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

Technical Paper

## 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, $$[{\text{s}}^{ - 1} ]$$

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