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Effect of Varying Viscosity on Two-Fluid Model of Blood Flow through Constricted Blood Vessels: A Comparative Study

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Abstract

Purpose

Most of the previously studied non-Newtonian blood flow models considered blood viscosity to be constant but for correct measurement of flow rate and flow resistance, the hematocrit dependent viscosity will be better as various literature suggested the variable nature of blood viscosity. Present work concerns the steady and pulsatile nature of blood flow through constricted blood vessels. Two-fluid model for blood is considered with the suspension of all the RBCs (erythrocytes) in the core region as a non-Newtonian (Herschel–Bulkley) fluid and the plasma in the cell free region near wall as a Newtonian fluid. No slip condition on the wall and radially varying viscosity has been taken.

Methods

For steady flow the analytical approach has been taken to obtain the exact solution. Regular perturbation expansion method has been used to solve the governing equations for pulsatile flow up to first order of approximation by assuming the pulsatile Reynolds number to be very small (much less than unity).

Results

Flow rate, wall shear stress and velocity profile have been graphically analyzed and compared with constant viscosity model. A noteworthy observation of the present study is that rise in viscosity index leads to decay in velocity, velocity of plug flow region, flow rate while flow resistance increases with rising viscosity index (m). The results for Power-law fluid (PL), Bingham-plastic fluid (BP), Newtonian fluid (NF) are found as special cases from this model. Like the constant viscosity model, it has been also observed that the velocity, flow rate and plug core velocity of two-fluid model are higher than the single-fluid model for variable viscosity.

Conclusions

The two-phase fluid model is more significant than the single-fluid model. Effect of viscosity parameter on various hemodynamical quantities has been obtained. It is also concluded that a rising viscosity parameter (varying nature of viscosity) significantly distinguishes the single and two-fluid models in terms of changes in blood flow resistance. The outcome of present study may leave a significant impact on analyzing blood flow through small blood vessels with constriction, where correct measurement of flow rate and flow resistance for medical treatment is very important.

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Abbreviations

\('-'\) :

The dimensional quantities

r :

Distance in radial direction

z :

Distance in axial direction

t :

Time

\(R_p\) :

Plug core (flow) radius

\(R_{p0}, R_{p1}\) :

Zeroth and first order approximation of plug core (flow) radius, respectively

\(u_H, u_N\) :

Axial velocities in core and plasma region, respectively

\(u_{H0}, u_{N0}\) :

Zeroth order approximation of axial velocity in core and plasma region, respectively

\(u_{H1}, u_{N1}\) :

First order approximation of axial velocities in core and plasma region, respectively

\(u_p\) :

Plug core velocity

\(u_{p0}, u_{p1}\) :

Zeroth and first order approximation of plug core velocity, respectively

n :

Herschel–Bulkley fluid parameter

m :

Viscosity index

K :

Constant in viscosity relation

p :

Pressure

\(p_s, q(z)\) :

Steady state pressure gradients for steady and pulsatile flow, respectively

\(Q_s, Q(z,t)\) :

Volumetric flow rate for steady and pulsatile flow, respectively

\(R_1(t, z)\) :

Radius of artery with time-dependent stenosis in core region

R(t, z):

Radius of artery with time-dependent stenosis in plama region

\(R_1(z)\) :

Radius of artery with stenosis in steady state case (core region)

R(z):

Radius of artery with stenosis in steady state case (plama region)

\(R_0\) :

Radius of normal artery

A :

Amplitude

L :

Length of the constricted blood vessel

\(\tau_H, \tau_N\) :

Shear stresses in core and plasma region, respectively

\(\tau_{H0}, \tau_{N0}\) :

Zeroth order approximation of shear stress in core and plasma region, respectively

\(\tau_{H1}, \tau_{N1}\) :

First order approximation of shear stress in core and plasma region, respectively

\(\tau_y\) :

Yield stress

\(\theta\) :

Dimensionless yield stress

\(\tau_w\) :

Wall shear stress

\(\rho_H, \rho_N\) :

Densities of blood in core and plasma region, respectively

\(\mu_H, \mu_N\) :

Constant viscosities of the blood in core and plasma region, respectively

\(\mu_H^{\prime}(r)\) :

Variable viscosity of the blood in core region

\(\rho_0\) :

The ratio of densities in plasma and core region

\(\alpha\) :

Womersley frequency parameter

\(\delta_H, \delta_N\) :

Peak height of stenosis in core and plasma region, respectively

\(\omega\) :

Angular frequency

\(\lambda_s, \lambda\) :

Flow resistance for steady and pulsatile flow, respectively

0:

Zeroth order approximation (for \(R_{p0}, u_{H0}, u_{p0}, \tau_{H0}, \tau_{N0}\))

1:

First order approximation (for \(R_{p1}, u_{H1}, u_{p1}, \tau_{H1}, \tau_{N1}\))

H :

Herschel–Bulkley fluid (for \(u_H, \tau_H, \delta_H, \rho_H\))

N :

Newtonian fluid (for \(u_N, \tau_N, \delta_N, \rho_N\))

p :

Plug flow value(for \(u_p, R_p\))

s :

Steady flow value (for \(p_s, Q_s, \lambda_s\))

w :

Value at wall (for \(\tau_w\))

y :

Value at yield stress (for \(\tau_y\))

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Acknowledgments

Authors acknowledge their sincere thank to the reviewers for their valuable suggestions and comments. Authors also acknowledge their sincere thank to SERB, New Delhi, Government of India, for providing the financial assistance grant under the research Grant SR/FTP/MS-038/2011.

Conflict of interest

Dr. Ashish Tiwari has received research grant from SERB, New Delhi, Government of India. Mr. Satyendra Singh Chauhan is getting fellowship from SERB, New Delhi, under the above research grant.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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  1. Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani, Rajasthan, 333 031, India

    Ashish Tiwari & Satyendra Singh Chauhan

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  1. Ashish Tiwari
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Correspondence to Ashish Tiwari.

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Associate Editors Ajit P. Yoganathan and David Elad oversaw the review of this article.

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Tiwari, A., Chauhan, S.S. Effect of Varying Viscosity on Two-Fluid Model of Blood Flow through Constricted Blood Vessels: A Comparative Study. Cardiovasc Eng Tech 10, 155–172 (2019). https://doi.org/10.1007/s13239-018-00379-x

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