# Unsteady blood flow of non-Newtonian fluid through a rigid artery in the presence of multi-irregular stenoses

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## Abstract

The pulsating flow of blood through a rigid artery in the presence of multi-irregular shaped stenoses is explored in this paper. Herschel–Bulkley fluid model is used for representing the characteristics of blood. The arterial wall is assumed to be rigid. The equations governing the flow are set up under appropriate assumptions and then solved by using standard perturbation method subject to the prescribed boundary conditions. The numerical values of steady component of pressure gradient are obtained by using Newton–Raphson method. The effects of yield stress, Womersley frequency parameter and severity of stenosis on the plug core radius, wall shear stress and resistance to the flow are shown diagrammatically and discussed in detail.

## Keywords

Herschel–Bulkley fluid Unsteady flow Multi-irregular stenoses Perturbation solution Yields stress## List of symbols

- \(A\)
Amplitude of the flow, \([ - ]\)

- \(\bar{d}\)
Axial position, \([{\text{cm}}]\)

- \(d\)
Dimensionless axial position, \([ - ]\)

- \(\bar{l}\)
Length of the stenosis, [cm]

- \(n\)
Power law index, \([ - ]\)

- \(\bar{p}\)
Pressure, [\({\text{dyn/cm}}^{2}\)]

- \(p\)
Dimensionless pressure, \([ - ]\)

- \( \bar{Q} \)
Flow rate, \([{\text{cm}}^{3} / {\text{s}}]\)

- \(Q\)
Dimensionless flow rate, \([ - ]\)

- \(Q_{s}\)
Dimensionless flow rate in case of steady flow, \([ - ]\)

- \(\bar{q}(\bar{z})\)
Steady-state pressure gradient, \([{\text{dyn/cm}}^{2} ]\)

- \(q(z)\)
Dimensionless steady-state pressure gradient, \([ - ]\)

- \(\bar{q}_{0}\)
Negative of the pressure gradient in the normal artery, [\({\text{dyn/cm}}^{2}\)]

- \(\bar{R}(\bar{z})\)
Radius of the stenotic blood vessel, \([{\text{cm}}]\)

- \(R(z)\)
Dimensionless radius of the stenotic blood vessel, \([ - ]\)

- \(\bar{R}_{0}\)
Radius of the normal artery, \([{\text{cm}}]\)

- \(R_{p}\)
Non-dimensional plug core radius, \([ - ]\)

- \(\bar{r}\)
Radial distance, \([{\text{cm}}]\)

- \(r\)
Dimensionless radial distance, \([ - ]\)

- \(\bar{t}\)
Time \([{\text{s}}]\)

- \(t\)
Dimensionless time,\([ - ]\)

- \(\bar{w}\)
Axial velocity \([{\text{cm/s}}]\)

- \(w\)
Dimensionless axial velocity, \([ - ]\)

- \(w_{p}\)
Dimensionless plug core velocity,\([ - ]\)

- \(\bar{z}\)
Axial distance, \([{\text{cm}}]\)

- \(z\)
Dimensionless axial distance, \([ - ]\)

## Greek Symbols

- \(\alpha^{2}\)
Womersley frequency parameter, \([ - ]\)

- \(\bar{\delta }\)
Height of the stenosis, \([{\text{cm}}]\)

- \(\delta\)
Dimensionless height of the stenosis, \([ - ]\)

- \(\Delta p\)
Dimensionless pressure drop, \([ - ]\)

- \(\theta\)
Dimensionless yield stress, \([ - ]\)

- \(\varLambda\)
Dimensionless resistance to the flow \([ - ]\)

- \(\bar{\mu }_{0}\)
Coefficient of viscosity for Newtonian fluid [\({\text{cP}}\)]

- \(\bar{\mu }_{H}\)
Coefficient of viscosity for Herschel–Bulkley fluid, \([({\text{cP}})^{n} / {\text{s}}^{n - 1} ]\)

