. https://doi.org/10.1098/rspa.2017.0427

Jaafar, Nurul Aini. "Mathematical Analysis of Herschel-Bulkley Fluid Model for Solute Dispersion in Blood Flow Through Narrow Conduits." PhD thesis, Universiti Sains Malaysia, 2017.

Rana, Jyotirmoy, and P. V. S. N. Murthy. "Unsteady solute dispersion in Herschel-Bulkley fluid in a tube with wall absorption." Physics of Fluids 28, no. 11 (2016): 111903. https://doi.org/10.1063/1.4967210

Jaafar, Nurul Aini, Yazariah Mohd Yatim, and D. S. Sankar. "Mathematical analysis for unsteady dispersion of solute with chemical reaction in blood flow." In AIP Conference Proceedings, vol. 1750, no. 1, p. 030033. AIP Publishing LLC, 2016. https://doi.org/10.1063/1.4954569

Miah, Md. Abdul Karim, Shorab Hossain, and Sayedus Salehin. "Effects of Severity and Dominance of Viscous Force on Stenosis and Aneurysm During Pulsatile Blood Flow Using Computational Modelling." CFD Letters 12, no. 8 (2020): 35-54. https://doi.org/10.37934/cfdl.12.8.3554

Ratchagar, Nirmala P., and R. Vijaya Kumar. "Dispersion of solute with chemical reaction in blood flow." Bulletin of Pure & Applied Sciences-Mathematics and Statistics 38, no. 1 (2019): 385-395. https://doi.org/10.5958/2320-3226.2019.00042.0

Patel, I. C., and J. A. Sirs. "Dispersion of solutes during blood flow through curved tubes." Medical and Biological Engineering and Computing 21, no. 2 (1983): 113-118. https://doi.org/10.1007/BF02441524

Al-Azawy, Mohammed Ghalib, Saleem Khalefa Kadhim, and Azzam Sabah Hameed. "Newtonian and Non-Newtonian Blood Rheology Inside a Model of Stenosis." CFD Letters 12, no. 11 (2020): 27-36. https://doi.org/10.37934/cfdl.12.11.2736

Sahak, Ahmad Sofianuddin A., Nor Azwadi Che Sidik, Siti Nurul Akmal Yusof, and Mahmoud Ahmed Alamir. "Numerical Study of Particle Behaviour in a Mixed Convection Channel Flow with Cavity using Cubic Interpolation Pseudo-Particle Navier-Stokes Formulation Method." Journal of Advanced Research in Numerical Heat Transfer 1, no. 1 (2020): 32-51.

Ewis, Karem Mahmoud. "Effects of Variable Thermal Conductivity and Grashof Number on Non-Darcian Natural Convection Flow of Viscoelastic Fluids with Non Linear Radiation and Dissipations." Journal of Advanced Research in Applied Sciences and Engineering Technology 22, no. 1 (2021): 69-80.

2021-08-20

## How to Cite

Jaafar, N. A. ., Mohd ZainulAbidin, S. N., Ismail, Z., & Mohamad, A. Q. (2021). Mathematical Analysis of Unsteady Solute Dispersion with Chemical Reaction Through a Stenosed Artery. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, 86(2), 56–73. https://doi.org/10.37934/arfmts.86.2.5673

Articles