# Effect of Varying Viscosity on Two-Layer Model of Pulsatile Flow through Blood Vessels with Porous Region near Walls

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## Abstract

The present work concerns the effect of hematocrit-dependent viscosity on pulsatile flow of blood through narrow tube with porous walls. Two-fluid model of blood is assumed to be consisting of a core region (Casson fluid) and a plasma region (Newtonian fluid). No slip condition is assumed on wall and pressure gradient has been considered as periodic function of time. The wall of the blood vessel composed of a thin porous (Brinkman) layer. The stress jump condition has been employed at the fluid–porous interface in the plasma region. Up to first order, approximate solutions of governing equations are obtained using perturbation approach. A comparative analysis for relative change in resistance offered against the flow between our model and previously studied single and two-fluid models without porous walls has also been done. Mathematical expressions for velocity, rate of flow and resistance offered against the flow have been obtained analytically for different regions and influence of plasma layer thickness, varying viscosity, stress jump parameter, permeability and viscosity ratio parameter on above quantities are pictorially discussed. It is perceived that the values of flow rate for two-fluid model with porous region near walls are higher in comparison with two-fluid model without porous region near walls. Dependency of hematocrit (Ht) on the porosity parameters is graphically discussed. The study reveals a significant impact of various parameters on hematocrit (Ht). A novel observation is that a slight increase in pressure wave amplitude leads to significant fluctuation in hematocrit (Ht) which also indicates how systole and diastole (which controls the pressure gradient amplitudes) leads to changes on blood hematocrit (Ht).

## Keywords

Casson fluid Porous media flow Porous tube Glycocalyx layer Permeability Variable viscosity## List of Symbols

- ‘\({-}\)’
Represents dimensional quantities

*r*,*z*Radial and axial distance, respectively

*t*Time

*h*Plasma layer thickness

- \(R_{\mathrm{p}}\)
Plug core radius

- \(v_1, v_2, v_3\)
Axial velocities in core, plasma and Brinkman region, respectively

- \(v_{\mathrm{p}}\)
Plug core velocity

*K*Constant in varying viscosity relation (viscosity parameter)

*n*Viscosity index

- \(p_1,p_2,p_3\)
Pressures in core, plasma and Brinkman region, respectively

*q*(*z*)Pressure gradient in steady flow state

*Q*Volumetric flow rate

- \(R_1, R_2, R_3\)
Radius of artery in core, plasma and Brinkman region, respectively

*k*Permeability in porous region

- \(R_0\)
Radius of blood vessel without porous walls

*L*Length of blood vessel

- SFM, TFM
Single-fluid model and two-fluid model, respectively

- CF, NF
Casson and Newtonian fluid, respectively

- PW
Porous region near walls

- Ht
Hematocrit (i.e., volume concentration of all RBCs in whole blood)

- \(C_{\mathrm{v}}(r)\)
Volume concentration of all RBCs

- \(c_{\mathrm{v}}\)
Constant in concentration relation

*H*(*r*)Heaviside unit function

## Greeks Letters

- \(\phi \)
Azimuthal angle

- \(\alpha \)
Womersley parameter

- \(\alpha _{\mathrm{p}}=1/\lambda _1^2\)
Porosity parameter (Srivastava and Srivastava 2005)

- \(\tau \)
Shear stress of Casson fluid

- \(\tau _{\mathrm{y}}\)
Yield stress

- \(\theta \)
Dimensionless yield stress

- \(\tau _{\mathrm{w}}\)
Wall shear stress

- \(\mu _1(r)\)
Variable viscosity of Casson fluid in core region

- \(\mu _2\)
Constant viscosity coefficient of peripheral region

- \(\mu _{\mathrm{e}}\)
Effective viscosity coefficient of Brinkman region (Srivastava and Srivastava 2005)

- \(\lambda _1\)
Viscosity ratio parameter in Brinkman region

- \(\beta \)
Stress jump parameter

- \(\lambda \)
Flow resistance

## Subscripts

- 1, 2, 3
Denote for core, plasma and Brinkman region, respectively (for \(p_i, v_i, R_i\))

- p
Represent plug flow value (for \(v_{\mathrm{p}}, R_{\mathrm{p}}\))

- e
Represent effective viscosity coefficient in Brinkman region (for \(\mu _{\mathrm{e}}\))

- w
Value at wall (for \(\tau _{\mathrm{w}}\))

- y
Value of yield stress (for \(\tau _{\mathrm{y}}\))

## Mathematics Subject Classification

76A05 76M45 76Z05## Notes

### Acknowledgements

Authors are thankful to reviewers for their valuable suggestions which motivated us for some new observations. Authors also acknowledge their sincere thanks to Department of Science and Technology (DST), New Delhi, India for providing support under FIST grant (SR/FST/MSI-090/2013(C)).

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