- \(\bar{\rho }\)
Density of the blood, \([{\text{Kg/cm}}^{3} ]\)

- \(\bar{\tau }\)
Shear stress, [\({\text{dyn/cm}}^{2}\)]\([ - ]\)

- \(\tau\)
Dimensionless shear stress, \([ - ]\)

- \(\bar{\tau }_{H}\)
Yield stress, [\({\text{dyn/cm}}^{2}\)]

- \(\tau_{p}\)
Dimensionless shear stress in the plug core region, \([ - ]\)

- \(\tau_{w}\)
Dimensionless wall shear stress, \([ - ]\)

- \(\bar{\omega }\)
Angular frequency, \([{\text{s}}^{ - 1} ]\)

## Notes

### Acknowledgements

The authors are very thankful to the anonymous reviewers for their valuable suggestions to improve the version of the article. The first author is very grateful to Higher Education Commission of Pakistan (HEC) for financial assistance.

## References

- 1.Chakravarty S (1987) Effect of stenosis on the flow behaviour of blood in an artery. Int J Eng Sci 25:1003–1018CrossRefGoogle Scholar
- 2.Peter R, Johnston Kilpatrick D (1991) Mathematical modelling of flow through an irregular arterial stenosis. J Ilimdda 24(11):1069–1077Google Scholar
- 3.Pincombe B, Mazumdar J, Hamilton-Craig I (1999) Effects of multiple stenoses and post-stenotic dilatation on non-Newtonian blood flow in small arteries. Med Biol Eng Comput 37:595–599CrossRefGoogle Scholar
- 4.Chaturani P, Palanisamy V (1989) Casson fluid model for pulsatile flow of blood under periodic body acceleration. Biorheology 27:619–630CrossRefGoogle Scholar
- 5.Liu B, Tang D (2000) A numerical simulation of viscous flows in collapsible tubes with stenoses. Appl Numer Math 32:87–101MathSciNetCrossRefGoogle Scholar
- 6.Chakravarty S, Mandal PK (1994) Mathematical modelling of blood flow through an overlapping arterial stenosis. Math Comput Model 19:59–70CrossRefGoogle Scholar
- 7.Gijsen FJH, Allanic E, Vosse FN, Janssen JD (1999) The influences of the non-Newtonian properties of blood on the flow in large arteries unsteady flow in a 90D curved tube. J Biol Med 32:705–713Google Scholar
- 8.Sankar DS, Hemalatha K (2006) Pulsatile flow of Herschel–Bulkley fluid through Stenosed Arterie—a mathematical model. Int J Non-Linear Mech 41:979–990CrossRefGoogle Scholar
- 9.Srikanth D, Ramana Reddy JV, Shubha J, Anup K (2015) Unsteady polar fluid model of blood flow through tapered
*ω*-shape stenosed artery: effects of catheter and velocity slip. Ain Shams Eng J 6:1093–1104CrossRefGoogle Scholar - 10.Zaman A, Ali N, Bég OA (2016) Numerical simulation of unsteady micropolar hemodynamics in a tapered catheterized artery with a combination of stenosis and aneurysm. Med Biol Eng Comput 54:1423–1436CrossRefGoogle Scholar
- 11.Pincombe B, Mazumdar JN (1997) The effects of post-stenotic dilatations on the flow of a blood analogue through stenosed coronary arteries. Math Comput Model 25:57–70MathSciNetCrossRefGoogle Scholar
- 12.Ismail Z, Abdullah I, Mustapha N, Amin N (2008) A power-law model of blood flow through a tapered overlapping stenosed artery. Appl Math Comput 195:669–680MathSciNetzbMATHGoogle Scholar
- 13.Chakravarty S, Mandal PK (2000) Two-dimensional blood flow through tapered arteries under stenotic conditions. Int J Non-Linear Mech 35:779–793CrossRefGoogle Scholar
