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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.

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Abbreviations

\(A\) :

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, \([ - ]\)

\(\alpha^{2}\) :

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, \([{\text{s}}^{ - 1} ]\)

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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.

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Authors and Affiliations

  1. Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan

    M. S. Shabbir & Z. Abbas

  2. Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, 44000, Pakistan

    N. Ali

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  1. M. S. Shabbir
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  2. N. Ali
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  3. Z. Abbas
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Correspondence to M. S. Shabbir.

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Technical Editor: Jader Barbosa Jr.

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Shabbir, M.S., Ali, N. & Abbas, Z. Unsteady blood flow of non-Newtonian fluid through a rigid artery in the presence of multi-irregular stenoses. J Braz. Soc. Mech. Sci. Eng. 40, 413 (2018). https://doi.org/10.1007/s40430-018-1327-x

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