, 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


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.


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


represents dimensional quantities


radial distance


axial distance


radius of the artery in stenosed region


radius of the normal artery


location of stenosis


length of stenosis


length of the artery


shape parameter of stenosis

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



pressure gradient


radius of core region


radius of plasma region


radius of Brinkman region


radius of Darcy region


thickness of plasma layer


thickness of Brinkman layer


thickness of Darcy layer


velocity of fluid in core region


velocity of fluid in Plasma region


velocity of fluid in Brinkman region


velocity of fluid in Darcy region




Darcy number

\(I_0, K_0\)

modified Bessel functions


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

\(\phi \)


\(\tau _w\)

wall shear stress

\(\lambda \)

flow impedance



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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