- 14.Zaman A, Ali N (2016) Effects of peripheral layer thickness on pulsatile flow of Herschel–Bulkley fluid through a stenotic artery. Can J Phys 94:920–928CrossRefGoogle Scholar
- 15.Biswas D, Laskar RB (2011) Steady flow of blood through a stenosed artery—a non-Newtonian fluid model. Assam Univ J Sci Technol 7:144–153Google Scholar
- 16.Jabir E, Anil S (2016) Numerical analysis of blood flow through an elliptic stenosis using large eddy simulation. Proc Inst Mech Eng H 30(8):709–726. https://doi.org/10.1177/0954411916644474 CrossRefGoogle Scholar
- 17.Abbas Z, Shabbir MS, Ali N (2018) Numerical study of magnetohydrodynamic pulsatile flow of Sutterby fluid through an inclined overlapping arterial stenosis in the presence of periodic body acceleration. Results Phys 9:753–762CrossRefGoogle Scholar
- 18.Das S, Das S, Changdar S, De S (2014) Analysis of blood flow through multi-irregular shape stenosed artery. Int J Pharm Biol Sci 4(2):244–252Google Scholar
- 19.Sacks AH, Raman KR, Burnell JA, Tickner EG (1963) Auscultatory Versus direct pressure measurements for Newtonian fluids and for blood in simulated arteries. Vidya Corp., Palo Alto, California Report 119Google Scholar
- 20.Sankar DS, Hemalatha K (2007) A non-Newtonian fluid flow model for blood flow through a catheterized artery—steady flow. Appl Math Model 31:1847–1864CrossRefGoogle Scholar
- 21.Sankar DS, Lee U (2008) Two-fluid Herschel–Bulkley model for blood flow in catheterized arteries. J Mech Sci Technol 22:1008–1018CrossRefGoogle Scholar
- 22.Meena K, Gayathri P, Subramanian KR (2013) A non-Newtonian Herschel–Bulkley model for blood flow through catheterized tapered artery. Int J Curr Res 5:1473–1483Google Scholar
- 23.Kumar S, Garg NR, Gupta A (2015) Herschel–Bulkley model for blood flow through an arterial segment with stenosis. Int J Sci Technol Manag 4:93–100Google Scholar
- 24.Prasad KM, Vijaya B, Umadevi C (2014) A mathematical model of Herschel–Bulkley fluid through an overlapping stenosis. IOSR JM 10:41–46CrossRefGoogle Scholar
- 25.Abbas Z, Shabbir MS, Ali N (2017) Analysis of rheological properties of Herschel–Bulkley fluid for pulsating flow of blood in
*ω*-shaped stenosed artery. AIP Adv 7:105123CrossRefGoogle Scholar - 26.Siddiqui SU, Verma NK, Gupta RS (2015) A mathematical model for pulsatile flow of Herschel–Bulkley fluid through stenosed arteries. J Sci Technol 5(4):49–66Google Scholar
- 27.Chaturani P, Samy RP (1986) Pulsatile flow of Casson’s fluid through stenosed arteries with application to blood flow. Biorheology 23:499–511CrossRefGoogle Scholar
- 28.Jaafar NA, Yatim YM, Sankar DS (2016) Mathematical analysis for unsteady dispersion of solute with chemical reaction in blood flow. AIP Confer Proc 1750:030033CrossRefGoogle Scholar
- 29.Young DF (1968) Effect of time-dependent stenosis on flow through a tube. J Eng Indus 90:248–254CrossRefGoogle Scholar
- 30.Gayathri K, Shailendhra K (2014) Pulsatile blood flow in large arteries: comparative study of Burton’s and McDonald’s models. Appl Math Mech Engl Ed 35(5):575–590MathSciNetCrossRefGoogle Scholar
- 31.Mustapha N, Mandal PK, Johnston PR, Amin N (2010) A numerical simulation of unsteady blood flow through a multi-irregular stenoses. Appl Math Model 34:1559–1573MathSciNetCrossRefGoogle Scholar