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Sādhanā

, 44:158 | Cite as

A four-layered model for flow of non-Newtonian fluid in an artery with mild stenosis

  • R Ponalagusamy
  • Ramakrishna ManchiEmail author
Article

Abstract

The present article deals with a four-layered mathematical model for blood flow through an artery with mild stenosis. The four-layered model comprises a cell-rich core of suspension of all the erythrocytes described as a non-Newtonian (Jeffrey) fluid, a peripheral zone of cell-free plasma (Newtonian fluid) and the stenosed artery with porous wall consisting of a thin transition (Brinkman) layer followed by Darcy region. Analytical expressions have been obtained for velocity profiles in all the four regions, total volumetric flow rate, wall shear stress and flow impedance. MATLAB software is employed to compute numerical values of the pressure gradient. The influences of different parameters such as variable core fluid viscosity, hematocrit, thickness of the plasma layer, Brinkman and Darcy layer thickness, Darcy number, Jeffrey fluid parameter, and size and shape parameters of stenosis on the physiologically vital flow characteristics, specifically velocity profile, volume flow rate, wall shear stress and flow impedance, have been examined. It is observed that the wall shear stress and resistive impedance decrease with the increase of plasma layer thickness, Jeffrey fluid parameter, Darcy number and Darcy slip parameter, and increase with the rise of hematocrit. The results in the case of variable core viscosity and constant core viscosity are compared to investigate the impact of variable core viscosity in managing the flow of blood.

Keywords

Jeffrey fluid; porous wall; hematocrit; Brinkman layer; Darcy region; variable core fluid viscosity 

Nomenclatures

represents dimensional quantities

\({\overline{r}}\)

radial distance

\({\overline{z}}\)

axial distance

\({\overline{R}}({\overline{z}})\)

radius of the artery in stenosed region

\({\overline{R}}_0\)

radius of the normal artery

\({\overline{d}}\)

location of stenosis

\({\overline{L}}_0\)

length of stenosis

\({\overline{L}}\)

length of the artery

n

shape parameter of stenosis

\(p_i (i=C, P, B, D)\)

pressure

\(P_0\)

pressure gradient

\(R_C(z)\)

radius of core region

\(R_P(z)\)

radius of plasma region

\(R_B(z)\)

radius of Brinkman region

\(R_D(z)\)

radius of Darcy region

\(h_P\)

thickness of plasma layer

\(h_B\)

thickness of Brinkman layer

\(h_D\)

thickness of Darcy layer

\(u_C\)

velocity of fluid in core region

\(u_P\)

velocity of fluid in Plasma region

\(u_B\)

velocity of fluid in Brinkman region

\(u_D\)

velocity of fluid in Darcy region

\(h_m\)

haematocrit

\(D_a\)

Darcy number

\(I_0, K_0\)

modified Bessel functions

Q

total flow rate

Greek symbols

\(\delta _s\)

maximum height of stenosis

\({\overline{\mu }}_C\)

viscosity of Jeffrey fluid

\({\overline{\mu }}_N\)

viscosity of Newtonian fluid

\(\alpha _2\)

effective viscosity of Brinkman layer

\(\lambda _1\)

ratio of relaxation to retardation times

\(\alpha \)

Darcy slip parameter

\(\beta \)

stress jump parameter

\(\phi \)

porosity

\(\tau _w\)

wall shear stress

\(\lambda \)

flow impedance

Notes

Acknowledgements

Ramakrishna Manchi is thankful to the MHRD, the Government of India, for the grant of Fellowship for the Research Scholars administered by NIT, Tiruchirappalli.

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Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologyTiruchirappalliIndia

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